Chapter 2

Differential Calculus of Functions of One Variable

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Section 2.1Derivatives of Explicit Functions[221/97%]
Problem 821

If xx changes from 11 to 10001000, find the increment Δx\Delta x of the independent variable xx and the corresponding increment Δy\Delta y of the function y=lg⁡xy=\lg x.

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Problem 822

If xx changes from 0.010.01 to 0.0010.001, find the increment Δx\Delta x of the independent variable xx and the corresponding increment Δy\Delta y of the function y=1x2y=\frac{1}{x^{2}}.

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Problem 823

Let:

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(a)

y=ax+by=ax+b;

(b)

y=ax2+bx+cy=ax^{2}+bx+c;

(c)

y=axy=a^{x}.

If the variable xx receives an increment Δx\Delta x, find the increment Δy\Delta y.

Problem 824

Prove that:

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(a)

Δ[f(x)+g(x)]=Δf(x)+Δg(x)\Delta \left[f(x)+g(x)\right]=\Delta f(x)+\Delta g(x);

(b)

Δ[f(x)g(x)]=g(x+Δx)Δf(x)+f(x)Δg(x)\Delta \left[f(x)g(x)\right]=g(x+\Delta x)\Delta f(x)+f(x)\Delta g(x).

Problem 825

Through the two points A(2,4)A(2,4) and A′(2+Δx,4+Δy)A'(2+\Delta x, 4+\Delta y) on the curve y=x2y=x^{2} draw the secant line AA′AA', and find the slope of this secant, when:

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(a)

Δx=1\Delta x=1;

(b)

Δx=0.1\Delta x=0.1;

(c)

Δx=0.01\Delta x=0.01;

(d)

Δx\Delta x is arbitrarily small.

What is the slope of the tangent line to the curve at the point AA?

Problem 826

The segment 1⩽x⩽1+h1 \leqslant x \leqslant 1+h on the OxOx axis is mapped onto the OyOy axis by the functional relation y=x3y=x^{3}. Find its average coefficient of elongation, when:

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(a)

h=0.1h=0.1;

(b)

h=0.01h=0.01;

(c)

h=0.001h=0.001, computing the value of this coefficient.

What is the coefficient of elongation at x=1x=1?

Problem 827

A point moves along the OxOx axis according to the law

x=10t+5t2, x=10t+5t^{2},

where tt is the time in seconds and xx is the distance in meters. Find the average velocity of the motion over the time interval 20⩽t⩽20+Δt20 \leqslant t \leqslant 20+\Delta t. Let:

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(a)

Δt=1\Delta t=1;

(b)

Δt=0.1\Delta t=0.1;

(c)

Δt=0.01\Delta t=0.01, computing the value of this velocity.

What is the velocity of the motion at t=20t=20?

Problem 828

Using the definition of the derivative, find directly the derivatives of the following functions:

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(a)

x2x^{2};

(b)

x3x^{3};

(c)

1x\dfrac {1}{x};

(d)

x\sqrt{x};

(e)

x3\sqrt[3]{x};

(f)

tg⁡x\tg x;

(g)

ctg⁡x\ctg x;

(h)

arcsin⁡x\arcsin x;

(i)

arccos⁡x\arccos x;

(j)

arctg⁡x\arctg x.

Problem 829

Let

f(x)=(x−1)(x−2)2(x−3)3, f(x)=(x-1)(x-2)^{2}(x-3)^{3},

find f′(1)f'(1), f′(2)f'(2) and f′(3)f'(3).

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Problem 830

Let

f(x)=x2sin⁡(x−2), f(x)=x^{2}\sin (x-2),

find f′(2)f'(2).

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Problem 831

Let

f(x)=x+(x−1)arcsin⁡xx+1, f(x)=x+(x-1)\arcsin \sqrt{\frac{x}{x+1}},

find f′(1)f'(1).

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Problem 832

Suppose the function f(x)f(x) is differentiable at the point aa, find

lim⁡x→af(x)−f(a)x−a. \lim _{x \to a} \frac{f(x)-f(a)}{x-a} .
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Problem 833

Prove that if the function f(x)f(x) is differentiable and nn is a natural number, then

lim⁡n→∞n[f(x+1n)−f(x)]=f′(x). \lim _{n \to \infty } n\left[f\left(x+\frac{1}{n}\right)-f(x)\right]=f'(x) .

Conversely, if the limit (1) exists for a function f(x)f(x), can we conclude that this function has a derivative? Study the example of the Dirichlet function (see Chapter I, Problem 734: the Dirichlet function χ(x)\chi (x) is defined by χ(x)=1\chi (x)=1 when xx is rational, and χ(x)=0\chi (x)=0 when xx is irrational).

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Problem 834

Using the table of derivatives, find the derivative of the following function:y=2+x−x2y=2+x-x^{2}. What are y′(0)y'(0); y′(12)y'\left(\dfrac {1}{2}\right); y′(1)y'(1); y′(−10)y'(-10)?

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Problem 835

Using the table of derivatives, find the derivative of the following function:y=x33+x22−2xy=\dfrac {x^{3}}{3}+\dfrac {x^{2}}{2}-2x. For what value of xx is:

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(a)

y′(x)=0y'(x)=0;

(b)

y′(x)=−2y'(x)=-2;

(c)

y′(x)=10y'(x)=10?

Problem 836

Using the table of derivatives, find the derivative of the following function:y=a5+5a3x2−x5y=a^{5}+5a^{3}x^{2}-x^{5}.

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Problem 837

Using the table of derivatives, find the derivative of the following function:y=ax+ba+by=\dfrac {ax+b}{a+b}.

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Problem 838

Using the table of derivatives, find the derivative of the following function:y=(x−a)(x−b)y=(x-a)(x-b).

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Problem 839

Using the table of derivatives, find the derivative of the following function:y=(x+1)(x+2)2(x+3)3y=(x+1)(x+2)^{2}(x+3)^{3}.

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Problem 840

Using the table of derivatives, find the derivative of the following function:y=(xsin⁡α+cos⁡α)(xcos⁡α−sin⁡α)y=(x\sin \alpha +\cos \alpha )(x\cos \alpha -\sin \alpha ).

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Problem 841

Using the table of derivatives, find the derivative of the following function:y=(1+nxm)(1+mxn)y=(1+nx^{m})(1+mx^{n}).

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Problem 842

Using the table of derivatives, find the derivative of the following function:y=(1−x)(1−x2)2(1−x3)3y=(1-x)(1-x^{2})^{2}(1-x^{3})^{3}.

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Problem 842.1
[NO TASK]
Problem 842.1
[NO TASK]
Problem 843

Using the table of derivatives, find the derivative of the following function:y=1x+2x2+3x3y=\dfrac {1}{x}+\dfrac {2}{x^{2}}+\dfrac {3}{x^{3}}.

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Problem 844

Prove the formula

(ax+bcx+d)′=∣abcd∣(cx+d)2. \left(\frac{ax+b}{cx+d}\right)'=\frac{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}{(cx+d)^{2}} .
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Problem 845

Find the derivative of the following function:y=2x1−x2y=\dfrac {2x}{1-x^{2}}.

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Problem 846

Find the derivative of the following function:y=1+x−x21−x+x2y=\dfrac {1+x-x^{2}}{1-x+x^{2}}.

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Problem 847

Find the derivative of the following function:y=x(1−x)2(1+x)3y=\dfrac {x}{(1-x)^{2}(1+x)^{3}}.

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Problem 848

Find the derivative of the following function:y=(2−x2)(3−x3)(1−x)2y=\dfrac {(2-x^{2})(3-x^{3})}{(1-x)^{2}}.

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Problem 849

Find the derivative of the following function:y=(1−x)p(1+x)qy=\dfrac {(1-x)^{p}}{(1+x)^{q}}.

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Problem 850

Find the derivative of the following function:y=xp(1−x)q1+xy=\dfrac {x^{p}(1-x)^{q}}{1+x}.

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Problem 851

Find the derivative of the following function:y=x+x+x3y=x+\sqrt{x}+\sqrt[3]{x}.

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Problem 852

Find the derivative of the following function:y=1x+1x+1x3y=\dfrac {1}{x}+\dfrac {1}{\sqrt{x}}+\dfrac {1}{\sqrt[3]{x}}.

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Problem 853

Find the derivative of the following function:y=x23−2xy=\sqrt[3]{x^{2}}-\dfrac {2}{\sqrt{x}}.

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Problem 854

Find the derivative of the following function:y=x1+x2y=x\sqrt{1+x^{2}}.

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Problem 855

Find the derivative of the following function:y=(1+x)⋅2+x2⋅3+x33y=(1+x) \cdot \sqrt{2+x^{2}} \cdot \sqrt[3]{3+x^{3}}.

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Problem 856

Find the derivative of the following function:y=(1−x)m(1+x)nm+ny=\sqrt[m+n]{(1-x)^{m}(1+x)^{n}}.

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Problem 857

Find the derivative of the following function:y=xa2−x2y=\dfrac {x}{\sqrt{a^{2}-x^{2}}}.

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Problem 858

Find the derivative of the following function:y=1+x31−x33y=\sqrt[3]{\dfrac {1+x^{3}}{1-x^{3}}}.

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Problem 859

Find the derivative of the following function:y=11+x2(x+1+x2)y=\dfrac {1}{\sqrt{1+x^{2}}\left(x+\sqrt{1+x^{2}}\right)}.

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Problem 860

Find the derivative of the following function:y=x+x+xy=\sqrt{x+\sqrt{x+\sqrt{x}}}.

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Problem 861

Find the derivative of the following function:y=1+1+x333y=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{x}}}.

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Problem 862

Find the derivative of the following function:y=cos⁡2x−2sin⁡xy=\cos 2x-2\sin x.

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Problem 863

Find the derivative of the following function:y=(2−x2)cos⁡x+2xsin⁡xy=(2-x^{2})\cos x+2x\sin x.

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Problem 864

Find the derivative of the following function:y=sin⁡(cos⁡2x)cos⁡(sin⁡2x)y=\sin (\cos^{2}x)\cos (\sin^{2}x).

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Problem 865

Find the derivative of the following function:y=sin⁡nxcos⁡nxy=\sin^{n}x\cos nx.

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Problem 866

Find the derivative of the following function:y=sin⁡[sin⁡(sin⁡x)]y=\sin \left[\sin (\sin x)\right].

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Problem 867

Find the derivative of the following function:y=sin⁡2xsin⁡x2y=\dfrac {\sin^{2}x}{\sin x^{2}}.

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Problem 868

Find the derivative of the following function:y=cos⁡x2sin⁡2xy=\dfrac {\cos x}{2\sin^{2}x}.

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Problem 869

Find the derivative of the following function:y=1cos⁡nxy=\dfrac {1}{\cos^{n}x}.

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Problem 870

Find the derivative of the following function:y=sin⁡x−xcos⁡xcos⁡x+xsin⁡xy=\dfrac {\sin x-x\cos x}{\cos x+x\sin x}.

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Problem 871

Find the derivative of the following function:y=tg⁡x2−ctg⁡x2y=\tg \dfrac {x}{2}-\ctg \dfrac {x}{2}.

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Problem 872

Find the derivative of the following function:y=tg⁡x−13tg⁡3x+15tg⁡5xy=\tg x-\dfrac {1}{3}\tg^{3}x+\dfrac {1}{5}\tg^{5}x.

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Problem 873

Find the derivative of the following function:y=4ctg⁡2x4+ctg⁡8x3y=4\sqrt[4]{\ctg^{2}x}+\sqrt[3]{\ctg^{8}x}.

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Problem 874

Find the derivative of the following function:y=sec⁡2xa+csc⁡2xay=\sec^{2}\dfrac {x}{a}+\csc^{2}\dfrac {x}{a}.

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Problem 875

Find the derivative of the following function:y=sin⁡[cos⁡2(tg⁡3x)]y=\sin \left[\cos^{2}(\tg^{3}x)\right].

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Problem 876

Find the derivative of the following function:y=e−x2y=e^{-x^{2}}.

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Problem 877

Find the derivative of the following function:y=2tg⁡1xy=2^{\tg \frac{1}{x}}.

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Problem 878

Find the derivative of the following function:y=ex(x2−2x+2)y=e^{x}(x^{2}-2x+2).

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Problem 879

Find the derivative of the following function:y=(1−x22sin⁡x−(1+x)22cos⁡x)e−xy=\left(\dfrac {1-x^{2}}{2}\sin x-\dfrac {(1+x)^{2}}{2}\cos x\right)e^{-x}.

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Problem 880

Find the derivative of the following function:y=ex(1+ctg⁡x2)y=e^{x}\left(1+\ctg \dfrac {x}{2}\right).

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Problem 881

Find the derivative of the following function:y=ln⁡3⋅sin⁡x+cos⁡x3xy=\dfrac {\ln 3 \cdot \sin x+\cos x}{3^{x}}.

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Problem 882

Find the derivative of the following function:y=eaxasin⁡bx−bcos⁡bxa2+b2y=e^{ax}\dfrac {a\sin bx-b\cos bx}{\sqrt{a^{2}+b^{2}}}.

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Problem 883

Find the derivative of the following function:y=ex+eex+eeexy=e^{x}+e^{e^{x}}+e^{e^{e^{x}}}.

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Problem 884

Find the derivative of the following function:y=(ab)x⋅(bx)a⋅(xa)b (a>0,b>0)y=\left(\dfrac {a}{b}\right)^{x} \cdot \left(\dfrac {b}{x}\right)^{a} \cdot \left(\dfrac {x}{a}\right)^{b} \ (a>0, b>0).

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Problem 885

Find the derivative of the following function:y=xaa+axa+aax (a>0)y=x^{a^{a}}+a^{x^{a}}+a^{a^{x}} \ (a>0).

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Problem 886

Find the derivative of the following function:y=lg⁡3x2y=\lg^{3}x^{2}.

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Problem 887

Find the derivative of the following function:y=ln⁡[ln⁡(ln⁡x)]y=\ln \left[\ln (\ln x)\right].

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Problem 888

Find the derivative of the following function:y=ln⁡[ln⁡2(ln⁡3x)]y=\ln \left[\ln^{2}(\ln^{3}x)\right].

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Problem 889

Find the derivative of the following function:y=12ln⁡(1+x)−14ln⁡(1+x2)−12(1+x)y=\dfrac {1}{2}\ln (1+x)-\dfrac {1}{4}\ln (1+x^{2})-\dfrac {1}{2(1+x)}.

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Problem 890

Find the derivative of the following function:y=14ln⁡x2−1x2+1y=\dfrac {1}{4}\ln \dfrac {x^{2}-1}{x^{2}+1}.

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Problem 891

Find the derivative of the following function:y=14(1+x4)+14ln⁡x41+x4y=\dfrac {1}{4(1+x^{4})}+\dfrac {1}{4}\ln \dfrac {x^{4}}{1+x^{4}}.

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Problem 892

Find the derivative of the following function:y=126ln⁡x3−2x3+2y=\dfrac {1}{2\sqrt{6}}\ln \dfrac {x\sqrt{3}-\sqrt{2}}{x\sqrt{3}+\sqrt{2}}.

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Problem 893

Find the derivative of the following function:y=11−kln⁡1+x1−x−k1−kln⁡1+xk1−xk (0<k<1)y=\dfrac {1}{1-k}\ln \dfrac {1+x}{1-x}-\dfrac {\sqrt{k}}{1-k}\ln \dfrac {1+x\sqrt{k}}{1-x\sqrt{k}} \ (0<k<1).

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Problem 894

Find the derivative of the following function:y=x+1−ln⁡(1+x+1)y=\sqrt{x+1}-\ln \left(1+\sqrt{x+1}\right).

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Problem 895

Find the derivative of the following function:y=ln⁡(x+x2+1)y=\ln \left(x+\sqrt{x^{2}+1}\right).

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Problem 896

Find the derivative of the following function:y=xln⁡(x+1+x2)−1+x2y=x\ln \left(x+\sqrt{1+x^{2}}\right)-\sqrt{1+x^{2}}.

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Problem 897

Find the derivative of the following function:y=xln⁡2(x+1+x2)−21+x2ln⁡(x+1+x2)+2xy=x\ln^{2}\left(x+\sqrt{1+x^{2}}\right)-2\sqrt{1+x^{2}}\ln \left(x+\sqrt{1+x^{2}}\right)+2x.

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Problem 898

Find the derivative of the following function:y=x2x2+a2+a22ln⁡(x+x2+a2)y=\dfrac {x}{2}\sqrt{x^{2}+a^{2}}+\dfrac {a^{2}}{2}\ln \left(x+\sqrt{x^{2}+a^{2}}\right).

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Problem 899

Find the derivative of the following function:y=12abln⁡a+xba−xb (a>0,b>0)y=\dfrac {1}{2\sqrt{ab}}\ln \dfrac {\sqrt{a}+x\sqrt{b}}{\sqrt{a}-x\sqrt{b}} \ (a>0, b>0).

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Problem 900

Find the derivative of the following function:y=2+3x2x41−x2+3ln⁡1+1−x2xy=\dfrac {2+3x^{2}}{x^{4}}\sqrt{1-x^{2}}+3\ln \dfrac {1+\sqrt{1-x^{2}}}{x}.

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Problem 901

Find the derivative of the following function:y=ln⁡tg⁡x2y=\ln \tg \dfrac {x}{2}.

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Problem 902

Find the derivative of the following function:y=ln⁡tg⁡(x2+π4)y=\ln \tg \left(\dfrac {x}{2}+\dfrac {\pi }{4}\right).

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Problem 903

Find the derivative of the following function:y=12ctg⁡2x+ln⁡sin⁡xy=\dfrac {1}{2}\ctg^{2}x+\ln \sin x.

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Problem 904

Find the derivative of the following function:y=ln⁡1−sin⁡x1+sin⁡xy=\ln \sqrt{\dfrac {1-\sin x}{1+\sin x}}.

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Problem 905

Find the derivative of the following function:y=−cos⁡x2sin⁡2x+ln⁡1+cos⁡xsin⁡xy=-\dfrac {\cos x}{2\sin^{2}x}+\ln \sqrt{\dfrac {1+\cos x}{\sin x}}.

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Problem 906

Find the derivative of the following function:y=ln⁡b+acos⁡x+b2−a2sin⁡xa+bcos⁡x (0⩽∣a∣<∣b∣)y=\ln \dfrac {b+a\cos x+\sqrt{b^{2}-a^{2}}\sin x}{a+b\cos x} \ (0 \leqslant \left|a\right|<\left|b\right|).

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Problem 907

Find the derivative of the following function:y=1x(ln⁡3x+3ln⁡2x+6ln⁡x+6)y=\dfrac {1}{x}(\ln^{3}x+3\ln^{2}x+6\ln x+6).

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Problem 908

Find the derivative of the following function:y=14x4ln⁡1x−116x4y=\dfrac {1}{4x^{4}}\ln \dfrac {1}{x}-\dfrac {1}{16x^{4}}.

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Problem 909

Find the derivative of the following function:y=32(1−1+x23)2+3ln⁡(1+1+x23)y=\dfrac {3}{2}\left(1-\sqrt[3]{1+x^{2}}\right)^{2}+3\ln \left(1+\sqrt[3]{1+x^{2}}\right).

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Problem 910

Find the derivative of the following function:y=ln⁡(1x+ln⁡(1x+ln⁡1x))y=\ln \left(\dfrac {1}{x}+\ln \left(\dfrac {1}{x}+\ln \dfrac {1}{x}\right)\right).

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Problem 911

Find the derivative of the following function:y=x[sin⁡(ln⁡x)−cos⁡(ln⁡x)]y=x\left[\sin (\ln x)-\cos (\ln x)\right].

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Problem 912

Find the derivative of the following function:y=ln⁡tg⁡x2−cos⁡x⋅ln⁡tg⁡xy=\ln \tg \dfrac {x}{2}-\cos x \cdot \ln \tg x.

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Problem 913

Find the derivative of the following function:y=arcsin⁡x2y=\arcsin \dfrac {x}{2}.

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Problem 914

Find the derivative of the following function:y=arccos⁡1−x2y=\arccos \dfrac {1-x}{\sqrt{2}}.

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Problem 915

Find the derivative of the following function:y=arctg⁡x2ay=\arctg \dfrac {x^{2}}{a}.

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Problem 916

Find the derivative of the following function:y=12arcctg⁡2xy=\dfrac {1}{\sqrt{2}}\arcctg \dfrac {\sqrt{2}}{x}.

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Problem 917

Find the derivative of the following function:y=x−arctg⁡xy=\sqrt{x}-\arctg \sqrt{x}.

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Problem 918

Find the derivative of the following function:y=x+1−x2arccos⁡xy=x+\sqrt{1-x^{2}}\arccos x.

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Problem 919

Find the derivative of the following function:y=xarcsin⁡x1+x+arctg⁡x−xy=x\arcsin \sqrt{\dfrac {x}{1+x}}+\arctg \sqrt{x}-\sqrt{x}.

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Problem 920

Find the derivative of the following function:y=arccos⁡1xy=\arccos \dfrac {1}{x}.

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Problem 921

Find the derivative of the following function:y=arcsin⁡(sin⁡x)y=\arcsin (\sin x).

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Problem 922

Find the derivative of the following function:y=arccos⁡(cos⁡2x)y=\arccos (\cos^{2}x).

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Problem 923

Find the derivative of the following function:y=arcsin⁡(sin⁡x−cos⁡x)y=\arcsin (\sin x-\cos x).

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Problem 924

Find the derivative of the following function:y=arccos⁡1−x2y=\arccos \sqrt{1-x^{2}}.

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Problem 925

Find the derivative of the following function:y=arctg⁡1+x1−xy=\arctg \dfrac {1+x}{1-x}.

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Problem 926

Find the derivative of the following function:y=arcctg⁡(sin⁡x+cos⁡xsin⁡x−cos⁡x)y=\arcctg \left(\dfrac {\sin x+\cos x}{\sin x-\cos x}\right).

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Problem 927

Find the derivative of the following function:y=2a2−b2arctg⁡(a−ba+btg⁡x2) (a>b⩾0)y=\dfrac {2}{\sqrt{a^{2}-b^{2}}}\arctg \left(\sqrt{\dfrac {a-b}{a+b}}\tg \dfrac {x}{2}\right) \ (a>b \geqslant 0).

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Problem 928

Find the derivative of the following function:y=arcsin⁡1−x21+x2y=\arcsin \dfrac {1-x^{2}}{1+x^{2}}.

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Problem 929

Find the derivative of the following function:y=1arccos⁡2(x2)y=\dfrac {1}{\arccos^{2}(x^{2})}.

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Problem 930

Find the derivative of the following function:y=arctg⁡x+13arctg⁡(x3)y=\arctg x+\dfrac {1}{3}\arctg (x^{3}).

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Problem 931

Find the derivative of the following function:y=ln⁡(1+sin⁡2x)−2sin⁡xarctg⁡(sin⁡x)y=\ln (1+\sin^{2}x)-2\sin x \arctg (\sin x).

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Problem 932

Find the derivative of the following function:y=ln⁡(arccos⁡1x)y=\ln \left(\arccos \dfrac {1}{\sqrt{x}}\right).

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Problem 933

Find the derivative of the following function:y=ln⁡x+ax2+b2+abarctg⁡xby=\ln \dfrac {x+a}{\sqrt{x^{2}+b^{2}}}+\dfrac {a}{b}\arctg \dfrac {x}{b}.

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Problem 934

Find the derivative of the following function:y=x2a2−x2+a22arcsin⁡xa (a>0)y=\dfrac {x}{2}\sqrt{a^{2}-x^{2}}+\dfrac {a^{2}}{2}\arcsin \dfrac {x}{a} \ (a>0).

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Problem 935

Find the derivative of the following function:y=16ln⁡(x+1)2x2−x+1+13arctg⁡2x−13y=\dfrac {1}{6}\ln \dfrac {(x+1)^{2}}{x^{2}-x+1}+\dfrac {1}{\sqrt{3}}\arctg \dfrac {2x-1}{\sqrt{3}}.

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Problem 936

Find the derivative of the following function:y=142ln⁡x2+x2+1x2−x2+1−122arctg⁡x2x2−1y=\dfrac {1}{4\sqrt{2}}\ln \dfrac {x^{2}+x\sqrt{2}+1}{x^{2}-x\sqrt{2}+1}-\dfrac {1}{2\sqrt{2}}\arctg \dfrac {x\sqrt{2}}{x^{2}-1}.

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Problem 937

Find the derivative of the following function:y=x(arcsin⁡x)2+21−x2arcsin⁡x−2xy=x(\arcsin x)^{2}+2\sqrt{1-x^{2}}\arcsin x-2x.

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Problem 938

Find the derivative of the following function:y=arccos⁡xx+12ln⁡1−1−x21+1−x2y=\dfrac {\arccos x}{x}+\dfrac {1}{2}\ln \dfrac {1-\sqrt{1-x^{2}}}{1+\sqrt{1-x^{2}}}.

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Problem 939

Find the derivative of the following function:y=arctg⁡x2−1−ln⁡xx2−1y=\arctg \sqrt{x^{2}-1}-\dfrac {\ln x}{\sqrt{x^{2}-1}}.

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Problem 940

Find the derivative of the following function:y=arcsin⁡x1−x2+12ln⁡1−x1+xy=\dfrac {\arcsin x}{\sqrt{1-x^{2}}}+\dfrac {1}{2}\ln \dfrac {1-x}{1+x}.

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Problem 941

Find the derivative of the following function:y=112ln⁡x4−x2+1(x2+1)2−123arctg⁡32x2−1y=\dfrac {1}{12}\ln \dfrac {x^{4}-x^{2}+1}{(x^{2}+1)^{2}}-\dfrac {1}{2\sqrt{3}}\arctg \dfrac {\sqrt{3}}{2x^{2}-1}.

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Problem 942

Find the derivative of the following function:y=x61+x12−arcctg⁡x6y=\dfrac {x^{6}}{1+x^{12}}-\arcctg x^{6}.

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Problem 943

Find the derivative of the following function:y=ln⁡1−x31+x3+x23+3arctg⁡1+2x33y=\ln \dfrac {1-\sqrt[3]{x}}{\sqrt{1+\sqrt[3]{x}+\sqrt[3]{x^{2}}}}+\sqrt{3}\arctg \dfrac {1+2\sqrt[3]{x}}{\sqrt{3}}.

?
Problem 944

Find the derivative of the following function:y=arctg⁡x1+1−x2y=\arctg \dfrac {x}{1+\sqrt{1-x^{2}}}.

?
Problem 945

Find the derivative of the following function:y=arcctg⁡a−2x2ax−x2 (a>0)y=\arcctg \dfrac {a-2x}{2\sqrt{ax-x^{2}}} \ (a>0).

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Problem 946

Find the derivative of the following function:y=3−x21−2x−x2+2arcsin⁡1+x2y=\dfrac {3-x}{2}\sqrt{1-2x-x^{2}}+2\arcsin \dfrac {1+x}{\sqrt{2}}.

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Problem 947

Find the derivative of the following function:y=14ln⁡1+x44+x1+x44−x−12arctg⁡1+x44xy=\dfrac {1}{4}\ln \dfrac {\sqrt[4]{1+x^{4}}+x}{\sqrt[4]{1+x^{4}}-x}-\dfrac {1}{2}\arctg \dfrac {\sqrt[4]{1+x^{4}}}{x}.

?
Problem 948

Find the derivative of the following function:y=arctg⁡(tg⁡2x)y=\arctg (\tg^{2}x).

?
Problem 949

Find the derivative of the following function:y=1−x2⋅ln⁡1−x1+x+12ln⁡1−1−x21+1−x2+1−x2+arcsin⁡xy=\sqrt{1-x^{2}} \cdot \ln \sqrt{\dfrac {1-x}{1+x}}+\dfrac {1}{2}\ln \dfrac {1-\sqrt{1-x^{2}}}{1+\sqrt{1-x^{2}}}+\sqrt{1-x^{2}}+\arcsin x.

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Problem 950

Find the derivative of the following function:y=xarctg⁡x−12ln⁡(1+x2)−12(arctg⁡x)2y=x\arctg x-\dfrac {1}{2}\ln (1+x^{2})-\dfrac {1}{2}(\arctg x)^{2}.

?
Problem 951

Find the derivative of the following function:y=ln⁡(ex+1+e2x)y=\ln \left(e^{x}+\sqrt{1+e^{2x}}\right).

?
Problem 952

Find the derivative of the following function:y=arctg⁡(x+1+x2)y=\arctg \left(x+\sqrt{1+x^{2}}\right).

?
Problem 953

Find the derivative of the following function:y=arcsin⁡(sin⁡αsin⁡x1−cos⁡αcos⁡x)y=\arcsin \left(\dfrac {\sin \alpha \sin x}{1-\cos \alpha \cos x}\right).

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Problem 954

Find the derivative of the following function:y=143ln⁡x2+2−x3x2+2+x3+12arctg⁡x2+2xy=\dfrac {1}{4\sqrt{3}}\ln \dfrac {\sqrt{x^{2}+2}-x\sqrt{3}}{\sqrt{x^{2}+2}+x\sqrt{3}}+\dfrac {1}{2}\arctg \dfrac {\sqrt{x^{2}+2}}{x}.

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Problem 955

Find the derivative of the following function:y=122arctg⁡x21+x4−142ln⁡1+x4−x21+x4+x2y=\dfrac {1}{2\sqrt{2}}\arctg \dfrac {x\sqrt{2}}{\sqrt{1+x^{4}}}-\dfrac {1}{4\sqrt{2}}\ln \dfrac {\sqrt{1+x^{4}}-x\sqrt{2}}{\sqrt{1+x^{4}}+x\sqrt{2}}.

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Problem 956

Find the derivative of the following function:y=x1−x21+x2−32arcctg⁡x21−x2y=\dfrac {x\sqrt{1-x^{2}}}{1+x^{2}}-\dfrac {3}{\sqrt{2}}\arcctg \dfrac {x\sqrt{2}}{\sqrt{1-x^{2}}}.

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Problem 957

Find the derivative of the following function:y=arccos⁡(sin⁡x2−cos⁡x2)y=\arccos (\sin x^{2}-\cos x^{2}).

?
Problem 958

Find the derivative of the following function:y=arcsin⁡(sin⁡x2)+arccos⁡(cos⁡x2)y=\arcsin (\sin x^{2})+\arccos (\cos x^{2}).

?
Problem 959

Find the derivative of the following function:y=emarcsin⁡x[cos⁡(marcsin⁡x)+sin⁡(marcsin⁡x)]y=e^{m\arcsin x}\left[\cos (m\arcsin x)+\sin (m\arcsin x)\right].

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Problem 960

Find the derivative of the following function:y=arctg⁡ex−ln⁡e2xe2x+1y=\arctg e^{x}-\ln \sqrt{\dfrac {e^{2x}}{e^{2x}+1}}.

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Problem 960.1
[NO TASK]
Problem 960.1
[NO TASK]
Problem 960.2
[NO TASK]
Problem 960.2
[NO TASK]
Problem 960.3
[NO TASK]
Problem 960.3
[NO TASK]
Problem 961

Find the derivative of the following function:y=x+xx+xxx (x>0)y=x+x^{x}+x^{x^{x}} \ (x>0).

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Problem 962

Find the derivative of the following function:y=xxa+xax+axx (a>0,x>0)y=x^{x^{a}}+x^{a^{x}}+a^{x^{x}} \ (a>0, x>0).

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Problem 963

Find the derivative of the following function:y=xx(x>0)y=\sqrt[x]{x} \quad (x>0).

?
Problem 964

Find the derivative of the following function:y=(sin⁡x)cos⁡x+(cos⁡x)sin⁡xy=(\sin x)^{\cos x}+(\cos x)^{\sin x}.

?
Problem 965

Find the derivative of the following function:y=(ln⁡x)x÷xln⁡xy=(\ln x)^{x} \div x^{\ln x}.

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Problem 965.1
[NO TASK]
Problem 965.1
[NO TASK]
Problem 966

Find the derivative of the following function:y=log⁡xey=\log_{x}e.

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Problem 967

Find the derivative of the following function:y=ln⁡(ch⁡x)+12ch⁡2xy=\ln (\ch x)+\dfrac {1}{2\ch^{2}x}.

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Problem 968

Find the derivative of the following function:y=ch⁡xsh⁡2x−ln⁡(cth⁡x2)y=\dfrac {\ch x}{\sh^{2}x}-\ln \left(\cth \dfrac {x}{2}\right).

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Problem 969

Find the derivative of the following function:y=arctg⁡(th⁡x)y=\arctg (\th x).

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Problem 970

Find the derivative of the following function:y=arccos⁡(1ch⁡x)y=\arccos \left(\dfrac {1}{\ch x}\right).

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Problem 971

Find the derivative of the following function:y=bax+2a2−b2aarctg⁡(a−ba+bth⁡x2) (0⩽∣b∣<a)y=\dfrac {b}{a}x+\dfrac {2\sqrt{a^{2}-b^{2}}}{a}\arctg \left(\sqrt{\dfrac {a-b}{a+b}}\th \dfrac {x}{2}\right) \ (0 \leqslant \left|b\right|<a).

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Problem 972

Introduce the intermediate variable u=cos⁡2xu=\cos^{2}x to find the derivative of the function

y=ln⁡(cos⁡2x+1+cos⁡4x) y=\ln \left(\cos ^{2}x+\sqrt{1+\cos ^{4}x}\right)

.

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Problem 973

Using the method shown in Problem 972, find the derivative of the following function:y=(arccos⁡x)2[ln⁡2(arccos⁡x)−ln⁡(arccos⁡x)+12]y=(\arccos x)^{2}\left[\ln^{2}(\arccos x)-\ln (\arccos x)+\dfrac {1}{2}\right].

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Problem 974

Using the method shown in Problem 972, find the derivative of the following function:y=12arctg⁡(1+x44)+14ln⁡1+x44+11+x44−1y=\dfrac {1}{2}\arctg \left(\sqrt[4]{1+x^{4}}\right)+\dfrac {1}{4}\ln \dfrac {\sqrt[4]{1+x^{4}}+1}{\sqrt[4]{1+x^{4}}-1}.

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Problem 975

Using the method shown in Problem 972, find the derivative of the following function:y=e−x2arcsin⁡(e−x2)1−e−2x2+12ln⁡(1−e−2x2)y=\dfrac {e^{-x^{2}}\arcsin (e^{-x^{2}})}{\sqrt{1-e^{-2x^{2}}}}+\dfrac {1}{2}\ln (1-e^{-2x^{2}}).

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Problem 976

Using the method shown in Problem 972, find the derivative of the following function:y=ax1+a2x−1−a2x1+a2xarcctg⁡(a−x)y=\dfrac {a^{x}}{1+a^{2x}}-\dfrac {1-a^{2x}}{1+a^{2x}}\arcctg (a^{-x}).

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Problem 977

Find the derivative of the function and sketch the graphs of the function and its derivative, given:

?
(a)

y=∣x∣y=\left|x\right|;

(b)

y=x∣x∣y=x\left|x\right|;

(c)

y=ln⁡∣x∣y=\ln \left|x\right|.

Problem 978

Find the derivatives of the following functions:

?
(a)

y=∣(x−1)2(x+1)3∣y=\left|(x-1)^{2}(x+1)^{3}\right|;

(b)

y=∣sin⁡3x∣y=\left|\sin^{3}x\right|;

(c)

y=arccos⁡1∣x∣y=\arccos \dfrac {1}{\left|x\right|};

(d)

y=[x]sin⁡2πxy=[x]\sin^{2}\pi x.

Problem 979

Find the derivative and sketch the graphs of the function and its derivative:y={1−xwhen −∞<x<1;(1−x)(2−x)when 1⩽x⩽2;−(2−x)when 2<x<+∞.y= \begin{cases} 1-x & \text{when } -\infty <x<1; \\ (1-x)(2-x) & \text{when } 1 \leqslant x \leqslant 2; \\ -(2-x) & \text{when } 2<x<+\infty . \end{cases} (Fig. 2.8)

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Problem 980

Find the derivative and sketch the graphs of the function and its derivative:y={(x−a)2(x−b)2,when a⩽x⩽b;0,outside the segment [a,b].y= \begin{cases} (x-a)^{2}(x-b)^{2}, & \text{when } a \leqslant x \leqslant b; \\ 0, & \text{outside the segment } [a,b]. \end{cases} (Fig. 2.10)

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Problem 981

Find the derivative and sketch the graphs of the function and its derivative:y={xwhen x<0;ln⁡(1+x)when x⩾0.y= \begin{cases} x & \text{when } x<0; \\ \ln (1+x) & \text{when } x \geqslant 0 . \end{cases} (Fig. 2.12)

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Problem 982

Find the derivative and sketch the graphs of the function and its derivative:y={arctg⁡x,when ∣x∣⩽1;π4sign⁡x+x−12when ∣x∣>1.y= \begin{cases} \arctg x, & \text{when } \left|x\right| \leqslant 1; \\ \dfrac {\pi }{4}\operatorname {sign} x+\dfrac {x-1}{2} & \text{when } \left|x\right|>1 . \end{cases} (Fig. 2.14)

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Problem 983

Find the derivative and sketch the graphs of the function and its derivative:y={x2e−x2when ∣x∣⩽1;1ewhen ∣x∣>1.y= \begin{cases} x^{2}e^{-x^{2}} & \text{when } \left|x\right| \leqslant 1; \\ \dfrac {1}{e} & \text{when } \left|x\right|>1 . \end{cases} (Fig. 2.16)

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Problem 984

The derivative obtained from the logarithm of a known function is called the logarithmic derivative of that function:

ddxln⁡f(x)=f′(x)f(x). \frac{d}{dx}\ln f(x)=\frac{f'(x)}{f(x)} .

Given the function yy, find its logarithmic derivative:

?
(a)

y=x1−x1+xy=x\sqrt{\dfrac {1-x}{1+x}};

(b)

y=x21−x3−x(3+x)23y=\dfrac {x^{2}}{1-x}\sqrt[3]{\dfrac {3-x}{(3+x)^{2}}};

(c)

y=(x−a1)α1(x−a2)α2⋯(x−an)αny=(x-a_{1})^{\alpha_{1}}(x-a_{2})^{\alpha_{2}}\cdots (x-a_{n})^{\alpha_{n}};

(d)

y=(x+1+x2)ny=\left(x+\sqrt{1+x^{2}}\right)^{n}.

Problem 985

Let φ(x)\varphi (x) and ψ(x)\psi (x) be differentiable functions of xx. Find the derivative of the function yy, if:

?
(a)

y=φ2(x)+ψ2(x)y=\sqrt{\varphi^{2}(x)+\psi^{2}(x)};

(b)

y=arctg⁡φ(x)ψ(x)y=\arctg \dfrac {\varphi (x)}{\psi (x)};

(c)

y=ψ(x)φ(x)(φ(x)≠0,ψ(x)>0)y=\sqrt[\varphi (x)]{\psi (x)} \quad (\varphi (x) \neq 0, \psi (x)>0);

(d)

y=log⁡φ(x)ψ(x)(φ(x)>0,ψ(x)>0)y=\log_{\varphi (x)}\psi (x) \quad (\varphi (x)>0, \psi (x)>0).

Problem 986

Find y′y', given:

?
(a)
(b)

y=f(sin⁡2x)+f(cos⁡2x)y=f(\sin^{2}x)+f(\cos^{2}x);

(c)

y=f(ex)⋅ef(x)y=f(e^{x}) \cdot e^{f(x)};

(d)

y=f{f[f(x)]}y=f\big\{ f[f(x)]\big\}.

where f(u)f(u) denotes a differentiable function.

Problem 986.1

y=f(x2)y=f(x^{2});

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Problem 986.1

y=f(x2)y=f(x^{2});

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Problem 987

Prove the differentiation rule for an nn-th order determinant:

ddx∣f11(x)f12(x)⋯f1n(x)⋯⋯⋯⋯fi1(x)fi2(x)⋯fin(x)⋯⋯⋯⋯fn1(x)fn2(x)⋯fnn(x)∣=∑i=1n∣f11(x)f12(x)⋯f1n(x)⋯⋯⋯⋯ddxfi1(x)ddxfi2(x)⋯ddxfin(x)⋯⋯⋯⋯fn1(x)fn2(x)⋯fnn(x)∣.(1) \frac{d}{dx} \begin{vmatrix} f_{11}(x) & f_{12}(x) & \cdots & f_{1n}(x) \\ \cdots & \cdots & \cdots & \cdots \\ f_{i1}(x) & f_{i2}(x) & \cdots & f_{in}(x) \\ \cdots & \cdots & \cdots & \cdots \\ f_{n1}(x) & f_{n2}(x) & \cdots & f_{nn}(x) \end{vmatrix} =\sum _{i=1}^{n} \begin{vmatrix} f_{11}(x) & f_{12}(x) & \cdots & f_{1n}(x) \\ \cdots & \cdots & \cdots & \cdots \\ \dfrac {d}{dx}f_{i1}(x) & \dfrac {d}{dx}f_{i2}(x) & \cdots & \dfrac {d}{dx}f_{in}(x) \\ \cdots & \cdots & \cdots & \cdots \\ f_{n1}(x) & f_{n2}(x) & \cdots & f_{nn}(x) \end{vmatrix} . \tag {1}
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Problem 988

Let:

F(x)=∣x−112−3x3−2−3x+1∣, F(x)= \begin{vmatrix} x-1 & 1 & 2 \\ -3 & x & 3 \\ -2 & -3 & x+1 \end{vmatrix},

find F′(x)F'(x).

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Problem 989

Let:

F(x)=∣xx2x312x3x2026x∣, F(x)= \begin{vmatrix} x & x^{2} & x^{3} \\ 1 & 2x & 3x^{2} \\ 0 & 2 & 6x \end{vmatrix},

find F′(x)F'(x).

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Problem 990

Given the graph of a function. Sketch, approximately, the graph of its derivative.

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Problem 991

Prove that the function

f(x)={x2sin⁡1xwhen x≠0;0when x=0 f(x)= \begin{cases} x^{2}\sin \dfrac {1}{x} & \text{when } x \neq 0; \\ 0 & \text{when } x=0 \end{cases}

has a discontinuous derivative.

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Problem 992

Under what conditions is the function

f(x)=xnsin⁡1x (x≠0) and f(0)=0 f(x)=x^{n}\sin \frac{1}{x} \ (x \neq 0) \text{ and } f(0)=0
?
(a)

continuous at x=0x=0;

(b)

differentiable at x=0x=0;

(c)

such that its derivative is continuous at x=0x=0?

Problem 993

Under what conditions does the function

f(x)=∣x∣nsin⁡1∣x∣m (x≠0) and f(0)=0(m>0) f(x)=\left|x\right|^{n}\sin \frac{1}{\left|x\right|^{m}} \ (x \neq 0) \text{ and } f(0)=0 \quad (m>0)

have

?
(a)

a bounded derivative on a neighborhood of the origin;

(b)

an unbounded derivative on this domain.

Problem 994

Let:

f(x)=(x−a)φ(x), f(x)=(x-a)\varphi (x),

where the function φ(x)\varphi (x) is continuous at x=ax=a. Find f′(a)f'(a).

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Problem 995

Let:

f(x)=∣x−a∣φ(x), f(x)=\left|x-a\right|\varphi (x),

where φ(x)\varphi (x) is a continuous function and φ(a)≠0\varphi (a) \neq 0. Prove that this function has no derivative at the point aa.

What are the one-sided derivatives f−′(a)f'_{-}(a) and f+′(a)f'_{+}(a)?

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Problem 996

Give an example of a continuous function that has no derivative at the given points a1,a2,⋯ ,ana_{1}, a_{2}, \cdots , a_{n}.

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Problem 997

Prove that the function

f(x)=x2∣cos⁡πx∣ (x≠0) and f(0)=0 f(x)=x^{2}\left|\cos \frac{\pi }{x}\right| \ (x \neq 0) \text{ and } f(0)=0

has points that are not differentiable in every neighborhood of the point x=0x=0, but is differentiable at the point x=0x=0 itself.

Sketch a graph of this function.

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Problem 998

Prove that the function

f(x)={x2,if x is rational,0,if x is irrational f(x)= \begin{cases} x^{2}, & \text{if } x \text{ is rational}, \\ 0, & \text{if } x \text{ is irrational} \end{cases}

has a derivative only at x=0x=0.

?
(a)
(b)
Problem 999

Investigate the differentiability of the following functions:

?
(a)

y=∣(x−1)(x−2)2(x−3)3∣y=\left|(x-1)(x-2)^{2}(x-3)^{3}\right|;

(b)

y=∣cos⁡x∣y=\left|\cos x\right|;

(c)

y=∣π2−x2∣sin⁡2xy=\left|\pi^{2}-x^{2}\right|\sin^{2}x;

(d)

y=arcsin⁡(cos⁡x)y=\arcsin (\cos x);

(e)

y={x−14(x+1)2when ∣x∣⩽1;∣x∣−1when ∣x∣>1.y= \begin{cases} \dfrac {x-1}{4}(x+1)^{2} & \text{when } \left|x\right| \leqslant 1; \\ \left|x\right|-1 & \text{when } \left|x\right|>1 . \end{cases}

Problem 1000

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=∣x∣f(x)=\left|x\right|.

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Problem 1001

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=[x]sin⁡πxf(x)=[x]\sin \pi x.

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Problem 1002

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=x∣cos⁡πx∣ (x≠0)f(x)=x\left|\cos \dfrac {\pi }{x}\right| \ (x \neq 0), f(0)=0f(0)=0.

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Problem 1003

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=sin⁡x2f(x)=\sqrt{\sin x^{2}}.

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Problem 1004

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=x1+e1x (x≠0)f(x)=\dfrac {x}{1+e^{\frac{1}{x}}} \ (x \neq 0), f(0)=0f(0)=0.

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Problem 1005

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=1−e−x2f(x)=\sqrt{1-e^{-x^{2}}}.

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Problem 1006

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=∣ln⁡∣x∣∣ (x≠0)f(x)=\left|\ln \left|x\right|\right| \ (x \neq 0).

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Problem 1007

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=arcsin⁡2x1+x2f(x)=\arcsin \dfrac {2x}{1+x^{2}}.

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Problem 1008

Find the left-hand and right-hand derivatives f−′(x)f'_{-}(x) and f+′(x)f'_{+}(x) of the function f(x)f(x), wheref(x)=(x−2)arctg⁡1x−2 (x≠2)f(x)=(x-2)\arctg \dfrac {1}{x-2} \ (x \neq 2), f(2)=0f(2)=0.

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Problem 1009

Prove that the function f(x)=xsin⁡1xf(x)=x\sin \dfrac {1}{x} for x≠0x \neq 0, with f(0)=0f(0)=0 at x=0x=0, is continuous at this point, but has neither a left-hand nor a right-hand derivative at this point.

?
Problem 1009.1
(a)
(b)
(c)
Problem 1009.1
(a)
(b)
(c)
Problem 1010

Let:

f(x)={x2,if x⩽x0;ax+b,if x>x0. f(x)= \begin{cases} x^{2}, & \text{if } x \leqslant x_{0}; \\ ax+b, & \text{if } x>x_{0} . \end{cases}

How should the coefficients aa and bb be chosen so that the function f(x)f(x) is continuous and differentiable at the point x=x0x=x_{0}?

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Problem 1011

Let:

F(x)={f(x),if x⩽x0;ax+b,if x>x0, F(x)= \begin{cases} f(x), & \text{if } x \leqslant x_{0}; \\ ax+b, & \text{if } x>x_{0}, \end{cases}

where the function f(x)f(x) is left-differentiable at x=x0x=x_{0}. How should the coefficients aa and bb be chosen so that the function F(x)F(x) is continuous and differentiable at the point x0x_{0}?

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Problem 1012

Suitably choose the parameters AA and cc so that the cubic parabola

y=A(x−a)(x−b)(x−c) y=A(x-a)(x-b)(x-c)

smoothly joins the two half-lines

y=k1(x−a)(−∞<x<a) y=k_{1}(x-a) \quad (-\infty <x<a)

and

y=k2(x−b)(b<x<+∞) y=k_{2}(x-b) \quad (b<x<+\infty )

on the region a⩽x⩽ba \leqslant x \leqslant b.

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Problem 1013

Use the parabola y=a+bx2 (∣x∣⩽c)y=a+bx^{2} \ (\left|x\right| \leqslant c) (where aa and bb are unknown parameters) to extend the curve y=m2∣x∣ (∣x∣>c)y=\dfrac {m^{2}}{\left|x\right|} \ (\left|x\right|>c) so that the resulting curve is smooth.

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Problem 1014

If:

?
(a)

the function f(x)f(x) has a derivative at the point x0x_{0}, while the function g(x)g(x) does not have a derivative at this point;

(b)

neither of the functions f(x)f(x) and g(x)g(x) has a derivative at the point x0x_{0},

can we conclude that their sum

F(x)=f(x)+g(x) F(x)=f(x)+g(x)

does not have a derivative at the point x=x0x=x_{0}?

Problem 1015

If:

?
(a)

the function f(x)f(x) has a derivative at the point x0x_{0}, while the function g(x)g(x) does not have a derivative at this point;

(b)

at the point x0x_{0} neither of the functions f(x)f(x) and g(x)g(x) has a derivative,

can we conclude that their product

F(x)=f(x)g(x) F(x)=f(x)g(x)

does not have a derivative at the point x=x0x=x_{0}?

Problem 1016

If:

?
(a)

the function f(x)f(x) has a derivative at the point x=g(x0)x=g(x_{0}), while the function g(x)g(x) does not have a derivative at the point x=x0x=x_{0};

(b)

the function f(x)f(x) does not have a derivative at the point x=g(x0)x=g(x_{0}), while the function g(x)g(x) has a derivative at the point x=x0x=x_{0};

(c)

the function f(x)f(x) does not have a derivative at the point x=g(x0)x=g(x_{0}) and the function g(x)g(x) does not have a derivative at the point x=x0x=x_{0},

what can be said about the differentiability of the function

F(x)=f[g(x)] F(x)=f[g(x)]

at the given point x=x0x=x_{0}?

Problem 1017

At which points of the graph of the function

y=x+sin⁡x3 y=x+\sqrt[3]{\sin x}

is the tangent line vertical? Sketch the graph.

?
Problem 1018

Can the function f(x)f(x) have, at a point of discontinuity,

?
(a)

a finite derivative;

(b)

an infinite derivative?

Problem 1019

If the function f(x)f(x) is differentiable on the finite interval (a,b)(a,b), and

lim⁡x→a+0f(x)=∞, \lim _{x \to a+0} f(x)=\infty ,

must it then follow that

?
(a)

lim⁡x→a+0f′(x)=∞\lim \limits_{x \to a+0} f'(x)=\infty;

(b)

lim‾⁡x→a+0∣f′(x)∣=+∞\varlimsup \limits_{x \to a+0} \left|f'(x)\right|=+\infty?

Problem 1020

If the function f(x)f(x) is differentiable on the finite interval (a,b)(a,b) and

lim⁡x→a+0f′(x)=∞, \lim _{x \to a+0} f'(x)=\infty ,

must it then follow that

lim⁡x→a+0f(x)=∞? \lim _{x \to a+0} f(x)=\infty ?
?
Problem 1021

Suppose the function f(x)f(x) is differentiable on the interval (x0,+∞)(x_{0}, +\infty ) and lim⁡x→+∞f(x)\lim \limits_{x \to +\infty } f(x) exists. Does it follow that lim⁡x→+∞f′(x)\lim \limits_{x \to +\infty } f'(x) exists?

?
Problem 1022

Suppose the bounded function f(x)f(x) is differentiable on (x0,+∞)(x_{0}, +\infty ) and lim⁡x→+∞f′(x)\lim \limits_{x \to +\infty } f'(x) exists. Does it follow that a finite or infinite lim⁡x→+∞f(x)\lim \limits_{x \to +\infty } f(x) exists?

?
Problem 1023

Can an inequality be differentiated term by term?

?
Problem 1024

Derive formulas expressing the sums

Pn=1+2x+3x2+⋯+nxn−1 P_{n}=1+2x+3x^{2}+\cdots +nx^{n-1}

and

Qn=12+22x+32x2+⋯+n2xn−1 Q_{n}=1^{2}+2^{2}x+3^{2}x^{2}+\cdots +n^{2}x^{n-1}
?
Problem 1025

Derive formulas expressing the sums

Sn=sin⁡x+sin⁡2x+⋯+sin⁡nx S_{n}=\sin x+\sin 2x+\cdots +\sin nx

and

Tn=cos⁡x+2cos⁡2x+⋯+ncos⁡nx T_{n}=\cos x+2\cos 2x+\cdots +n\cos nx
?
Problem 1025.1
[NO TASK]
Problem 1025.1
[NO TASK]
Problem 1026

Using the identity

cos⁡x2cos⁡x4⋯cos⁡x2n=sin⁡x2nsin⁡x2n, \cos \frac{x}{2}\cos \frac{x}{4}\cdots \cos \frac{x}{2^{n}}=\frac{\sin x}{2^{n}\sin \dfrac {x}{2^{n}}},

derive a formula expressing the sum

Sn=12tg⁡x2+14tg⁡x4+⋯+12ntg⁡x2n S_{n}=\frac{1}{2}\tg \frac{x}{2}+\frac{1}{4}\tg \frac{x}{4}+\cdots +\frac{1}{2^{n}}\tg \frac{x}{2^{n}}
?
Problem 1027

Prove that the derivative of a differentiable even function is an odd function, and the derivative of a differentiable odd function is an even function.

Give a geometric interpretation of this fact.

?
Problem 1028

Prove that the derivative of a differentiable periodic function is again a periodic function with the same period.

?
Problem 1029

If the radius of a circle increases at the constant rate of 22 cm/sec, what is the rate of increase of the area of the circle when the radius R=10R=10 cm?

?
Problem 1030

One side of a rectangle is x=20x=20 meters, and the other side is y=15y=15 meters. If the first side decreases at a rate of 11 meter/sec, and the second side increases at a rate of 22 meters/sec, what are the rates of change of the area and the diagonal of the rectangle?

?
Problem 1031

Two boats AA and BB set out from the same dock at the same time. Boat AA travels north, and boat BB travels east. If the speed of boat AA is 3030 km/hour and the speed of boat BB is 4040 km/hour, what is the rate of increase of the distance between the two boats?

?
Problem 1032

Let

f(x)={x,if 0⩽x⩽2;2x−2,if 2<x<+∞, f(x)= \begin{cases} x, & \text{if } 0 \leqslant x \leqslant 2; \\ 2x-2, & \text{if } 2<x<+\infty , \end{cases}

and let S(x)S(x) denote the area enclosed by the curve y=f(x)y=f(x), the axis OxOx, and the line through the point x (x⩾0)x \ (x \geqslant 0) perpendicular to OxOx. Find the analytic expression for the function S(x)S(x), find its derivative S′(x)S'(x), and sketch the graph of the function y=S′(x)y=S'(x).

?
Problem 1033

The function S(x)S(x) is the area bounded by the circular arc y=a2−x2y=\sqrt{a^{2}-x^{2}}, the axis OxOx, and two straight lines passing through the point OO and the point x (∣x∣⩽a)x \ (\left|x\right| \leqslant a) and perpendicular to the axis OxOx. Find an analytic expression for the function S(x)S(x), find the derivative S′(x)S'(x), and sketch the graph of the derivative y=S′(x)y=S'(x).

?
Section 2.2Derivatives of Inverse Functions. Derivatives of Functions Given Parametrically. Derivatives of Implicit Functions[21/100%]
Problem 1034

Prove that the equation y3+3y=xy^{3}+3y=x defines a single-valued function y=y(x)y=y(x), and find its derivative yx′y'_{x}.

?
Problem 1035

Prove that the equation y−εsin⁡y=x (0⩽ε<1)y-\varepsilon \sin y=x \ (0 \leqslant \varepsilon <1) determines a single-valued function y=y(x)y=y(x), and find its derivative yx′y'_{x}.

?
Problem 1036

Let:

?
(a)

y=x+ln⁡x (x>0)y=x+\ln x \ (x>0);

(b)

y=x+exy=x+e^{x};

(c)

y=sh⁡xy=\sh x;

(d)

y=th⁡xy=\th x.

Find the domain of existence of the inverse function x=x(y)x=x(y) and find its derivative.

Problem 1037

Let:

?
(a)

y=2x2−x4y=2x^{2}-x^{4};

(b)

y=x21+x2y=\dfrac {x^{2}}{1+x^{2}};

(c)

y=2e−x−e−2xy=2e^{-x}-e^{-2x}.

Select the single-valued continuous branches of the inverse function x=x(y)x=x(y), find their derivatives, and sketch their graphs.

Problem 1038

Sketch the graph of the function y=y(x)y=y(x) and find its derivative yx′y'_{x}, given x=−1+2t−t2x=-1+2t-t^{2}, y=2−3t+t3y=2-3t+t^{3}. What is yx′(x)y'_{x}(x) at x=0x=0 and at x=−1x=-1? At what point M(x,y)M(x,y) is the derivative yx′(x)=0y'_{x}(x)=0?

?
Problem 1039

Find the derivative yx′y'_{x} (the parameters are positive), given:x=1−t3x=\sqrt[3]{1-\sqrt{t}}, y=1−t3y=\sqrt{1-\sqrt[3]{t}}.

?
Problem 1040

Find the derivative yx′y'_{x} (the parameters are positive), given:x=sin⁡2tx=\sin^{2}t, y=cos⁡2ty=\cos^{2}t.

?
Problem 1041

Find the derivative yx′y'_{x} (the parameters are positive), given:x=acos⁡tx=a\cos t, y=bsin⁡ty=b\sin t.

?
Problem 1042

Find the derivative yx′y'_{x} (the parameters are positive), given:x=ach⁡tx=a\ch t, y=bsh⁡ty=b\sh t.

?
Problem 1043

Find the derivative yx′y'_{x} (the parameters are positive), given:x=acos⁡3tx=a\cos^{3}t, y=asin⁡3ty=a\sin^{3}t.

?
Problem 1044

Find the derivative yx′y'_{x} (the parameters are positive), given:x=a(t−sin⁡t)x=a(t-\sin t), y=a(1−cos⁡t)y=a(1-\cos t).

?
Problem 1045

Find the derivative yx′y'_{x} (the parameters are positive), given:x=e2tcos⁡2tx=e^{2t}\cos^{2}t, y=e2tsin⁡2ty=e^{2t}\sin^{2}t.

?
Problem 1046

Find the derivative yx′y'_{x} (the parameters are positive), given:x=arcsin⁡t1+t2x=\arcsin \dfrac {t}{\sqrt{1+t^{2}}}, y=arccos⁡11+t2y=\arccos \dfrac {1}{\sqrt{1+t^{2}}}.

?
Problem 1047

Prove that the function y=y(x)y=y(x) determined by the system of equations

x=2t+∣t∣, y=5t2+4t∣t∣, x=2t+\left|t\right|, \ y=5t^{2}+4t\left|t\right|,

is differentiable at t=0t=0, but its derivative cannot be found using the ordinary formula.

?
Problem 1048

Find the derivative yx′y'_{x} of the following implicitly defined function:x2+2xy−y2=2xx^{2}+2xy-y^{2}=2x. What is y′y' when x=2x=2 and y=4y=4, and when x=2x=2 and y=0y=0?

?
Problem 1049

Find the derivative yx′y'_{x} of the following implicitly defined function:y2=2pxy^{2}=2px (parabola).

?
Problem 1050

Find the derivative yx′y'_{x} of the following implicitly defined function:x2a2+y2b2=1\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1 (ellipse).

?
Problem 1051

Find the derivative yx′y'_{x} of the following implicitly defined function:x+y=a\sqrt{x}+\sqrt{y}=\sqrt{a} (parabola).

?
Problem 1052

Find the derivative yx′y'_{x} of the following implicitly defined function:x2/3+y2/3=a2/3x^{2/3}+y^{2/3}=a^{2/3} (astroid).

?
Problem 1053

Find the derivative yx′y'_{x} of the following implicitly defined function:arctg⁡yx=ln⁡x2+y2\arctg \dfrac {y}{x}=\ln \sqrt{x^{2}+y^{2}} (logarithmic spiral).

?
Problem 1054

Find yx′y'_{x}, given:

?
(a)

r=aφr=a\varphi (Archimedean spiral);

(b)

r=a(1+cos⁡φ)r=a(1+\cos \varphi ) (cardioid);

(c)

r=aemφr=ae^{m\varphi } (logarithmic spiral),

where r=x2+y2r=\sqrt{x^{2}+y^{2}} and φ=arctg⁡yx\varphi =\arctg \dfrac {y}{x} denote polar coordinates.

Section 2.3Geometric Meaning of the Derivative[30/93%]
Problem 1055

Write the equations of the tangent and normal lines to the curve y=(x+1)3−x3y=(x+1)\sqrt[3]{3-x} at the points A(−1,0)A(-1,0), B(2,3)B(2,3), C(3,0)C(3,0).

?
(a)
(b)
(c)
Problem 1056

On the curve

y=2+x−x2 y=2+x-x^{2}

at which points is the tangent line:

?
(a)

parallel to the axis OxOx;

(b)

parallel to the bisector of the angle in the first quadrant?

Problem 1057

Prove that the parabola

y=a(x−x1)(x−x2) (a≠0,x1<x2) y=a(x-x_{1})(x-x_{2}) \ (a \neq 0, x_{1}<x_{2})

forms equal angles α\alpha and β\beta (0<α<π2,0<β<π2)\left(0<\alpha <\dfrac {\pi }{2}, 0<\beta <\dfrac {\pi }{2}\right) where it intersects the OxOx axis.

?
Problem 1058

On the curve

y=2sin⁡x (−π⩽x⩽π) y=2\sin x \ (-\pi \leqslant x \leqslant \pi )

find the region where the "slope of the curve" (that is, ∣y′∣\left|y'\right|) is greater than 11.

?
Problem 1059

The functions y=xy=x and y1=x+0.01⋅sin⁡1000πxy_{1}=x+0.01 \cdot \sin 1000\pi x differ by no more than 0.010.01; what is the maximum value of the difference of the derivatives of these functions? Sketch the corresponding graphs.

?
Problem 1060

What is the angle at which the curve y=ln⁡xy=\ln x meets the OxOx axis?

?
Problem 1061

What is the angle at which the curves y=x2y=x^{2} and x=y2x=y^{2} intersect?

?
Problem 1062

What is the angle at which the curves y=sin⁡xy=\sin x and y=cos⁡xy=\cos x intersect?

?
Problem 1063

How should the parameter nn be chosen so that the curve

y=arctg⁡nx (n>0) y=\arctg nx \ (n>0)

meets the OxOx axis at an angle greater than 89°89°?

?
Problem 1063.1
(a)
(b)
Problem 1063.1
(a)
(b)
Problem 1063.2
[NO TASK]
Problem 1063.2
[NO TASK]
Problem 1064

Find the angle between the left and right tangent lines of the curve:

?
(a)

y=1−e−a2x2y=\sqrt{1-e^{-a^{2}x^{2}}} at the point x=0x=0,

(b)

y=arcsin⁡2x1+x2y=\arcsin \dfrac {2x}{1+x^{2}} at the point x=1x=1

Problem 1065

Prove that the tangent line to the logarithmic spiral r=aemφr=ae^{m\varphi } (aa and mm constants) makes a constant angle with the radius vector to the point of tangency.

?
Problem 1066

Find the length of the subtangent of the curve y=axny=ax^{n}, and use this to give a method for constructing the tangent line to this curve.

?
Problem 1067

Prove that for the parabola y2=2pxy^{2}=2px:

?
(a)

the length of the subtangent equals twice the abscissa of the point of tangency;

(b)

the subnormal is a constant.

Give a method for constructing the tangent line to the parabola.

Problem 1068

Prove that the exponential curve y=ax (a>0,anda≠1)y=a^{x} \ (a>0, \text{and} a \neq 1) has a subtangent of constant length. Give a method for constructing the tangent line to the exponential curve.

?
Problem 1069

Find the length of the normal at an arbitrary point M(x0,y0)M(x_{0}, y_{0}) on the catenary

y=ach⁡xa y=a\ch \frac{x}{a}
?
Problem 1070

Prove that the length of the portion of the tangent line to the astroid

x2/3+y2/3=a2/3 (a>0) x^{2/3}+y^{2/3}=a^{2/3} \ (a>0)

that lies between the coordinate axes is a constant.

?
Problem 1071

If the parabola y=ax2+bx+cy=ax^{2}+bx+c is tangent to the OxOx axis, what relation holds among the coefficients a,b,ca,b,c?

?
Problem 1072

Under what condition is the cubic parabola

y=x3+px+q y=x^{3}+px+q

tangent to the OxOx axis?

?
Problem 1073

For what value of the parameter aa is the parabola y=ax2y=ax^{2} tangent to the curve y=ln⁡xy=\ln x?

?
Problem 1074

Prove that the curves

y=f(x) [f(x)>0] y=f(x) \ [f(x)>0]

and

y=f(x)sin⁡ax, y=f(x)\sin ax,

where f(x)f(x) is a differentiable function, are tangent to each other at their common points.

?
Problem 1075

Prove that the families of hyperbolas x2−y2=ax^{2}-y^{2}=a and xy=bxy=b form an orthogonal net, that is, the curves of these two families intersect at right angles.

?
Problem 1076

Prove that the families of parabolas

y2=4a(a−x) (a>0) y^{2}=4a(a-x) \ (a>0)

and

y2=4b(b+x) (b>0) y^{2}=4b(b+x) \ (b>0)

form an orthogonal net.

?
Problem 1077

Write the equations of the tangent line and the normal line to the curve x=2t−t2x=2t-t^{2} and y=3t−t3y=3t-t^{3} at the points where (1) t=0t=0, (2) t=1t=1.

?
(a)
(b)
Problem 1078

Write the equations of the tangent line and the normal line to the curve

x=2t+t21+t3,y=2t−t21+t3 x=\frac{2t+t^{2}}{1+t^{3}}, \quad y=\frac{2t-t^{2}}{1+t^{3}}

at the points where (1) t=0t=0, (2) t=1t=1, (3) t=∞t=\infty.

?
(a)
(b)
(c)
Problem 1079

Write the equation of the tangent line

x=a(t−sin⁡t), y=a(1−cos⁡t) x=a(t-\sin t), \ y=a(1-\cos t)

to the cycloid (rolling curve) at an arbitrary point t=t0t=t_{0}. Give a construction for the tangent line of the cycloid.

?
Problem 1080

Prove that the tractrix

x=a(ln⁡tg⁡t2+cos⁡t), x=a\left(\ln \tg \frac{t}{2}+\cos t\right), y=asin⁡t (a>0,0<t<π) y=a\sin t \ (a>0, 0<t<\pi )

has a tangent segment of constant length.

?
Problem 1081

Write the equations of the tangent line and the normal line to the following curve at the given point:x2100+y264=1\dfrac {x^{2}}{100}+\dfrac {y^{2}}{64}=1, M(6,6.4)M(6, 6.4).

?
Problem 1082

Write the equations of the tangent line and the normal line to the following curve at the given point:xy+ln⁡y=1xy+\ln y=1, M(1,1)M(1,1).

?
Section 2.4Differentials of Functions[29/100%]
Problem 1083

Let:

?
(a)

Δx=1\Delta x=1;

(b)

Δx=0.1\Delta x=0.1;

(c)

Δx=0.01\Delta x=0.01,

for the function

f(x)=x3−2x+1 f(x)=x^{3}-2x+1

find: (1) Δf(1)\Delta f(1), (2) df(1)df(1), and compare them.

Problem 1084

The equation of motion is

x=5t2, x=5t^{2},

where tt is measured in seconds and xx in meters. Let (1) Δt=1\Delta t=1 second, (2) Δt=0.1\Delta t=0.1 second, (3) Δt=0.001\Delta t=0.001 second; at the instant t=2t=2 seconds, find the increment Δx\Delta x of the path and the differential dxdx of the path, and compare them.

?
(a)
(b)
(c)
Problem 1085

Find the differential of the function yy, given:y=1xy=\dfrac {1}{x}.

?
Problem 1086

Find the differential of the function yy, given:y=1aarctg⁡xa (a≠0)y=\dfrac {1}{a}\arctg \dfrac {x}{a} \ (a \neq 0).

?
Problem 1087

Find the differential of the function yy, given:y=12aln⁡∣x−ax+a∣y=\dfrac {1}{2a}\ln \left|\dfrac {x-a}{x+a}\right|.

?
Problem 1088

Find the differential of the function yy, given:y=ln⁡∣x+x2+a∣y=\ln \left|x+\sqrt{x^{2}+a}\right|.

?
Problem 1089

Find the differential of the function yy, given:y=arcsin⁡xa (a≠0)y=\arcsin \dfrac {x}{a} \ (a \neq 0).

?
Problem 1090

Find:

?
(a)

d(xex)d(xe^{x});

(b)

d(sin⁡x−xcos⁡x)d(\sin x-x\cos x);

(c)

d(1x3)d\left(\dfrac {1}{x^{3}}\right);

(d)

d(ln⁡xx)d\left(\dfrac {\ln x}{\sqrt{x}}\right);

(e)

d(a2+x2)d\left(\sqrt{a^{2}+x^{2}}\right);

(f)

d(x1−x2)d\left(\dfrac {x}{\sqrt{1-x^{2}}}\right);

(g)

dln⁡(1−x2)d\ln (1-x^{2});

(h)

d(arccos⁡1∣x∣)d\left(\arccos \dfrac {1}{\left|x\right|}\right);

(i)

d(sin⁡x2cos⁡2x+12ln⁡∣tg⁡(x2+π4)∣)d\left(\dfrac {\sin x}{2\cos^{2}x}+\dfrac {1}{2}\ln \left|\tg \left(\dfrac {x}{2}+\dfrac {\pi }{4}\right)\right|\right).

Problem 1091

Let uu, vv, ww be differentiable functions of xx. Find the differential of the function yy, given:y=uvwy=uvw.

?
Problem 1092

Let uu, vv, ww be differentiable functions of xx. Find the differential of the function yy, given:y=uv2y=\dfrac {u}{v^{2}}.

?
Problem 1093

Let uu, vv, ww be differentiable functions of xx. Find the differential of the function yy, given:y=1u2+v2y=\dfrac {1}{\sqrt{u^{2}+v^{2}}}.

?
Problem 1094

Let uu, vv, ww be differentiable functions of xx. Find the differential of the function yy, given:y=arctg⁡uvy=\arctg \dfrac {u}{v}.

?
Problem 1095

Let uu, vv, ww be differentiable functions of xx. Find the differential of the function yy, given:y=ln⁡u2+v2y=\ln \sqrt{u^{2}+v^{2}}.

?
Problem 1096

Find

?
(a)

ddx3(x3−2x6−x9)\dfrac {d}{dx^{3}}(x^{3}-2x^{6}-x^{9});

(b)

ddx2(sin⁡xx)+\dfrac {d}{dx^{2}}\left(\dfrac {\sin x}{x}\right)^{+};

(c)

d(sin⁡x)d(cos⁡x)\dfrac {d(\sin x)}{d(\cos x)};

(d)

d(tg⁡x)d(ctg⁡x)\dfrac {d(\tg x)}{d(\ctg x)};

(e)

d(arcsin⁡x)d(arccos⁡x)\dfrac {d(\arcsin x)}{d(\arccos x)}.

Problem 1097

There is a circular sector with radius R=100R=100 cm and central angle α=60°\alpha =60°. If (1) its radius RR increases by 11 cm; (2) the angle α\alpha decreases by 30′30', what is the change in the area of the sector? Find both the exact and the approximate solutions.

?
(a)
(b)
Problem 1098

The period of oscillation of a simple pendulum (measured in seconds) is determined by the formula:

T=2πlg, T=2\pi \sqrt{\frac{l}{g}},

where ll is the pendulum length measured in centimeters, and g=981g=981 cm/sec2^{2} is the acceleration due to gravity.

In order to increase the period TT by 0.050.05 seconds, how much modification is needed to the pendulum length l=20l=20 cm?

?
Problem 1099

Replacing the increment of the function by its differential, find the approximate value of the following:1.023\sqrt[3]{1.02}.

?
Problem 1100

Replacing the increment of the function by its differential, find the approximate value of the following:sin⁡29°\sin 29°.

?
Problem 1101

Replacing the increment of the function by its differential, find the approximate value of the following:cos⁡151°\cos 151°.

?
Problem 1102

Replacing the increment of the function by its differential, find the approximate value of the following:arctg⁡1.05\arctg 1.05.

?
Problem 1103

Replacing the increment of the function by its differential, find the approximate value of the following:lg⁡11\lg 11.

?
Problem 1104

Prove the approximate formula:

a2+x≈a+x2a (a>0), \sqrt{a^{2}+x} \approx a+\frac{x}{2a} \ (a>0),

where ∣x∣≪a\left|x\right| \ll a (the relation A≪BA \ll B between positive numbers AA and BB means that when AA is compared with BB, AA is a higher-order infinitesimal).

Use this formula to approximately compute:

?
(a)
(b)

34\sqrt{34};

(c)

120\sqrt{120}

and compare with the values in the table.

Problem 1104.1

5\sqrt{5};

?
Problem 1104.1

5\sqrt{5};

?
Problem 1105

Prove the approximate formula:

an+xn≈a+xnan−1 (a>0). \sqrt[n]{a^{n}+x} \approx a+\frac{x}{na^{n-1}} \ (a>0) .

where ∣x∣≪a\left|x\right| \ll a. Use this formula to approximately compute:

?
(a)

93\sqrt[3]{9};

(b)

804\sqrt[4]{80};

(c)

1007\sqrt[7]{100};

(d)

100010\sqrt[10]{1000}.

Problem 1106

The side of a square is x=2.4x=2.4 m ±0.05\pm 0.05 m; what are the relative error and absolute error of the resulting computed area of the square?

?
Problem 1107

In order to compute the volume of a sphere accurate to 1%1\%, what relative error is permitted when measuring the radius RR of the sphere?

?
Problem 1108

Using the oscillation of a simple pendulum, the formula g=4π2lT2g=\dfrac {4\pi^{2}l}{T^{2}} (where ll is the pendulum length and TT is the full period of the pendulum's oscillation) is used to find the acceleration due to gravity. How does the relative error δ\delta when measuring (1) the pendulum length ll, (2) the period TT, affect the value gg?

?
(a)
(b)
Problem 1109

Find the absolute error of the common logarithm of a number x (x>0)x \ (x>0), given that the relative error of this number equals δ\delta.

?
Problem 1110

Prove: an angle found from a table of logarithms of tangents is more precise than an angle found from a table of logarithms of sines having the same number of decimal places.

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Section 2.5Higher-Order Derivatives and Differentials[126/98%]
Problem 1111

Find y′′y'', given: y=x1+x2y=x\sqrt{1+x^{2}}.

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Problem 1112

Find y′′y'', given:y=x1−x2y=\dfrac {x}{\sqrt{1-x^{2}}}.

?
Problem 1113

Find y′′y'', given:y=e−x2y=e^{-x^{2}}.

?
Problem 1114

Find y′′y'', given:y=tg⁡xy=\tg x.

?
Problem 1115

Find y′′y'', given:y=(1+x2)arctg⁡xy=(1+x^{2})\arctg x.

?
Problem 1116

Find y′′y'', given:y=arcsin⁡x1−x2y=\dfrac {\arcsin x}{\sqrt{1-x^{2}}}.

?
Problem 1117

Find y′′y'', given:y=xln⁡xy=x\ln x.

?
Problem 1118

Find y′′y'', given:y=ln⁡f(x)y=\ln f(x).

?
Problem 1119

Find y′′y'', given:y=x[sin⁡(ln⁡x)+cos⁡(ln⁡x)]y=x\left[\sin (\ln x)+\cos (\ln x)\right].

?
Problem 1120

Let y=esin⁡xcos⁡(sin⁡x)y=e^{\sin x}\cos (\sin x), find y(0)y(0), y′(0)y'(0) and y′′(0)y''(0).

?
Problem 1121

Let u=φ(x)u=\varphi (x) and v=ψ(x)v=\psi (x) be twice differentiable functions. Find y′′y'', given:y=u2y=u^{2}.

?
Problem 1122

Let u=φ(x)u=\varphi (x) and v=ψ(x)v=\psi (x) be twice differentiable functions. Find y′′y'', given:y=ln⁡uvy=\ln \dfrac {u}{v}.

?
Problem 1123

Let u=φ(x)u=\varphi (x) and v=ψ(x)v=\psi (x) be twice differentiable functions. Find y′′y'', given:y=u2+v2y=\sqrt{u^{2}+v^{2}}.

?
Problem 1124

Let u=φ(x)u=\varphi (x) and v=ψ(x)v=\psi (x) be twice differentiable functions. Find y′′y'', given:y=uv (u>0)y=u^{v} \ (u>0).

?
Problem 1125

Let f(x)f(x) be a thrice differentiable function. Find y′′y'' and y′′′y''', given:y=f(x2)y=f(x^{2}).

?
Problem 1126

Let f(x)f(x) be a thrice differentiable function. Find y′′y'' and y′′′y''', given:y=f(1x)y=f\left(\dfrac {1}{x}\right).

?
Problem 1127

Let f(x)f(x) be a thrice differentiable function. Find y′′y'' and y′′′y''', given:y=f(ex)y=f(e^{x}).

?
Problem 1128

Let f(x)f(x) be a thrice differentiable function. Find y′′y'' and y′′′y''', given:y=f(ln⁡x)y=f(\ln x).

?
Problem 1129

Let f(x)f(x) be a thrice differentiable function. Find y′′y'' and y′′′y''', given:y=f[φ(x)]y=f[\varphi (x)], where φ(x)\varphi (x) is a function differentiable as many times as needed.

?
Problem 1130

Find d2yd^{2}y for the function y=exy=e^{x} in the following two cases:

?
(a)

xx is the independent variable;

(b)

xx is an intermediate variable.

Problem 1131

Treating xx as the independent variable, find d2yd^{2}y, given: y=1+x2y=\sqrt{1+x^{2}}.

?
Problem 1132

Treating xx as the independent variable, find d2yd^{2}y, given: y=ln⁡xxy=\dfrac {\ln x}{x}.

?
Problem 1133

Treating xx as the independent variable, find d2yd^{2}y, given:y=xxy=x^{x}.

?
Problem 1134

Let uu and vv be twice differentiable functions of the variable xx. Find d2yd^{2}y, given:y=uvy=uv.

?
Problem 1135

Let uu and vv be twice differentiable functions of the variable xx. Find d2yd^{2}y, given:y=uvy=\dfrac {u}{v}.

?
Problem 1136

Let uu and vv be twice differentiable functions of the variable xx. Find d2yd^{2}y, given:y=umvny=u^{m}v^{n} (mm and nn are constants).

?
Problem 1137

Let uu and vv be twice differentiable functions of the variable xx. Find d2yd^{2}y, given:y=au (a>0)y=a^{u} \ (a>0).

?
Problem 1138

Let uu and vv be twice differentiable functions of the variable xx. Find d2yd^{2}y, given:y=ln⁡u2+v2y=\ln \sqrt{u^{2}+v^{2}}.

?
Problem 1139

Let uu and vv be twice differentiable functions of the variable xx. Find d2yd^{2}y, given:y=arctg⁡uvy=\arctg \dfrac {u}{v}.

?
Problem 1140

Find the derivatives yx′y'_{x}, yx2′′y''_{x^{2}}, yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given parametrically:x=2t−t2x=2t-t^{2}, y=3t−t3y=3t-t^{3}.

?
Problem 1141

Find the derivatives yx′y'_{x}, yx2′′y''_{x^{2}}, yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given parametrically:x=acos⁡tx=a\cos t, y=asin⁡ty=a\sin t.

?
Problem 1142

Find the derivatives yx′y'_{x}, yx2′′y''_{x^{2}}, yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given parametrically:x=a(t−sin⁡t)x=a(t-\sin t), y=a(1−cos⁡t)y=a(1-\cos t).

?
Problem 1143

Find the derivatives yx′y'_{x}, yx2′′y''_{x^{2}}, yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given parametrically:x=etcos⁡tx=e^{t}\cos t, y=etsin⁡ty=e^{t}\sin t.

?
Problem 1144

Find the derivatives yx′y'_{x}, yx2′′y''_{x^{2}}, yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given parametrically:x=f′(t)x=f'(t), y=tf′(t)−f(t)y=tf'(t)-f(t).

?
Problem 1145

Let the function y=f(x)y=f(x) be differentiable a certain number of times. Find the derivatives x′x', x′′x'', x′′′x''', x(4)x^{(4)} of the inverse function x=f−1(y)x=f^{-1}(y) (assuming all these derivatives exist).

?
Problem 1146

Find yx′y'_{x}, yx2′′y''_{x^{2}} and yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given implicitly:x2+y2=25x^{2}+y^{2}=25. What are y′y', y′′y'', and y′′′y''' at the point M(3,4)M(3,4)?

?
Problem 1147

Find yx′y'_{x}, yx2′′y''_{x^{2}} and yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given implicitly:y2=2pxy^{2}=2px.

?
Problem 1148

Find yx′y'_{x}, yx2′′y''_{x^{2}} and yx3′′′y'''_{x^{3}} of the function y=y(x)y=y(x) given implicitly:x2−xy+y2=1x^{2}-xy+y^{2}=1.

?
Problem 1149

Find yx′y'_{x} and yx2′′y''_{x^{2}}, given:y2+2ln⁡y=x4y^{2}+2\ln y=x^{4}.

?
Problem 1150

Find yx′y'_{x} and yx2′′y''_{x^{2}}, given:x2+y2=aearctg⁡yx (a>0)\sqrt{x^{2}+y^{2}}=ae^{\arctg \frac{y}{x}} \ (a>0).

?
Problem 1151

Suppose the function f(x)f(x) is defined and twice differentiable for x⩽x0x \leqslant x_{0}. How should the coefficients a,ba,b and cc be chosen so that the function

F(x)={f(x),if x⩽x0,a(x−x0)2+b(x−x0)+c,if x>x0 F(x)= \begin{cases} f(x), & \text{if } x \leqslant x_{0}, \\ a(x-x_{0})^{2}+b(x-x_{0})+c, & \text{if } x>x_{0} \end{cases}

is twice differentiable?

?
Problem 1152

A point moves in a straight line according to the law

s=10+20t−5t2. s=10+20t-5t^{2} .

Find the velocity and acceleration of the motion. What are the velocity and acceleration at the instant t=2t=2?

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Problem 1153

A point M(x,y)M(x,y) moves uniformly along the circle x2+y2=a2x^{2}+y^{2}=a^{2}, completing one revolution every TT seconds. Find the velocity vv and acceleration jj of the projection of point MM onto the OxOx axis, given that at t=0t=0 the position of the point is M0(a,0)M_{0}(a,0).

?
Problem 1154

A particle M(x,y)M(x,y) is thrown in the vertical plane OxyOxy with initial velocity v0v_{0} in a direction making angle α\alpha with the horizontal. Set up the equations of motion (neglecting air resistance) and compute the magnitudes of the velocity vv and acceleration jj, and the trajectory of the motion. What are the maximum height and range?

?
Problem 1155

A point moves according to the equations x=4sin⁡ωt−3cos⁡ωtx=4\sin \omega t-3\cos \omega t, y=3sin⁡ωt+4cos⁡ωty=3\sin \omega t+4\cos \omega t (ω\omega is a constant). Find the trajectory of the motion and the magnitudes of the velocity and acceleration.

?
Problem 1156

Find the derivative of the indicated order: y=x(2x−1)2(x+3)3y=x(2x-1)^{2}(x+3)^{3}; find y(6)y^{(6)} and y(7)y^{(7)}.

?
Problem 1157

Find the derivative of the indicated order: y=axmy=\dfrac {a}{x^{m}}, find y′′′y'''.

?
Problem 1158

Find the derivative of the indicated order: y=xy=\sqrt{x}, find y(10)y^{(10)}.

?
Problem 1159

Find the derivative of the indicated order: y=x21−xy=\dfrac {x^{2}}{1-x}, find y(8)y^{(8)}.

?
Problem 1160

Find the derivative of the indicated order: y=1+x1−xy=\dfrac {1+x}{\sqrt{1-x}}, find y(100)y^{(100)}.

?
Problem 1161

Find the derivative of the indicated order: y=x2e2xy=x^{2}e^{2x}, find y(20)y^{(20)}.

?
Problem 1162

Find the derivative of the indicated order: y=exxy=\dfrac {e^{x}}{x}, find y(10)y^{(10)}.

?
Problem 1163

Find the derivative of the indicated order: y=xln⁡xy=x\ln x, find y(5)y^{(5)}.

?
Problem 1164

Find the derivative of the indicated order: y=ln⁡xxy=\dfrac {\ln x}{x}, find y(5)y^{(5)}.

?
Problem 1165

Find the derivative of the indicated order: y=x2sin⁡2xy=x^{2}\sin 2x, find y(50)y^{(50)}.

?
Problem 1166

Find the derivative of the indicated order: y=cos⁡3x1−3x3y=\dfrac {\cos 3x}{\sqrt[3]{1-3x}}, find y′′′y'''.

?
Problem 1167

Find the derivative of the indicated order: y=sin⁡xsin⁡2xsin⁡3xy=\sin x\sin 2x\sin 3x, find y(10)y^{(10)}.

?
Problem 1168

Find the derivative of the indicated order: y=xsh⁡xy=x\sh x, find y(100)y^{(100)}.

?
Problem 1169

Find the derivative of the indicated order: y=excos⁡xy=e^{x}\cos x, find y(4)y^{(4)}.

?
Problem 1170

Find the derivative of the indicated order: y=sin⁡2xln⁡xy=\sin^{2}x\ln x, find y(6)y^{(6)}.

?
Problem 1171

In each of the following examples, treating xx as the independent variable, find the differential of the indicated order: y=x5y=x^{5}, find d5yd^{5}y.

?
Problem 1172

In each of the following examples, treating xx as the independent variable, find the differential of the indicated order: y=1xy=\dfrac {1}{\sqrt{x}}, find d3yd^{3}y.

?
Problem 1173

In each of the following examples, treating xx as the independent variable, find the differential of the indicated order: y=xcos⁡2xy=x\cos 2x, find d10yd^{10}y.

?
Problem 1174

In each of the following examples, treating xx as the independent variable, find the differential of the indicated order: y=exln⁡xy=e^{x}\ln x, find d4yd^{4}y.

?
Problem 1175

In each of the following examples, treating xx as the independent variable, find the differential of the indicated order: y=cos⁡x⋅ch⁡xy=\cos x \cdot \ch x, find d6yd^{6}y.

?
Problem 1176

Let uu be a function of xx that is differentiable a sufficient number of times; in each of the following examples find the differential of the indicated order: y=u2y=u^{2}, find d10yd^{10}y.

?
Problem 1177

Let uu be a function of xx that is differentiable a sufficient number of times; in each of the following examples find the differential of the indicated order: y=euy=e^{u}, find d4yd^{4}y.

?
Problem 1178

Let uu be a function of xx that is differentiable a sufficient number of times; in each of the following examples find the differential of the indicated order: y=ln⁡uy=\ln u, find d3yd^{3}y.

?
Problem 1179

Treating xx as a function of some independent variable, find d2y,d3yd^{2}y,d^{3}y and d4yd^{4}y for the function y=f(x)y=f(x).

?
Problem 1180

Express the second and third derivatives y′′y'' and y′′′y''' of the function y=f(x)y=f(x) in terms of successive differentials of the variables xx and yy, without assuming xx to be the independent variable.

?
Problem 1181

Prove: the function

y=C1cos⁡x+C2sin⁡x, y=C_{1}\cos x+C_{2}\sin x,

where C1C_{1} and C2C_{2} are arbitrary constants, satisfies the equation

y′′+y=0. y''+y=0 .
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Problem 1182

Prove: the function

y=C1ch⁡x+C2sh⁡x, y=C_{1}\ch x+C_{2}\sh x,

where C1C_{1} and C2C_{2} are arbitrary constants, satisfies the equation

y′′−y=0. y''-y=0 .
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Problem 1183

Prove: the function

y=C1eλ1x+C2eλ2x, y=C_{1}e^{\lambda _{1}x}+C_{2}e^{\lambda _{2}x},

where C1C_{1} and C2C_{2} are arbitrary constants and λ1\lambda_{1} and λ2\lambda_{2} are constants, satisfies the equation

y′′−(λ1+λ2)y′+λ1λ2y=0. y''-(\lambda _{1}+\lambda _{2})y'+\lambda _{1}\lambda _{2}y=0 .
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Problem 1184

Prove: the function

y=xn[C1cos⁡(ln⁡x)+C2sin⁡(ln⁡x)], y=x^{n}\left[C_{1}\cos (\ln x)+C_{2}\sin (\ln x)\right],

where C1C_{1} and C2C_{2} are arbitrary constants and nn is a constant, satisfies the equation:

x2y′′+(1−2n)xy′+(1+n2)y=0. x^{2}y''+(1-2n)xy'+(1+n^{2})y=0 .
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Problem 1185

Prove: the function

y=e−x2(C1cos⁡x2+C2sin⁡x2)+e−x2(C3cos⁡x2+C4sin⁡x2), \begin{align} y & =e^{-\frac{x}{\sqrt{2}}}\left(C_{1}\cos \frac{x}{\sqrt{2}}+C_{2}\sin \frac{x}{\sqrt{2}}\right) \\ & \quad +e^{-\frac{x}{\sqrt{2}}}\left(C_{3}\cos \frac{x}{\sqrt{2}}+C_{4}\sin \frac{x}{\sqrt{2}}\right), \end{align}

where C1,C2,C3C_{1},C_{2},C_{3} and C4C_{4} are arbitrary constants, satisfies the equation

y(4)+y=0. y^{(4)}+y=0 .
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Problem 1186

Prove: if the function f(x)f(x) has a derivative of order nn, then

[f(ax+b)](n)=anf(n)(ax+b). \left[f(ax+b)\right]^{(n)}=a^{n}f^{(n)}(ax+b) .
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Problem 1187

If P(x)=a0xn+a1xn−1+⋯+anP(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}, find P(n)(x)P^{(n)}(x).

?
Problem 1188

Find y(n)y^{(n)}, given: y=ax+bcx+dy=\dfrac {ax+b}{cx+d}.

?
Problem 1189

Find y(n)y^{(n)}, given: y=1x(1−x)y=\dfrac {1}{x(1-x)}.

?
Problem 1190

Find y(n)y^{(n)}, given: y=1x2−3x+2y=\dfrac {1}{x^{2}-3x+2}.

?
Problem 1191

Find y(n)y^{(n)}, given: y=11−2xy=\dfrac {1}{\sqrt{1-2x}}.

?
Problem 1192

Find y(n)y^{(n)}, given: y=x1+x3y=\dfrac {x}{\sqrt[3]{1+x}}.

?
Problem 1193

Find y(n)y^{(n)}, given: y=sin⁡2xy=\sin^{2}x.

?
Problem 1194

Find y(n)y^{(n)}, given: y=cos⁡2xy=\cos^{2}x.

?
Problem 1195

Find y(n)y^{(n)}, given: y=sin⁡3xy=\sin^{3}x.

?
Problem 1196

Find y(n)y^{(n)}, given: y=cos⁡3xy=\cos^{3}x.

?
Problem 1197

Find y(n)y^{(n)}, given: y=sin⁡axsin⁡bxy=\sin ax\sin bx.

?
Problem 1198

Find y(n)y^{(n)}, given: y=cos⁡axcos⁡bxy=\cos ax\cos bx.

?
Problem 1199

Find y(n)y^{(n)}, given: y=sin⁡axcos⁡bxy=\sin ax\cos bx.

?
Problem 1200

Find y(n)y^{(n)}, given: y=sin⁡2axcos⁡bxy=\sin^{2}ax\cos bx.

?
Problem 1201

Find y(n)y^{(n)}, given: y=sin⁡4x+cos⁡4xy=\sin^{4}x+\cos^{4}x.

?
Problem 1202

Find y(n)y^{(n)}, given: y=xcos⁡axy=x\cos ax.

?
Problem 1203

Find y(n)y^{(n)}, given: y=x2sin⁡axy=x^{2}\sin ax.

?
Problem 1204

Find y(n)y^{(n)}, given: y=(x2+2x+2)e−xy=(x^{2}+2x+2)e^{-x}.

?
Problem 1205

Find y(n)y^{(n)}, given: y=exxy=\dfrac {e^{x}}{x}.

?
Problem 1206

Find y(n)y^{(n)}, given: y=excos⁡xy=e^{x}\cos x.

?
Problem 1207

Find y(n)y^{(n)}, given: y=exsin⁡xy=e^{x}\sin x.

?
Problem 1208

Find y(n)y^{(n)}, given: y=ln⁡a+bxa−bxy=\ln \dfrac {a+bx}{a-bx}.

?
Problem 1209

Find y(n)y^{(n)}, given: y=eaxP(x)y=e^{ax}P(x), where P(x)P(x) is a polynomial.

?
Problem 1210

Find y(n)y^{(n)}, given: y=xsh⁡xy=x\sh x.

?
Problem 1211

Find dnyd^{n}y, given y=xnexy=x^{n}e^{x}.

?
Problem 1212

Find dnyd^{n}y, given y=ln⁡xxy=\dfrac {\ln x}{x}.

?
Problem 1213

Prove the identities:

?
(a)

[eaxsin⁡(bx+c)](n)\left[e^{ax}\sin (bx+c)\right]^{(n)}

=eax(a2+b2)n2sin⁡(bx+c+nφ) =e^{ax}(a^{2}+b^{2})^{\frac{n}{2}}\sin (bx+c+n\varphi )
(b)

[eaxcos⁡(bx+c)](n)\left[e^{ax}\cos (bx+c)\right]^{(n)}

=eax(a2+b2)n2cos⁡(bx+c+nφ), =e^{ax}(a^{2}+b^{2})^{\frac{n}{2}}\cos (bx+c+n\varphi ),

where sin⁡φ=ba2+b2\sin \varphi =\dfrac {b}{\sqrt{a^{2}+b^{2}}} and cos⁡φ=aa2+b2\cos \varphi =\dfrac {a}{\sqrt{a^{2}+b^{2}}}.

Problem 1214

Find y(n)y^{(n)}, given:

?
(a)

y=ch⁡axcos⁡bxy=\ch ax\cos bx;

(b)

y=ch⁡axsin⁡bxy=\ch ax\sin bx;

(c)

y=sh⁡axcos⁡bxy=\sh ax\cos bx;

(d)

y=sh⁡axsin⁡bxy=\sh ax\sin bx.

Problem 1215

Express the function

f(x)=sin⁡2px, f(x)=\sin ^{2p}x,

where pp is a natural number, as a trigonometric polynomial:

f(x)=∑k=0pAkcos⁡2kx, f(x)=\sum _{k=0}^{p}A_{k}\cos 2kx,

in order to find f(n)(x)f^{(n)}(x).

?
Problem 1216

Given:

?
(a)

f(x)=sin⁡2p+1xf(x)=\sin^{2p+1}x;

(b)

f(x)=cos⁡2pxf(x)=\cos^{2p}x;

(c)

f(x)=cos⁡2p+1xf(x)=\cos^{2p+1}x,

where pp is a positive integer, find f(n)(x)f^{(n)}(x).

Problem 1217

Using the identity

1x2+1=12i(1x−i−1x+i) \frac{1}{x^{2}+1}=\frac{1}{2i}\left(\frac{1}{x-i}-\frac{1}{x+i}\right)

prove that

(1x2+1)(n)=(−1)nn!(1+x2)n+12sin⁡[(n+1)arcctg⁡x]. \left(\frac{1}{x^{2}+1}\right)^{(n)}=\frac{(-1)^{n}n!}{(1+x^{2})^{\frac{n+1}{2}}}\sin \left[(n+1)\arcctg x\right] .
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Problem 1218

Find the nn-th order derivative of the function f(x)=arctg⁡xf(x)=\arctg x.

?
Problem 1219

Find f(n)(0)f^{(n)}(0), given:

?
(a)

f(x)=1(1−2x)(1+x)f(x)=\dfrac {1}{(1-2x)(1+x)};

(b)

f(x)=x1−xf(x)=\dfrac {x}{\sqrt{1-x}}.

Problem 1220

Given:

?
(a)

f(x)=x2eaxf(x)=x^{2}e^{ax};

(b)

f(x)=arctg⁡xf(x)=\arctg x;

(c)

f(x)=arcsin⁡xf(x)=\arcsin x.

Find f(n)(0)f^{(n)}(0).

Problem 1221

Given:

?
(a)

f(x)=cos⁡(marcsin⁡x)f(x)=\cos (m\arcsin x);

(b)

f(x)=sin⁡(marcsin⁡x)f(x)=\sin (m\arcsin x).

Problem 1222

Given:

?
(a)

f(x)=(arctg⁡x)2f(x)=(\arctg x)^{2};

(b)

f(x)=(arcsin⁡x)2f(x)=(\arcsin x)^{2}.

Problem 1223

Given

f(x)=(x−a)nφ(x), f(x)=(x-a)^{n}\varphi (x),

where the function φ(x)\varphi (x) has a continuous derivative of order (n−1)(n-1) in a neighborhood of the point aa, find f(n)(a)f^{(n)}(a).

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Problem 1224

Prove that the function

f(x)={x2nsin⁡1x,if x≠0,0,if x=0 f(x)= \begin{cases} x^{2n}\sin \dfrac {1}{x}, & \text{if } x \neq 0, \\ 0, & \text{if } x=0 \end{cases}

(nn a natural number) has, at the point x=0x=0, derivatives up to order nn, but no derivative of order (n+1)(n+1).

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Problem 1225

Prove that the function

f(x)={e−1x2,if x≠0,0,if x=0, f(x)= \begin{cases} e^{-\frac{1}{x^{2}}}, & \text{if } x \neq 0, \\ 0, & \text{if } x=0, \end{cases}

is infinitely differentiable at x=0x=0. Sketch the graph of this function.

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Problem 1226

Prove that the Chebyshev polynomials

Tm(x)=12m−1cos⁡(marccos⁡x) (m=0,1,2,⋯ ) T_{m}(x)=\frac{1}{2^{m-1}}\cos (m\arccos x) \ (m=0,1,2,\cdots )

satisfy the equation

(1−x2)Tm′′(x)−xTm′(x)+m2Tm(x)=0. (1-x^{2})T''_{m}(x)-xT'_{m}(x)+m^{2}T_{m}(x)=0 .
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Problem 1227

Prove that the Legendre polynomials

Pm(x)=12mm![(x2−1)m](m) (m=0,1,2,⋯ ) P_{m}(x)=\frac{1}{2^{m}m!}\left[(x^{2}-1)^{m}\right]^{(m)} \ (m=0,1,2,\cdots )

satisfy the equation

(1−x2)Pm′′(x)−2xPm′(x)+m(m+1)Pm(x)=0. (1-x^{2})P''_{m}(x)-2xP'_{m}(x)+m(m+1)P_{m}(x)=0 .
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Problem 1228

The Chebyshev--Laguerre polynomials are defined as follows:

Lm(x)=ex(xme−x)(m) (m=0,1,2,⋯ ). L_{m}(x)=e^{x}(x^{m}e^{-x})^{(m)} \ (m=0,1,2,\cdots ) .

Find an explicit expression for the polynomial Lm(x)L_{m}(x).

Prove that Lm(x)L_{m}(x) satisfies the equation

xLm′′(x)+(1−x)Lm′(x)+mLm(x)=0. xL''_{m}(x)+(1-x)L'_{m}(x)+mL_{m}(x)=0 .
?
Problem 1229

Let y=f(u)y=f(u) and u=φ(x)u=\varphi (x), where f(u)f(u) and φ(x)\varphi (x) are functions differentiable nn times.

Prove that dnydxn=∑k=1nAk(x)f(k)(u)\dfrac {d^{n}y}{dx^{n}}=\displaystyle \sum_{k=1}^{n}A_{k}(x)f^{(k)}(u), where the coefficients Ak(x) (k=0,1,⋯ ,n)A_{k}(x) \ (k=0,1,\cdots ,n) do not depend on the function f(u)f(u).

?
Problem 1230

Prove that for the nn-th derivative of the composite function y=f(x2)y=f(x^{2}), the following formula holds

dnydxn=(2x)nf(n)(x2)−n(n−1)1!(2x)n−2f(n−1)(x2)+n(n−1)(n−2)(n−3)2!(2x)n−4f(n−2)(x2)+⋯ . \begin{align} \frac{d^{n}y}{dx^{n}} & =(2x)^{n}f^{(n)}(x^{2})-\frac{n(n-1)}{1!}(2x)^{n-2}f^{(n-1)}(x^{2}) \\ & \quad +\frac{n(n-1)(n-2)(n-3)}{2!}(2x)^{n-4}f^{(n-2)}(x^{2}) \\ & \quad +\cdots . \end{align}
?
Problem 1231

The Chebyshev--Hermite polynomials are defined as follows:

Hm(x)=(−1)mex2(e−x2)(m) (m=0,1,2,⋯ ). H_{m}(x)=(-1)^{m}e^{x^{2}}(e^{-x^{2}})^{(m)} \ (m=0,1,2,\cdots ) .

Find an explicit expression for the polynomial Hm(x)H_{m}(x).

Prove that Hm(x)H_{m}(x) satisfies the equation

Hm′′(x)−2xHm′(x)+2mHm(x)=0. H''_{m}(x)-2xH'_{m}(x)+2mH_{m}(x)=0 .
?
Problem 1232

Prove the identity

(xn−1e1x)n=(−1)nxn+1e1x. (x^{n-1}e^{\frac{1}{x}})^{n}=\frac{(-1)^{n}}{x^{n+1}}e^{\frac{1}{x}} .
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Problem 1232.1
[NO TASK]
Problem 1232.1
[NO TASK]
Problem 1232.2
[NO TASK]
Problem 1232.2
[NO TASK]
Problem 1233

Let ddx=D\dfrac {d}{dx}=D denote the differential operator and let

f(D)=∑k=0npk(x)Dk f(D)=\sum _{k=0}^{n}p_{k}(x)D^{k}

be a polynomial in the differential symbol, where pk(x) (k=0,1,⋯ ,n)p_{k}(x) \ (k=0,1,\cdots ,n) are certain continuous functions of xx.

Prove that

f(D){eλxu(x)}=eλxf(D+λ)u(x), f(D)\left\{ e^{\lambda x}u(x)\right\} =e^{\lambda x}f(D+\lambda )u(x),

where λ\lambda is a constant.

?
Problem 1234

Prove that in the equation

∑k=0nakxkdkydxk=0 \sum _{k=0}^{n}a_{k}x^{k}\frac{d^{k}y}{dx^{k}}=0

if we set

x=et, x=e^{t},

where tt is the independent variable, then this equation takes the form

∑k=0nakD(D−1)⋯(D−k+1)y=0, \sum _{k=0}^{n}a_{k}D(D-1)\cdots (D-k+1)y=0,

where D=ddtD=\dfrac {d}{dt}.

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Section 2.6Rolle's, Lagrange's, and Cauchy's Theorems[36/97%]
Problem 1235

Verify the correctness of Rolle's theorem for the function

f(x)=(x−1)(x−2)(x−3) f(x)=(x-1)(x-2)(x-3)

Rolle's theorem: if the function f(x)f(x): (1) is defined and continuous on the closed interval [a,b][a,b]; (2) has a finite derivative f′(x)f'(x) on this interval; (3) f(a)=f(b)f(a)=f(b), then there exists at least one number cc in the interval (a,b)(a,b) such that f′(c)=0f'(c)=0, where a<c<ba<c<b.

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Problem 1236

The function

f(x)=1−x23 f(x)=1-\sqrt[3]{x^{2}}

is zero at x1=−1x_{1}=-1 and x2=1x_{2}=1, but for −1⩽x⩽1-1 \leqslant x \leqslant 1, f′(x)≠0f'(x) \neq 0. Explain this apparent contradiction with Rolle's theorem.

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Problem 1237

Let the function f(x)f(x) have a finite derivative f′(x)f'(x) at every point of the finite or infinite interval (a,b)(a,b), and let

lim⁡x→a+0f(x)=lim⁡x→b−0f(x). \lim _{x \to a+0}f(x)=\lim _{x \to b-0}f(x) .

Prove that

f′(c)=0, f'(c)=0,

where cc is some point in the interval (a,b)(a,b).

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Problem 1238

Let the function f(x)f(x): (1) be defined on the closed interval [x0,xn][x_{0},x_{n}] and have a continuous (n−1)(n-1)-th derivative f(n−1)(x)f^{(n-1)}(x); (2) have an nn-th derivative f(n)(x)f^{(n)}(x) on the interval (x0,xn)(x_{0},x_{n}); (3) satisfy the following equalities

f(x0)=f(x1)=⋯=f(xn) (x0<x1<⋯<xn). f(x_{0})=f(x_{1})=\cdots =f(x_{n}) \ (x_{0}<x_{1}<\cdots <x_{n}) .

Prove that there is at least one point ξ\xi in the interval (x0,xn)(x_{0},x_{n}) such that

f(n)(ξ)=0. f^{(n)}(\xi )=0 .
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Problem 1239

Let the function f(x)f(x): (1) be defined on the closed interval [a,b][a,b] and have a continuous (p+q)(p+q)-th derivative f(p+q)(x)f^{(p+q)}(x); (2) have a (p+q+1)(p+q+1)-th derivative f(p+q+1)(x)f^{(p+q+1)}(x) on the interval (a,b)(a,b); (3) satisfy the following equalities:

f(a)=f′(a)=⋯=f(p)(a)=0 f(a)=f'(a)=\cdots =f^{(p)}(a)=0

and f(b)=f′(b)=⋯=f(q)(b)=0f(b)=f'(b)=\cdots =f^{(q)}(b)=0.

Prove that in this case

f(p+q+1)(c)=0, f^{(p+q+1)}(c)=0,

where cc is some point in the interval (a,b)(a,b).

?
Problem 1240

Prove that if the polynomial with real coefficients ak (k=0,1,⋯ ,n)a_{k} \ (k=0,1,\cdots ,n)

Pn(x)=a0xn+a1xn−1+⋯+an (a0≠0) P_{n}(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n} \ (a_{0} \neq 0)

has only real roots, then its successive derivatives Pn′(x),Pn′′(x),⋯ ,Pn(n−1)(x)P'_{n}(x),P''_{n}(x),\cdots ,P_{n}^{(n-1)}(x) also have only real roots.

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Problem 1241

Prove that the Legendre polynomial

Pn(x)=12nn!dndxn{(x2−1)n} P_{n}(x)=\frac{1}{2^{n}n!}\frac{d^{n}}{dx^{n}}\left\{ (x^{2}-1)^{n}\right\}

has all its roots real and contained in the interval (−1,1)(-1,1).

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Problem 1242

Prove that the Chebyshev--Laguerre polynomial

Ln(x)=exdndxn(xne−x) L_{n}(x)=e^{x}\frac{d^{n}}{dx^{n}}(x^{n}e^{-x})

has all its roots positive.

?
Problem 1243

Prove: all the roots of the Chebyshev--Hermite polynomial

Hn(x)=(−1)nex2dndxn(e−x2) H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}(e^{-x^{2}})

are real.

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Problem 1244

Find the point on the curve y=x3y=x^{3} at which the tangent line is parallel to the chord joining the points A(−1,−1)A(-1,-1) and B(2,8)B(2,8).

?
Problem 1245

If ab<0ab<0, does the finite-increment formula hold for the function

f(x)=1x f(x)=\frac{1}{x}

on the closed interval [a,b][a,b]?

?
Problem 1246

Let:

?
(a)
(b)
(c)

f(x)=1xf(x)=\dfrac {1}{x};

(d)

f(x)=exf(x)=e^{x}.

Find the function θ=θ(x,Δx)\theta =\theta (x,\Delta x) satisfying

f(x+Δx)−f(x)=Δxf′(x+θΔx) (0<θ<1) f(x+\Delta x)-f(x)=\Delta x f'(x+\theta \Delta x) \ (0<\theta <1)

for each of these.

Problem 1246.1

f(x)=ax2+bx+c (a≠0)f(x)=ax^{2}+bx+c \ (a \neq 0);

?
Problem 1246.1

f(x)=ax2+bx+c (a≠0)f(x)=ax^{2}+bx+c \ (a \neq 0);

?
Problem 1246.2

f(x)=x3f(x)=x^{3};

?
Problem 1246.2

f(x)=x3f(x)=x^{3};

?
Problem 1247

Prove: if x⩾0x \geqslant 0, then

x+1−x=12x+θ(x), \sqrt{x+1}-\sqrt{x}=\frac{1}{2\sqrt{x+\theta (x)}},

where

14⩽θ(x)⩽12, \frac{1}{4} \leqslant \theta (x) \leqslant \frac{1}{2},

and lim⁡x→−0θ(x)=14,lim⁡x→+∞θ(x)=12\lim \limits_{x \to -0}\theta (x)=\dfrac {1}{4}, \lim \limits_{x \to +\infty }\theta (x)=\dfrac {1}{2}.

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Problem 1248

Let

f(x)={3−x22when 0⩽x⩽1,1xwhen 1<x<+∞. f(x)= \begin{cases} \dfrac {3-x^{2}}{2} & \text{when } 0 \leqslant x \leqslant 1, \\ \dfrac {1}{x} & \text{when } 1<x<+\infty . \end{cases}

On the closed interval [0,2][0,2], find the intermediate value cc in the finite-increment formula for the function f(x)f(x).

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Problem 1249

Given:

f(x)−f(0)=xf′[ξ(x)], f(x)-f(0)=xf'[\xi (x)],

where 0<ξ(x)<x0<\xi (x)<x.

Prove: if

f(x)=xsin⁡(ln⁡x) (x>0) f(x)=x\sin (\ln x) \ (x>0)

and

f(0)=0 f(0)=0

then the function ξ=ξ(x)\xi =\xi (x) is discontinuous on every arbitrarily small interval (0,ε)(0,\varepsilon ) (with ε>0\varepsilon >0).

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Problem 1250

Suppose the function f(x)f(x) has a continuous derivative f′(x)f'(x) on the interval (a,b)(a,b). For any point ξ\xi in (a,b)(a,b), can one always find two other points x1x_{1} and x2x_{2} in this interval such that

f(x2)−f(x1)x2−x1=f′(ξ) (x1<ξ<x2)? \frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}=f'(\xi ) \ (x_{1}<\xi <x_{2}) ?
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Problem 1251

Prove the following inequalities:

?
(a)

∣sin⁡x−sin⁡y∣⩽∣x−y∣\left|\sin x-\sin y\right| \leqslant \left|x-y\right|;

(b)

if 0<y<x0<y<x and p>1p>1, then pyp−1(x−y)<xp−yp<pxp−1(x−y)py^{p-1}(x-y)<x^{p}-y^{p}<px^{p-1}(x-y);

(c)

∣arctg⁡a−arctg⁡b∣⩽∣a−b∣\left|\arctg a-\arctg b\right| \leqslant \left|a-b\right|;

(d)

a−ba<ln⁡ab<a−bb\dfrac {a-b}{a}<\ln \dfrac {a}{b}<\dfrac {a-b}{b}, where 0<b<a0<b<a.

Problem 1252

Explain why Cauchy's theorem fails on the closed interval [−1,1][-1,1] for the functions f(x)=x2f(x)=x^{2} and g(x)=x3g(x)=x^{3}.

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Problem 1253

Suppose the function f(x)f(x) is differentiable on the closed interval [x1,x2][x_{1},x_{2}], and x1x2>0x_{1}x_{2}>0. Prove that

1x1−x2∣x1x2f(x1)f(x2)∣=f(ξ)−ξf′(ξ), \frac{1}{x_{1}-x_{2}}\begin{vmatrix} x_{1} & x_{2} \\ f(x_{1}) & f(x_{2}) \end{vmatrix}=f(\xi )-\xi f'(\xi ),

where x1<ξ<x2x_{1}<\xi <x_{2}.

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Problem 1254

Prove: if the function f(x)f(x) is differentiable but unbounded on a finite interval (a,b)(a,b), then its derivative f′(x)f'(x) is also unbounded on (a,b)(a,b). The converse is false; give an example.

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Problem 1255

Prove: if the function f(x)f(x) has a bounded derivative f′(x)f'(x) on a finite or infinite interval (a,b)(a,b), then f(x)f(x) is uniformly continuous on (a,b)(a,b).

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Problem 1256

Prove: if the function f(x)f(x) is differentiable on the infinite interval (x0,+∞)(x_{0},+\infty ), and

lim⁡x→+∞f′(x)=0, \lim _{x \to +\infty }f'(x)=0,

then

lim⁡x→+∞f(x)x=0, \lim _{x \to +\infty }\frac{f(x)}{x}=0,

that is: as x→+∞x \to +\infty, f(x)=o(x)f(x)=o(x).

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Problem 1257

Prove: if the function f(x)f(x) is differentiable on the infinite interval (x0,+∞)(x_{0},+\infty ) and, as

x→+∞,f(x)=o(x); x \to +\infty , f(x)=o(x);

then lim⁡x→+∞∣f′(x)∣=0\lim \limits_{x \to +\infty }\left|f'(x)\right|=0.

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Problem 1258

Suppose:

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(a)

Prove: if the function f(x)f(x): (1) is defined and continuous on the closed interval [x0,X][x_{0},X]; (2) has a finite derivative f′(x)f'(x) on the interval (x0,X)(x_{0},X); (3) has a finite or infinite limit

lim⁡x→x0+0f′(x)=f′(x0+0), \lim _{x \to x_{0}+0}f'(x)=f'(x_{0}+0),

then the finite or infinite one-sided derivative f+′(x0)f'_{+}(x_{0}) exists, and

f+′(x0)=f′(x0+0). f'_{+}(x_{0})=f'(x_{0}+0) .
(b)

Prove that the function

f(x)=arctg⁡1+x1−x (x≠1) and f(1)=0 f(x)=\arctg \frac{1+x}{1-x} \ (x \neq 1) \ \text{and} \ f(1)=0

has a finite limit

lim⁡x→1f′(x), \lim _{x \to 1}f'(x),

but the function f(x)f(x) has neither one-sided derivative f−′(1)f'_{-}(1) nor f+′(1)f'_{+}(1).

Give a geometric illustration of this fact.

Problem 1259

Prove: if f′(x)=0f'(x)=0 for a<x<ba<x<b, then f(x)=constantf(x)=\text{constant} for a<x<ba<x<b.

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Problem 1260

Prove: the only function f(x) (−∞<x<+∞)f(x) \ (-\infty <x<+\infty ) whose derivative is the constant

f′(x)=k f'(x)=k

is the linear function:

f(x)=kx+b. f(x)=kx+b .
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Problem 1261

Let f(n)(x)=0f^{(n)}(x)=0; what property does the function f(x)f(x) then have?

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Problem 1261.1
[NO TASK]
Problem 1261.1
[NO TASK]
Problem 1262

Prove that the only function y=y(x) (−∞<x<+∞)y=y(x) \ (-\infty <x<+\infty ) satisfying the equation

y′=λy (λ=a constant) y'=\lambda y \ (\lambda =\text{a constant})

is the exponential function:

y=Ceλx, y=Ce^{\lambda x},

where CC is an arbitrary constant.

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Problem 1263

Verify that the functions

f(x)=arctg⁡x+a1−ax f(x)=\arctg \frac{x+a}{1-ax}

and

g(x)=arctg⁡x g(x)=\arctg x

have the same derivative on the ranges: (1) ax<1ax<1 and (2) ax>1ax>1.

Deduce the relation between these functions.

?
(a)
(b)
Problem 1264

Prove the following identities:

?
(a)

2arctg⁡x+arcsin⁡2x1+x2=πsign⁡x2\arctg x+\arcsin \dfrac {2x}{1+x^{2}}=\pi \operatorname {sign} x,

when ∣x∣⩾1\left|x\right| \geqslant 1;

(b)

3arccos⁡x−arccos⁡(3x−4x3)=π3\arccos x-\arccos (3x-4x^{3})=\pi, when ∣x∣⩽12\left|x\right| \leqslant \dfrac {1}{2}.

Problem 1265

Prove that if the function f(x)f(x): (1) is continuous on the closed interval [a,b][a,b]; (2) has a finite derivative f′(x)f'(x) on this interval; (3) is not a linear function, then there exists at least one point cc in the interval (a,b)(a,b) satisfying

∣f′(c)∣>∣f(b)−f(a)b−a∣. \left|f'(c)\right|>\left|\frac{f(b)-f(a)}{b-a}\right| .

Give a geometric interpretation of this fact.

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Problem 1266

If the function f(x)f(x): (1) has a second derivative f′′(x)f''(x) on the interval [a,b][a,b] and (2) f′(a)=f′(b)=0f'(a)=f'(b)=0, then there exists at least one point cc in the interval (a,b)(a,b) satisfying

∣f′′(c)∣⩾4(b−a)2∣f(b)−f(a)∣. \left|f''(c)\right| \geqslant \frac{4}{(b-a)^{2}}\left|f(b)-f(a)\right| .
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Problem 1267

A car starts moving from a certain point, and within tt seconds it has covered a distance of ss meters during that time. Prove that the absolute value of the car's acceleration is not less than 4st2 meterssecond2\dfrac {4s}{t^{2}} \ \dfrac {\text{meters}}{\text{second}^{2}} at some instant.

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Section 2.7Increase and Decrease of a Function. Inequalities[30/100%]
Problem 1268

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=2+x−x2y=2+x-x^{2}.

?
Problem 1269

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=3x−x3y=3x-x^{3}.

?
Problem 1270

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=2x1+x2y=\dfrac {2x}{1+x^{2}}.

?
Problem 1271

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=xx+100 (x⩾0)y=\dfrac {\sqrt{x}}{x+100} \ (x \geqslant 0).

?
Problem 1272

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=x+sin⁡xy=x+\sin x.

?
Problem 1273

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=x+∣sin⁡2x∣y=x+\left|\sin 2x\right|.

?
Problem 1274

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=cos⁡πxy=\cos \dfrac {\pi }{x}.

?
Problem 1275

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=x22xy=\dfrac {x^{2}}{2^{x}}.

?
Problem 1276

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=xne−x (n>0,x⩾0)y=x^{n}e^{-x} \ (n>0, x \geqslant 0).

?
Problem 1277

Find the intervals of strict monotonicity (increase or decrease) of the following function: y=x2−ln⁡x2y=x^{2}-\ln x^{2}.

?
Problem 1278

If x>0,f(x)=x(32+sin⁡ln⁡x)x>0, f(x)=x\left(\sqrt{\dfrac {3}{2}}+\sin \ln x\right) and f(0)=0f(0)=0.

Discuss the monotonicity of the function f(x)f(x).

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Problem 1279

Prove that for a regular nn-gon inscribed in a circle, as the number of sides nn increases, its perimeter pnp_{n} increases, while the perimeter PnP_{n} of the regular nn-gon circumscribed about this circle decreases. Use this to prove that as n→∞n \to \infty, pnp_{n} and PnP_{n} have the same limit.

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Problem 1280

Prove that the function

(1+1x)x \left(1+\frac{1}{x}\right)^{x}

is increasing on the intervals (−∞,−1)(-\infty ,-1) and (0,+∞)(0,+\infty ).

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Problem 1281

Prove that the polynomial

P(x)=a0+a1x+⋯+anxn (n⩾1,an≠0) P(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n} \ (n \geqslant 1, a_{n} \neq 0)

is monotonic (in the strict sense!) on the intervals (−∞,−x0)(-\infty ,-x_{0}) and (x0,+∞)(x_{0},+\infty ), where x0x_{0} is a sufficiently large positive number.

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Problem 1282

Prove that the rational function

R(x)=a0+a1x+⋯+anxnb0+b1x+⋯+bmxm R(x)=\frac{a_{0}+a_{1}x+\cdots +a_{n}x^{n}}{b_{0}+b_{1}x+\cdots +b_{m}x^{m}} (m+n⩾1,m≠n,anbm≠0) (m+n \geqslant 1, m \neq n, a_{n}b_{m} \neq 0)

is monotonic (in the strict sense!) on the intervals (−∞,−x0)(-\infty ,-x_{0}) and (x0,+∞)(x_{0},+\infty ), where x0x_{0} is a sufficiently large positive number.

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Problem 1283

Must the derivative of a monotonic function also be monotonic?

?
Problem 1284

Prove that if φ(x)\varphi (x) is a monotonically increasing differentiable function, and when

x⩾x0,∣f′(x)∣⩽φ(x), x \geqslant x_{0}, \left|f'(x)\right| \leqslant \varphi (x),

then when x⩾x0x \geqslant x_{0}, ∣f(x)−f(x0)∣⩽φ(x)−φ(x0)\left|f(x)-f(x_{0})\right| \leqslant \varphi (x)-\varphi (x_{0}).

Give a geometric interpretation of this fact.

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Problem 1285

Suppose the function f(x)f(x) is continuous on the interval a⩽x<+∞a \leqslant x<+\infty, and when x>ax>a, f′(x)>k>0f'(x)>k>0, where kk is a constant. Prove: if f(a)<0f(a)<0, then on the interval (a,a−f(a)k)\left(a,a-\dfrac {f(a)}{k}\right) the equation f(x)=0f(x)=0 has one and only one real root.

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Problem 1286

If in some neighborhood ∣x−x0∣<δ\left|x-x_{0}\right|<\delta, the increment of the function Δf(x0)=f(x)−f(x0)\Delta f(x_{0})=f(x)-f(x_{0}) has the same sign as the increment of the independent variable Δx0=x−x0\Delta x_{0}=x-x_{0}, the function f(x)f(x) is said to be increasing at x0x_{0}.

Prove: if the function f(x) (a<x<b)f(x) \ (a<x<b) is increasing at every point of the finite or infinite interval (a,b)(a,b), then it is an increasing function on this interval.

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Problem 1287

Prove that the function

f(x)=x+x2sin⁡1x, if x≠0, and f(0)=0, f(x)=x+x^{2}\sin \frac{1}{x}, \text{ if } x \neq 0, \text{ and } f(0)=0,

is increasing at the point x=0x=0, but is not increasing on any interval (−ε,ε)(-\varepsilon ,\varepsilon ) containing this point, where ε>0\varepsilon >0 is an arbitrarily small number. Sketch this function.

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Problem 1288

Prove the theorem: suppose (1) the functions φ(x)\varphi (x) and ψ(x)\psi (x) are nn times differentiable; (2) φ(k)(x0)=ψ(k)(x0) (k=0,1,2,⋯ ,n−1)\varphi^{(k)}(x_{0})=\psi^{(k)}(x_{0}) \ (k=0,1,2,\cdots ,n-1); (3) when x>x0x>x_{0}, φ(n)(x)>ψ(n)(x)\varphi^{(n)}(x)>\psi^{(n)}(x); then when x>x0x>x_{0} the inequality

φ(x)>ψ(x) \varphi (x)>\psi (x)

holds.

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Problem 1289

Prove the following inequalities:

?
(a)

When x≠0x \neq 0, ex>1+xe^{x}>1+x;

(b)

When x>0x>0, x−x22<ln⁡(1+x)<xx-\dfrac {x^{2}}{2}<\ln (1+x)<x;

(c)

When x>0x>0, x−x36<sin⁡x<xx-\dfrac {x^{3}}{6}<\sin x<x;

(d)

When 0<x<π20<x<\dfrac {\pi }{2}, tg⁡x>x+x33\tg x>x+\dfrac {x^{3}}{3};

(e)

When x>0,y>0x>0, y>0 and 0<α<β0<\alpha <\beta,

(xα+yα)1α>(xβ+yβ)1β. (x^{\alpha }+y^{\alpha })^{\frac{1}{\alpha }}>(x^{\beta }+y^{\beta })^{\frac{1}{\beta }} .

Give a geometric interpretation of inequalities (a)--(4).

Problem 1290

Prove the inequality

2πx<sin⁡x<x, \frac{2}{\pi }x<\sin x<x,

holds when 0<x<π20<x<\dfrac {\pi }{2}.

?
Problem 1291

Prove that when x>0x>0 the inequality

(1+1x)x<e<(1+1x)x+1 \left(1+\frac{1}{x}\right)^{x}<e<\left(1+\frac{1}{x}\right)^{x+1}

holds.

?
Problem 1292

An arithmetic progression and a geometric progression have the same number of terms and the same first and last terms, and all their terms are positive. Prove that the sum of the terms of the arithmetic progression is greater than or equal to the sum of the terms of the geometric progression.

?
Problem 1293

Using the inequality

∑k=1n(akx+bk)2⩾0, \sum _{k=1}^{n}(a_{k}x+b_{k})^{2} \geqslant 0 ,

where x,ak,bk (k=1,2,⋯ ,n)x,a_{k},b_{k} \ (k=1,2,\cdots ,n) are real numbers, prove the Cauchy--Bunyakovsky inequality

(∑k=1nakbk)2⩽∑k=1nak2⋅∑k=1nbk2. \left(\sum _{k=1}^{n}a_{k}b_{k}\right)^{2} \leqslant \sum _{k=1}^{n}a_{k}^{2} \cdot \sum _{k=1}^{n}b_{k}^{2} .
?
Problem 1294

Prove: the arithmetic mean of positive numbers is not greater than the mean of the squares of these numbers, that is,

1n∑k=1nxk⩽1n∑k=1nxk2. \frac{1}{n}\sum _{k=1}^{n}x_{k} \leqslant \sqrt{\frac{1}{n}\sum _{k=1}^{n}x_{k}^{2}} .
?
Problem 1295

Prove: the geometric mean of positive numbers is not greater than the arithmetic mean of these numbers, that is,

x1x2⋯xnn⩽1n(x1+x2+⋯+xn). \sqrt[n]{x_{1}x_{2}\cdots x_{n}} \leqslant \frac{1}{n}(x_{1}+x_{2}+\cdots +x_{n}) .
?
Problem 1296

Let aa and bb be two positive numbers; then the function defined by the equalities below

if s≠0,Δs(a,b)=(as+bs2)1s, \text{if } s \neq 0, \Delta _{s}(a,b)=\left(\frac{a^{s}+b^{s}}{2}\right)^{\frac{1}{s}} , Δ0(a,b)=lim⁡s→0Δs(a,b) \Delta _{0}(a,b)=\lim _{s \to 0}\Delta _{s}(a,b)

is called the mean of order ss of the positive numbers aa and bb.

In particular, when s=−1s=-1 we get the harmonic mean, when s=0s=0 we get the geometric mean (try to prove this!); when s=1s=1 we get the arithmetic mean; when s=2s=2 we get the quadratic mean.

Prove:

  1. min⁡(a,b)⩽Δs(a,b)⩽max⁡(a,b)\min (a,b) \leqslant \Delta_{s}(a,b) \leqslant \max (a,b);

  2. when a≠ba \neq b, the function Δs(a,b)\Delta_{s}(a,b) is an increasing function of the variable ss;

  3. lim⁡s→−∞Δs(a,b)=min⁡(a,b)\displaystyle \lim_{s \to -\infty }\Delta_{s}(a,b)=\min (a,b);

    lim⁡s→+∞Δs(a,b)=max⁡(a,b). \lim _{s \to +\infty }\Delta _{s}(a,b)=\max (a,b) .
?
(a)
(b)
(c)
Problem 1297

Let f(x) (−∞<x<+∞)f(x) \ (-\infty <x<+\infty ) be a twice differentiable function, and

Mk=sup⁡−∞<x<+∞∣f(k)(x)∣<+∞ (k=0,1,2). M_{k}=\sup _{-\infty <x<+\infty }\left|f^{(k)}(x)\right|<+\infty \ (k=0,1,2) .

Prove the inequality:

M12⩽2M0M2. M_{1}^{2} \leqslant 2M_{0}M_{2} .
?
(a)
(b)
(c)
Section 2.8Concavity and Convexity. Inflection Points[21/100%]
Problem 1298

Examine the concavity or convexity of the curve

y=1+x3 y=1+\sqrt[3]{x}

at the points A(−1,0),B(1,2)A(-1,0), B(1,2) and C(0,0)C(0,0).

?
Problem 1299

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=3x2−x3. y=3x^{2}-x^{3} .
?
Problem 1300

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=a3a2+x2 (a>0). y=\frac{a^{3}}{a^{2}+x^{2}} \ (a>0) .
?
Problem 1301

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x+x53. y=x+x^{\frac{5}{3}} .
?
Problem 1302

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=1+x2. y=\sqrt{1+x^{2}} .
?
Problem 1303

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x+sin⁡x. y=x+\sin x .
?
Problem 1304

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=e−x2. y=e^{-x^{2}} .
?
Problem 1305

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=ln⁡(1+x2). y=\ln (1+x^{2}) .
?
Problem 1306

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=xsin⁡(ln⁡x) (x>0). y=x\sin (\ln x) \ (x>0) .
?
Problem 1307

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=xx (x>0). y=x^{x} \ (x>0) .
?
Problem 1308

Prove that the curve

y=x+1x2+1 y=\frac{x+1}{x^{2}+1}

has three inflection points lying on the same straight line. Sketch the graph of this function.

?
Problem 1309

How should the parameter hh be chosen so that the "probability curve"

y=hπe−h2x2 (h>0) y=\frac{h}{\sqrt{\pi }}e^{-h^{2}x^{2}} \ (h>0)

has inflection points x=±σx=\pm \sigma?

?
Problem 1310

Examine the concavity or convexity of the cycloid

x=a(t−sin⁡t),y=a(1−cos⁡t) (a>0) x=a(t-\sin t), y=a(1-\cos t) \ (a>0)
?
Problem 1311

Let the function f(x)f(x) be twice differentiable on the interval a⩽x<+∞a \leqslant x<+\infty, and:

  1. f(a)=A>0f(a)=A>0;

  2. f′(a)<0f'(a)<0;

  3. when x>ax>a, f′′(x)⩽0f''(x) \leqslant 0. Prove: the equation f(x)=0f(x)=0 has one and only one real root in the interval (a,+∞)(a,+\infty ).

?
Problem 1312

If for any two points x1x_{1} and x2x_{2} in the interval (a,b)(a,b) and any two numbers λ1\lambda_{1} and λ2 (λ1>0,λ2>0,λ1+λ2=1)\lambda_{2} \ (\lambda_{1}>0, \lambda_{2}>0, \lambda_{1}+\lambda_{2}=1) we have the inequality:

f(λ1x1+λ2x2)<λ1f(x1)+λ2f(x2) f(\lambda _{1}x_{1}+\lambda _{2}x_{2})<\lambda _{1}f(x_{1})+\lambda _{2}f(x_{2})

(or correspondingly, the reverse inequality f(λ1x1+λ2x2)>λ1f(x1)+λ2f(x2)f(\lambda_{1}x_{1}+\lambda_{2}x_{2})>\lambda_{1}f(x_{1})+\lambda_{2}f(x_{2})), then the function f(x)f(x) is said to be concave (convex) on the interval (a,b)(a,b).

Prove that the function f(x)f(x): (1) if f′′(x)>0f''(x)>0 for a<x<ba<x<b, then it is concave on (a,b)(a,b); (2) if f′′(x)<0f''(x)<0 for a<x<ba<x<b, then it is convex on (a,b)(a,b).

?
(a)
(b)
Problem 1313

Prove that the functions

xn (n>1),ex,xln⁡x x^{n} \ (n>1), e^{x}, x\ln x

are concave on the interval (0,+∞)(0,+\infty ); while the functions

xn (0<n<1),ln⁡x. x^{n} \ (0<n<1), \ln x .

are convex on the interval (0,+∞)(0,+\infty ).

?
Problem 1314

Prove the following inequalities, and explain their geometric meaning:

?
(a)
(b)

ex+ey2>ex+y2 (x≠y)\dfrac {e^{x}+e^{y}}{2}>e^{\frac{x+y}{2}} \ (x \neq y);

(c)

xln⁡x+yln⁡y>(x+y)ln⁡x+y2x\ln x+y\ln y>(x+y)\ln \dfrac {x+y}{2}

(x>0,y>0). (x>0, y>0) .
Problem 1314.1

12(xn+yn)>(x+y2)n, (x>0,y>0,x≠y,n>1)\dfrac {1}{2}(x^{n}+y^{n})>\left(\dfrac {x+y}{2}\right)^{n}, \ (x>0, y>0, x \neq y, n>1);

?
Problem 1314.1

12(xn+yn)>(x+y2)n, (x>0,y>0,x≠y,n>1)\dfrac {1}{2}(x^{n}+y^{n})>\left(\dfrac {x+y}{2}\right)^{n}, \ (x>0, y>0, x \neq y, n>1);

?
Problem 1315

Prove that a bounded convex function is continuous everywhere, and has left-hand and right-hand derivatives.

?
Problem 1316

Let the function f(x)f(x) be twice differentiable on the interval (a,b)(a,b), with f′′(ξ)≠0f''(\xi ) \neq 0, where a<ξ<ba<\xi <b. Prove that two values x1x_{1} and x2x_{2} can be found in the interval (a,b)(a,b) satisfying

f(x2)−f(x1)x2−x1=f′(ξ). \frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}=f'(\xi ) .
?
Problem 1317

Prove that if the function f(x)f(x) is twice differentiable on the infinite interval (x0,+∞)(x_{0},+\infty ), and

lim⁡x→x0+0f(x)=0,lim⁡x→+∞f(x)=0, \lim _{x \to x_{0}+0}f(x)=0, \quad \lim _{x \to +\infty }f(x)=0 ,

then there is at least one point ξ\xi in the interval (x0,+∞)(x_{0},+\infty ) satisfying

f′′(ξ)=0. f''(\xi )=0 .
?
Section 2.9Methods for Evaluating Indeterminate Forms[65/89%]
Problem 1318

Find the value of the following:

lim⁡x→0sin⁡axsin⁡bx. \lim _{x \to 0}\frac{\sin ax}{\sin bx} .
?
Problem 1319

Find the value of the following:

lim⁡x→0ch⁡x−cos⁡xx2. \lim _{x \to 0}\frac{\ch x-\cos x}{x^{2}} .
?
Problem 1320

Find the value of the following:

lim⁡x→0tg⁡x−xx−sin⁡x. \lim _{x \to 0}\frac{\tg x-x}{x-\sin x} .
?
Problem 1321

Find the value of the following:

lim⁡x→03tg⁡4x−12tg⁡x3sin⁡4x−12sin⁡x. \lim _{x \to 0}\frac{3\tg 4x-12\tg x}{3\sin 4x-12\sin x} .
?
Problem 1322

Find the value of the following:

lim⁡x→π2tg⁡3xtg⁡x. \lim _{x \to \frac{\pi }{2}}\frac{\tg 3x}{\tg x} .
?
Problem 1323

Find the value of the following:

lim⁡x→0xctg⁡x−1x2. \lim _{x \to 0}\frac{x\ctg x-1}{x^{2}} .
?
Problem 1324

Find the value of the following:

lim⁡x→π4tg⁡x3−12sin⁡2x−1. \lim _{x \to \frac{\pi }{4}}\frac{\sqrt[3]{\tg x}-1}{2\sin ^{2}x-1} .
?
Problem 1325

Find the value of the following:

lim⁡x→0x(ex+1)−2(ex−1)x3. \lim _{x \to 0}\frac{x(e^{x}+1)-2(e^{x}-1)}{x^{3}} .
?
Problem 1326

Find the value of the following:

lim⁡x→01−cos⁡x2x2sin⁡x2. \lim _{x \to 0}\frac{1-\cos x^{2}}{x^{2}\sin x^{2}} .
?
Problem 1327

Find the value of the following:

lim⁡x→0arcsin⁡2x−2arcsin⁡xx3. \lim _{x \to 0}\frac{\arcsin 2x-2\arcsin x}{x^{3}} .
?
Problem 1328

Find the value of the following:

lim⁡x→01xx(aarctg⁡xa−barctg⁡xb). \lim _{x \to 0}\frac{1}{x\sqrt{x}}\left(\sqrt{a}\arctg \sqrt{\frac{x}{a}}-\sqrt{b}\arctg \sqrt{\frac{x}{b}}\right) .
?
Problem 1329

Find the value of the following:

lim⁡x→0ax−asin⁡xx3. \lim _{x \to 0}\frac{a^{x}-a^{\sin x}}{x^{3}} .
?
Problem 1330

Find the value of the following:

lim⁡x→1(xx−xln⁡x−x+1). \lim _{x \to 1}\left(\frac{x^{x}-x}{\ln x-x+1}\right) .
?
Problem 1331

Find the value of the following:

lim⁡x→0ln⁡(sin⁡ax)ln⁡(sin⁡bx). \lim _{x \to 0}\frac{\ln (\sin ax)}{\ln (\sin bx)} .
?
Problem 1332

Find the value of the following:

lim⁡x→0ln⁡(cos⁡ax)ln⁡(cos⁡bx). \lim _{x \to 0}\frac{\ln (\cos ax)}{\ln (\cos bx)} .
?
Problem 1333

Find the value of the following:

lim⁡x→0cos⁡(sin⁡x)−cos⁡xx4. \lim _{x \to 0}\frac{\cos (\sin x)-\cos x}{x^{4}} .
?
Problem 1334

Find the value of the following:

lim⁡x→01x(1th⁡x−1tg⁡x). \lim _{x \to 0}\frac{1}{x}\left(\frac{1}{\th x}-\frac{1}{\tg x}\right) .
?
Problem 1335

Find the value of each of the following expressions:

lim⁡x→0\arsh(sh⁡x)−\arsh(sin⁡x)sh⁡x−sin⁡x, \lim _{x \to 0}\frac{\arsh (\sh x)-\arsh (\sin x)}{\sh x-\sin x} ,

where \arshx=ln⁡(x+1+x2)\arsh x=\ln (x+\sqrt{1+x^{2}}).

?
Problem 1336

Find the value of each of the following expressions:

lim⁡x→+∞ln⁡xxε (ε<0). \lim _{x \to +\infty }\frac{\ln x}{x^{\varepsilon }} \ (\varepsilon <0) .
?
Problem 1337

Find the value of each of the following expressions:

lim⁡x→+∞xneax (a>0,n>0). \lim _{x \to +\infty }\frac{x^{n}}{e^{ax}} \ (a>0, n>0) .
?
Problem 1338

Find the value of each of the following expressions:

lim⁡x→0e−1x2x100. \lim _{x \to 0}\frac{e^{-\frac{1}{x^{2}}}}{x^{100}} .
?
Problem 1339

Find the value of each of the following expressions:

lim⁡x→+∞x2e−0.01x. \lim _{x \to +\infty }x^{2}e^{-0.01x} .
?
Problem 1340

Find the value of each of the following expressions:

lim⁡x→1−0ln⁡x⋅ln⁡(1−x). \lim _{x \to 1-0}\ln x \cdot \ln (1-x) .
?
Problem 1341

Find the value of each of the following expressions:

lim⁡x→+0xεln⁡x (ε>0). \lim _{x \to +0}x^{\varepsilon }\ln x \ (\varepsilon >0) .
?
Problem 1342

Find the value of each of the following expressions:

lim⁡x→+0xx. \lim _{x \to +0}x^{x} .
?
Problem 1343

Find the value of each of the following expressions:

lim⁡x→+0xxx−1. \lim _{x \to +0}x^{x^{x}-1} .
?
Problem 1344

Find the value of each of the following expressions:

lim⁡x→+0(xxx−1). \lim _{x \to +0}(x^{x^{x}}-1) .
?
Problem 1345

Find the value of each of the following expressions:

lim⁡x→+0xk1+ln⁡x. \lim _{x \to +0}x^{\frac{k}{1+\ln x}} .
?
Problem 1346

Find the value of each of the following expressions:

lim⁡x→1x11−x. \lim _{x \to 1}x^{\frac{1}{1-x}} .
?
Problem 1347

Find the value of each of the following expressions:

lim⁡x→1(2−x)tg⁡πx2. \lim _{x \to 1}(2-x)^{\tg \frac{\pi x}{2}} .
?
Problem 1348

Find the value of each of the following expressions:

lim⁡x→π4(tg⁡x)tg⁡2x. \lim _{x \to \frac{\pi }{4}}(\tg x)^{\tg 2x} .
?
Problem 1349

Find the value of each of the following expressions:

lim⁡x→0(ctg⁡x)sin⁡x. \lim _{x \to 0}(\ctg x)^{\sin x} .
?
Problem 1350

Find the value of each of the following expressions:

lim⁡x→+0(ln⁡1x)x. \lim _{x \to +0}\left(\ln \frac{1}{x}\right)^{x} .
?
Problem 1351

Find the value of each of the following expressions:

lim⁡x→∞(tg⁡πx2x+1)1x. \lim _{x \to \infty }\left(\tg \frac{\pi x}{2x+1}\right)^{\frac{1}{x}} .
?
Problem 1352

Find the value of each of the following expressions:

lim⁡x→a(tg⁡xtg⁡a)ctg⁡(x−a). \lim _{x \to a}\left(\frac{\tg x}{\tg a}\right)^{\ctg (x-a)} .
?
Problem 1353

Find the value of each of the following expressions:

lim⁡x→0(ax−xln⁡abx−xln⁡b)1x2. \lim _{x \to 0}\left(\frac{a^{x}-x\ln a}{b^{x}-x\ln b}\right)^{\frac{1}{x^{2}}} .
?
Problem 1354

Find the value of each of the following expressions:

lim⁡x→0(1x−1ex−1). \lim _{x \to 0}\left(\frac{1}{x}-\frac{1}{e^{x}-1}\right) .
?
Problem 1355

Find the value of each of the following expressions:

lim⁡x→1(1ln⁡x−1x−1). \lim _{x \to 1}\left(\frac{1}{\ln x}-\frac{1}{x-1}\right) .
?
Problem 1356

Find the value of each of the following expressions:

lim⁡x→0(ctg⁡x−1x). \lim _{x \to 0}\left(\ctg x-\frac{1}{x}\right) .
?
Problem 1357

Find the value of each of the following expressions:

lim⁡x→0(1ln⁡(x+1+x2)−1ln⁡(1+x)). \lim _{x \to 0}\left(\frac{1}{\ln (x+\sqrt{1+x^{2}})}-\frac{1}{\ln (1+x)}\right) .
?
Problem 1358

Find the value of each of the following expressions:

lim⁡x→aax−xax−a (a>0). \lim _{x \to a}\frac{a^{x}-x^{a}}{x-a} \ (a>0) .
?
Problem 1359

Find the value of each of the following expressions:

lim⁡x→0(1+x)1x−ex. \lim _{x \to 0}\frac{(1+x)^{\frac{1}{x}}-e}{x} .
?
Problem 1360

Find the value of each of the following expressions:

lim⁡x→0(a+x)x−axx2 (a>0). \lim _{x \to 0}\frac{(a+x)^{x}-a^{x}}{x^{2}} \ (a>0) .
?
Problem 1361

Find the value of each of the following expressions:

lim⁡x→+∞(2πarctg⁡x)x. \lim _{x \to +\infty }\left(\frac{2}{\pi }\arctg x\right)^{x} .
?
Problem 1362

Find the value of each of the following expressions:

lim⁡x→+∞(th⁡x)x. \lim _{x \to +\infty }(\th x)^{x} .
?
Problem 1363

Find the value of each of the following expressions:

lim⁡x→0(arcsin⁡xx)1x2. \lim _{x \to 0}\left(\frac{\arcsin x}{x}\right)^{\frac{1}{x^{2}}} .
?
Problem 1363.1
[NO TASK]
Problem 1363.1
[NO TASK]
Problem 1363.2
[NO TASK]
Problem 1363.2
[NO TASK]
Problem 1363.3
[NO TASK]
Problem 1363.3
[NO TASK]
Problem 1363.4
[NO TASK]
Problem 1363.4
[NO TASK]
Problem 1364

Find the value of each of the following expressions:

lim⁡x→0((1+x)1xe)1x. \lim _{x \to 0}\left(\frac{(1+x)^{\frac{1}{x}}}{e}\right)^{\frac{1}{x}} .
?
Problem 1365

Find the value of each of the following expressions:

lim⁡x→0(2πarccos⁡x)1x. \lim _{x \to 0}\left(\frac{2}{\pi }\arccos x\right)^{\frac{1}{x}} .
?
Problem 1366

Find the value of each of the following expressions:

lim⁡x→0(cos⁡xch⁡x)1x2. \lim _{x \to 0}\left(\frac{\cos x}{\ch x}\right)^{\frac{1}{x^{2}}} .
?
Problem 1367

Find the value of each of the following expressions:

lim⁡x→0ln⁡ch⁡xch⁡xm−ch⁡xn. \lim _{x \to 0}\frac{\ln \ch x}{\sqrt[m]{\ch x}-\sqrt[n]{\ch x}} .
?
Problem 1368

Find the value of each of the following expressions:

lim⁡x→0(1+ex2)ctg⁡x. \lim _{x \to 0}\left(\frac{1+e^{x}}{2}\right)^{\ctg x} .
?
Problem 1368.1
[NO TASK]
Problem 1368.1
[NO TASK]
Problem 1369

Find the value of each of the following expressions:

lim⁡x→+∞(x3+x2+x+13−x2+x+1⋅ln⁡(ex+x)x). \lim _{x \to +\infty }\left(\sqrt[3]{x^{3}+x^{2}+x+1}-\sqrt{x^{2}+x+1} \cdot \frac{\ln (e^{x}+x)}{x}\right) .
?
Problem 1370

Find the value of each of the following expressions:

lim⁡x→+∞[(x+a)1+1x−x1+1x+a]. \lim _{x \to +\infty }\left[(x+a)^{1+\frac{1}{x}}-x^{1+\frac{1}{x+a}}\right] .
?
Problem 1371

If, as x→0x \to 0, the curve y=f(x)y=f(x) passes through the origin (0,0) [lim⁡x→0f(x)=f(0)=0](0,0) \ \left[\lim_{x \to 0}f(x)=f(0)=0\right], and has inclination angle α\alpha there, find

lim⁡x→0yx. \lim _{x \to 0}\frac{y}{x} .
?
Problem 1372

If, as x→+0x \to +0, the curve y=f(x)y=f(x) passes through the origin (0,0) [lim⁡x→+0f(x)=0](0,0) \ \left[\lim_{x \to +0}f(x)=0\right], and when 0<x<ε0<x<\varepsilon, this curve lies entirely within the acute angle formed by the two lines y=−kxy=-kx and y=kx (k≠∞)y=kx \ (k \neq \infty ), prove that

lim⁡x→+0xf(x)=1. \lim _{x \to +0}x^{f(x)}=1 .
?
Problem 1373

Prove: if the second-order derivative f′′(x)f''(x) of the function f(x)f(x) exists, then

f′′(x)=lim⁡h→0f(x+h)+f(x−h)−2f(x)h2. f''(x)=\lim _{h \to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^{2}} .
?
Problem 1373.1
[NO TASK]
Problem 1373.1
[NO TASK]
Problem 1373.2
[NO TASK]
Problem 1373.2
[NO TASK]
Problem 1374

Investigate the possibility of applying L'Hospital's rule to the following examples:

?
(a)

lim⁡x→0x2sin⁡1xsin⁡x\displaystyle \lim_{x \to 0}\frac{x^{2}\sin \dfrac {1}{x}}{\sin x};

(b)

lim⁡x→∞x−sin⁡xx+sin⁡x\displaystyle \lim_{x \to \infty }\frac{x-\sin x}{x+\sin x};

(c)

lim⁡x→+∞e−2x(cos⁡x+2sin⁡x)+e−x2sin⁡2xe−x(cos⁡x+sin⁡x)\displaystyle \lim_{x \to +\infty }\frac{e^{-2x}(\cos x+2\sin x)+e^{-x^{2}}\sin^{2}x}{e^{-x}(\cos x+\sin x)};

(d)

lim⁡x→+∞1+x+sin⁡xcos⁡x(x+sin⁡xcos⁡x)esin⁡x\displaystyle \lim_{x \to +\infty }\frac{1+x+\sin x\cos x}{(x+\sin x\cos x)e^{\sin x}}.

Problem 1375

Given a circular segment with chord bb and height (sagitta) hh, and an isosceles triangle inscribed in this segment; if RR is held fixed while the arc of the segment tends to zero, find the ratio of the area of the segment to the area of the inscribed isosceles triangle. Use the result obtained to derive the approximate formula for the area of the segment:

S≈23bh. S \approx \frac{2}{3}bh .
?
Section 2.10Taylor's Formula[47/81%]
Problem 1376

Express the polynomial

P(x)=1+3x+5x2−2x3 P(x)=1+3x+5x^{2}-2x^{3}

as a polynomial in positive integer powers of the binomial x+1x+1.

?
Problem 1377

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

1+x+x21−x+x2 up to the term in x4. What is f(4)(0)? \frac{1+x+x^{2}}{1-x+x^{2}} \text{ up to the term in } x^{4} \text{. What is } f^{(4)}(0) \text{?}
?
Problem 1378

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

(1+x)100(1−2x)40(1+2x)60 up to the term in x2. \frac{(1+x)^{100}}{(1-2x)^{40}(1+2x)^{60}} \text{ up to the term in } x^{2} \text{.}
?
Problem 1379

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

am+xm (a>0) up to the term in x2. \sqrt[m]{a^{m}+x} \ (a>0) \text{ up to the term in } x^{2} \text{.}
?
Problem 1380

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

1−2x+x3−1−3x+x23 up to the term in x3. \sqrt{1-2x+x^{3}}-\sqrt[3]{1-3x+x^{2}} \text{ up to the term in } x^{3} \text{.}
?
Problem 1381

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

e2x−x2 up to the term in x5. e^{2x-x^{2}} \text{ up to the term in } x^{5} \text{.}
?
Problem 1382

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

xex−1 up to the term in x4. \frac{x}{e^{x}-1} \text{ up to the term in } x^{4} \text{.}
?
Problem 1383

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

sin⁡x33 up to the term in x13. \sqrt[3]{\sin x^{3}} \text{ up to the term in } x^{13} \text{.}
?
Problem 1384

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

ln⁡cos⁡x up to the term in x6. \ln \cos x \text{ up to the term in } x^{6} \text{.}
?
Problem 1385

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

sin⁡(sin⁡x) up to the term in x3. \sin (\sin x) \text{ up to the term in } x^{3} \text{.}
?
Problem 1386

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

tg⁡x up to the term in x5. \tg x \text{ up to the term in } x^{5} \text{.}
?
Problem 1387

Write out the expansion of the following function in positive integer powers of the variable xx, up to the term of the indicated order:

ln⁡sin⁡xx up to the term in x6. \ln \frac{\sin x}{x} \text{ up to the term in } x^{6} \text{.}
?
Problem 1388

Find the first three terms of the expansion of the function f(x)=xf(x)=\sqrt{x} in positive integer powers of the difference (x−1)(x-1).

?
Problem 1389

Expand the function f(x)=xx−1f(x)=x^{x}-1 in positive integer powers of (x−1)(x-1) up to the term containing (x−1)3(x-1)^{3}.

?
Problem 1390

In a neighborhood of the point x=0x=0, approximate the function

y=ach⁡xa (a>0) y=a\ch \frac{x}{a} \ (a>0)

by a second-order parabola.

?
Problem 1391

Expand the function f(x)=1+x2−x (x>0)f(x)=\sqrt{1+x^{2}}-x \ (x>0) in positive integer powers of 1x\dfrac {1}{x}, up to the term in 1x3\dfrac {1}{x^{3}}.

?
Problem 1392

Expand the function f(h)=ln⁡(x+h) (x>0)f(h)=\ln (x+h) \ (x>0) in positive integer powers of the increment hh, up to the term in hnh^{n} (nn a natural number).

?
Problem 1393

Let:

f(x+h)=f(x)+hf′(x)+⋯+hnn!f(n)(x+θh) (0<θ<1). \begin{align} f(x+h) & =f(x)+hf'(x)+\cdots +\frac{h^{n}}{n!}f^{(n)}(x+\theta h) \ (0<\theta <1) . \end{align}

and f(n+1)(x)≠0f^{(n+1)}(x) \neq 0. Prove that:

lim⁡h→0θ=1n+1. \lim _{h \to 0}\theta =\frac{1}{n+1} .
?
Problem 1393.1
[NO TASK]
Problem 1393.1
[NO TASK]
Problem 1393.2
[NO TASK]
Problem 1393.2
[NO TASK]
Problem 1393.3
[NO TASK]
Problem 1393.3
[NO TASK]
Problem 1394

Estimate the absolute error of the following approximate formulas:

?
(a)

ex≈1+x+x22!+⋯+xnn!e^{x} \approx 1+x+\dfrac {x^{2}}{2!}+\cdots +\dfrac {x^{n}}{n!} for 0⩽x⩽10 \leqslant x \leqslant 1;

(b)

sin⁡x≈x−x36\sin x \approx x-\dfrac {x^{3}}{6} for ∣x∣⩽12\left|x\right| \leqslant \dfrac {1}{2};

(c)

tg⁡x≈x+x33\tg x \approx x+\dfrac {x^{3}}{3} for ∣x∣⩽0.1\left|x\right| \leqslant 0.1;

(d)

1+x≈1+x2−x28\sqrt{1+x} \approx 1+\dfrac {x}{2}-\dfrac {x^{2}}{8} for 0⩽x⩽10 \leqslant x \leqslant 1.

Problem 1395

For what values of xx is the approximate formula

cos⁡x=1−x22 \cos x=1-\frac{x^{2}}{2}

accurate to 0.00010.0001.

?
Problem 1395.1
[NO TASK]
Problem 1395.1
[NO TASK]
Problem 1396

Using Taylor's formula, compute approximately, and estimate the error:

?
(a)

303\sqrt[3]{30};

(b)

2505\sqrt[5]{250};

(c)

400015\sqrt[15]{4000};

(d)

e\sqrt{e};

(e)

sin⁡18°\sin 18°;

(f)

ln⁡1.2\ln 1.2;

(g)

arctg⁡0.8\arctg 0.8;

(h)

arcsin⁡0.45\arcsin 0.45;

(i)

(1.1)1.2(1.1)^{1.2}.

Problem 1397

Compute:

?
(a)

ee to within 10−910^{-9};

(b)

sin⁡1°\sin 1° to within 10−810^{-8};

(c)

cos⁡9°\cos 9° to within 10−510^{-5};

(d)

5\sqrt{5} to within 10−410^{-4};

(e)

log⁡1011\log_{10}11 to within 10−510^{-5}.

Problem 1398

Using expansions I--V, find the following limits:

lim⁡x→0cos⁡x−e−x22x4. \lim _{x \to 0}\frac{\cos x-e^{-\frac{x^{2}}{2}}}{x^{4}} .
?
Problem 1399

Using expansions I--V, find the following limits:

lim⁡x→0exsin⁡x−x(1+x)x3. \lim _{x \to 0}\frac{e^{x}\sin x-x(1+x)}{x^{3}} .
?
Problem 1400

Using expansions I--V, find the following limits:

lim⁡x→+∞x32(x+1+x−1−2x). \lim _{x \to +\infty }x^{\frac{3}{2}}(\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}) .
?
Problem 1401

Using expansions I--V, find the following limits:

lim⁡x→+∞(x5+x56−x6−x56). \lim _{x \to +\infty }(\sqrt[6]{x^{5}+x^{5}}-\sqrt[6]{x^{6}-x^{5}}) .
?
Problem 1402

Using expansions I--V, find the following limits:

lim⁡x→+∞[(x3−x2+x2)e1x−x6+1]. \lim _{x \to +\infty }\left[\left(x^{3}-x^{2}+\frac{x}{2}\right)e^{\frac{1}{x}}-\sqrt{x^{6}+1}\right] .
?
Problem 1403

Using expansions I--V, find the following limits:

lim⁡x→+0ax+a−x−2x2 (a>0). \lim _{x \to +0}\frac{a^{x}+a^{-x}-2}{x^{2}} \ (a>0) .
?
Problem 1404

Using expansions I--V, find the following limits:

lim⁡x→∞[x−x2ln⁡(1+1x)]. \lim _{x \to \infty }\left[x-x^{2}\ln \left(1+\frac{1}{x}\right)\right] .
?
Problem 1405

Using expansions I--V, find the following limits:

lim⁡x→0(1x−1sin⁡x). \lim _{x \to 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right) .
?
Problem 1406

Using expansions I--V, find the following limits:

lim⁡x→01x(1x−ctg⁡x). \lim _{x \to 0}\frac{1}{x}\left(\frac{1}{x}-\ctg x\right) .
?
Problem 1406.1
[NO TASK]
Problem 1406.1
[NO TASK]
Problem 1406.2
[NO TASK]
Problem 1406.2
[NO TASK]
Problem 1406.3
[NO TASK]
Problem 1406.3
[NO TASK]
Problem 1407

As x→0x \to 0, find the leading term of the form CxnCx^{n} (CC a constant) of the infinitesimal yy, given:

y=tg⁡(sin⁡x)−sin⁡(tg⁡x). y=\tg (\sin x)-\sin (\tg x) .
?
Problem 1408

As x→0x \to 0, find the leading term of the form CxnCx^{n} (CC a constant) of the infinitesimal yy, given:

y=(1+x)x−1. y=(1+x)^{x}-1 .
?
Problem 1409

As x→0x \to 0, find the leading term of the form CxnCx^{n} (CC a constant) of the infinitesimal yy, given:

y=1−(1+x)1xe. y=1-\frac{(1+x)^{\frac{1}{x}}}{e} .
?
Problem 1410

For what choice of coefficients aa and bb is the quantity

x−(a+bcos⁡x)sin⁡x x-(a+b\cos x)\sin x

an infinitesimal of order 55 with respect to xx?

?
Problem 1410.1
[NO TASK]
Problem 1410.1
[NO TASK]
Problem 1410.2
[NO TASK]
Problem 1410.2
[NO TASK]
Problem 1411

For small ∣x∣\left|x\right|, derive simple approximate formulas for the following expressions:

?
(a)

1R2−1(R+x)2 (R>0)\dfrac {1}{R^{2}}-\dfrac {1}{(R+x)^{2}} \ (R>0);

(b)

1+x1−x3−1−x1+x3\sqrt[3]{\dfrac {1+x}{1-x}}-\sqrt[3]{\dfrac {1-x}{1+x}};

(c)

Ax[1−(1+x100)−n]\dfrac {A}{x}\left[1-\left(1+\dfrac {x}{100}\right)^{-n}\right];

(d)

ln⁡2ln⁡(1+x100)\dfrac {\ln 2}{\ln \left(1+\dfrac {x}{100}\right)}.

Problem 1412

For small absolute value of xx, derive an approximate formula of the form

x=αsin⁡x+βtg⁡x x=\alpha \sin x+\beta \tg x

accurate to the x5x^{5} term.

Apply this formula to the approximate computation of the arc length for a small angle.

?
Problem 1413

Estimate the relative error of the following Chebyshev rule: a circular arc is approximately equal to the sum of the two equal legs of an isosceles triangle standing on the chord subtended by the arc, whose height is 43\sqrt{\dfrac {4}{3}} times the sagitta of the circular segment.

?
Section 2.11Extrema of a Function. Maximum and Minimum Values of a Function[59/97%]
Problem 1414

Investigate the extrema of the following functions:

y=2−x−x2. y=2-x-x^{2} .
?
Problem 1415

Investigate the extrema of the following functions:

y=(x−1)3. y=(x-1)^{3} .
?
Problem 1416

Investigate the extrema of the following functions:

y=(x−1)4. y=(x-1)^{4} .
?
Problem 1417

Investigate the extrema of the following functions:

y=xm(1−x)n (m and n are positive integers). y=x^{m}(1-x)^{n} \ (m \text{ and } n \text{ are positive integers}) .
?
Problem 1418

Investigate the extrema of the following functions:

y=cos⁡x+ch⁡x. y=\cos x+\ch x .
?
Problem 1419

Investigate the extrema of the following functions:

y=(x+1)10e−x. y=(x+1)^{10}e^{-x} .
?
Problem 1420

Investigate the extrema of the following functions:

y=(1+x+x22!+⋯+xnn!)e−x (n a natural number). y=\left(1+x+\frac{x^{2}}{2!}+\cdots +\frac{x^{n}}{n!}\right)e^{-x} \ (n \text{ a natural number}) .
?
Problem 1421

Investigate the extrema of the following functions:

y=∣x∣. y=\left|x\right| .
?
Problem 1422

Investigate the extrema of the following functions:

y=x13(1−x)23. y=x^{\frac{1}{3}}(1-x)^{\frac{2}{3}} .
?
Problem 1423

The function

f(x)=(x−x0)nφ(x) f(x)=(x-x_{0})^{n}\varphi (x)

(nn a natural number), where the function φ(x)\varphi (x) is continuous at x=x0x=x_{0} and φ(x0)≠0\varphi (x_{0}) \neq 0. Investigate the extremum of this function at the point x=x0x=x_{0}.

?
Problem 1424

Let:

f(x)=P(x)Q(x),f′(x)=P1(x)Q2(x), f(x)=\frac{P(x)}{Q(x)}, f'(x)=\frac{P_{1}(x)}{Q^{2}(x)} ,

and let x0x_{0} be a stationary point of the function f(x)f(x), that is,

P1(x0)=0,Q(x0)≠0. P_{1}(x_{0})=0, Q(x_{0}) \neq 0 .

Prove that:

\sgnf′′(x0)=\sgnP1′(x0). \sgn f''(x_{0})=\sgn P_{1}'(x_{0}) .
?
Problem 1425

Can one assert the following: if the function f(x)f(x) has a maximum at the point x0x_{0}, then in some sufficiently small neighborhood of this point, f(x)f(x) is increasing to the left of x0x_{0} and decreasing to the right?

?
Problem 1426

Given the function

f(x)=e−1x2 when x≠0;f(0)=0. f(x)=e^{-\frac{1}{x^{2}}} \text{ when } x \neq 0 ; f(0)=0 .

Prove that although

f(n)(0)=0 (n=1,2,⋯ ), f^{(n)}(0)=0 \ (n=1,2,\cdots ) ,

the function f(x)f(x) has a minimum at the point x=0x=0.

Sketch the graph of this function.

?
Problem 1427

Investigate the extrema of the following functions:

?
(a)

when x≠0x \neq 0, f(x)=e−1∣x∣(2+sin⁡1x)f(x)=e^{-\frac{1}{\left|x\right|}}\left(\sqrt{2}+\sin \dfrac {1}{x}\right) and f(0)=0f(0)=0;

(b)

when x≠0x \neq 0, f(x)=e−1∣x∣(2+cos⁡1x)f(x)=e^{-\frac{1}{\left|x\right|}}\left(\sqrt{2}+\cos \dfrac {1}{x}\right) and f(0)=0f(0)=0.

Sketch the graphs of these functions.

Problem 1428

Given the function

f(x)=∣x∣(2+cos⁡1x) when x≠0;f(0)=0. f(x)=\left|x\right|\left(2+\cos \frac{1}{x}\right) \text{ when } x \neq 0 ; f(0)=0 .

Investigate the extremum of this function at the point x=0x=0, and sketch the graph of this function.

?
Problem 1429

Find the extrema of the following function:

y=x3−6x2+9x−4. y=x^{3}-6x^{2}+9x-4 .
?
Problem 1430

Find the extrema of the following function:

y=2x2−x4. y=2x^{2}-x^{4} .
?
Problem 1431

Find the extrema of the following function:

y=x(x−1)2(x−2)3. y=x(x-1)^{2}(x-2)^{3} .
?
Problem 1432

Find the extrema of the following function:

y=x+1x. y=x+\dfrac {1}{x} .
?
Problem 1433

Find the extrema of the following function:

y=2x1+x2. y=\dfrac {2x}{1+x^{2}} .
?
Problem 1434

Find the extrema of the following function:

y=x2−3x+2x2+2x+1. y=\dfrac {x^{2}-3x+2}{x^{2}+2x+1} .
?
Problem 1435

Find the extrema of the following function:

y=2x−x2. y=\sqrt{2x-x^{2}} .
?
Problem 1436

Find the extrema of the following function:

y=xx−13. y=x\sqrt[3]{x-1} .
?
Problem 1437

Find the extrema of the following function:

y=xe−x. y=xe^{-x} .
?
Problem 1438

Find the extrema of the following function:

y=xln⁡x. y=\sqrt{x}\ln x .
?
Problem 1439

Find the extrema of the following function:

y=ln⁡2xx. y=\dfrac {\ln ^{2}x}{x} .
?
Problem 1440

Find the extrema of the following function:

y=cos⁡x+12cos⁡2x. y=\cos x+\dfrac {1}{2}\cos 2x .
?
Problem 1441

Find the extrema of the following function:

y=101+sin⁡2x. y=\dfrac {10}{1+\sin ^{2}x} .
?
Problem 1442

Find the extrema of the following function:

y=arctg⁡x−12ln⁡(1+x2). y=\operatorname {arctg}x-\dfrac {1}{2}\ln (1+x^{2}) .
?
Problem 1443

Find the extrema of the following function:

y=exsin⁡x. y=e^{x}\sin x .
?
Problem 1444

Find the extrema of the following function:

y=∣x∣e−∣x−1∣. y=\left|x\right|e^{-\left|x-1\right|} .
?
Problem 1445

Find the maximum and minimum of the following function on the closed interval [−1,5]\left[-1,5\right]:

f(x)=2x. f(x)=2^{x} .
?
Problem 1446

Find the maximum and minimum of the following function on the closed interval [−3,10]\left[-3,10\right]:

f(x)=x2−4x+6. f(x)=x^{2}-4x+6 .
?
Problem 1447

Find the maximum and minimum of the following function on the closed interval [−10,10]\left[-10,10\right]:

f(x)=∣x2−3x+2∣. f(x)=\left|x^{2}-3x+2\right| .
?
Problem 1448

Find the maximum and minimum of the following function on the closed interval [0.01,100]\left[0.01,100\right]:

f(x)=x+1x. f(x)=x+\dfrac {1}{x} .
?
Problem 1449

Find the maximum and minimum of the following function on the closed interval [−1,1]\left[-1,1\right]:

f(x)=5−4x. f(x)=\sqrt{5-4x} .
?
Problem 1450

Find the infimum (inf⁡)(\inf ) and supremum (sup⁡)(\sup ) of the following function on the interval (0,+∞)(0,+\infty ):

f(x)=xe−0.01x. f(x)=xe^{-0.01x} .
?
Problem 1451

Find the infimum (inf⁡)(\inf ) and supremum (sup⁡)(\sup ) of the following function on the interval (0,+∞)(0,+\infty ):

f(x)=(1+x+x22!+⋯+xnn!)e−x. f(x)=\left(1+x+\dfrac {x^{2}}{2!}+\cdots +\dfrac {x^{n}}{n!}\right)e^{-x} .
?
Problem 1452

Find the infimum (inf⁡)(\inf ) and supremum (sup⁡)(\sup ) of the following function on the interval (0,+∞)(0,+\infty ):

f(x)=1+x21+x3. f(x)=\dfrac {1+x^{2}}{1+x^{3}} .
?
Problem 1453

Find the infimum (inf⁡)(\inf ) and supremum (sup⁡)(\sup ) of the following function on the interval (−∞,+∞)(-\infty ,+\infty ):

f(x)=e−x2cos⁡x2. f(x)=e^{-x^{2}}\cos x^{2} .
?
Problem 1454

Find the infimum and supremum of the function

f(ξ)=1+ξ3+ξ2 f(\xi )=\dfrac {1+\xi }{3+\xi ^{2}}

on the interval x<ξ<+∞x<\xi <+\infty, and sketch the graphs of the following functions:

m(x)=inf⁡x<ξ<+∞f(ξ) and M(x)=sup⁡x<ξ<+∞f(ξ). m(x)=\inf _{x<\xi <+\infty }f(\xi ) \text{ and } M(x)=\sup _{x<\xi <+\infty }f(\xi ) .
?
Problem 1454.1
[NO TASK]
Problem 1454.1
[NO TASK]
Problem 1455

Find the largest term of each of the following sequences:

?
(a)

n102n (n=1,2,⋯ )\dfrac {n^{10}}{2^{n}} \ (n=1,2,\cdots );

(b)

nn+10000 (n=1,2,⋯ )\dfrac {\sqrt{n}}{n+10000} \ (n=1,2,\cdots );

(c)

nn (n=1,2,⋯ )\sqrt[n]{n} \ (n=1,2,\cdots ).

Problem 1456

Prove the following inequalities:

?
(a)
(b)

if 0≤x≤10 \leq x \leq 1 and p>1p>1, then 12p−1≤xp+(1−x)p≤1\dfrac {1}{2^{p-1}} \leq x^{p}+(1-x)^{p} \leq 1;

(c)

when m>0,n>0m>0, n>0 and 0≤x≤a0 \leq x \leq a,

xm(a−x)n≤mmnn(m+n)m+nam+n; x^{m}(a-x)^{n} \leq \dfrac {m^{m}n^{n}}{(m+n)^{m+n}}a^{m+n} ;
(d)

x+a2n−1n≤xn+ann≤x+a (x>0,a>0,n>1)\dfrac {x+a}{2^{\frac{n-1}{n}}} \leq \sqrt[n]{x^{n}+a^{n}} \leq x+a \ (x>0,a>0,n>1);

(e)

∣asin⁡x+bcos⁡x∣≤a2+b2\left|a\sin x+b\cos x\right| \leq \sqrt{a^{2}+b^{2}}.

Problem 1456.1

when ∣x∣≤2\left|x\right| \leq 2, ∣3x−x3∣≤2\left|3x-x^{3}\right| \leq 2;

?
Problem 1456.1

when ∣x∣≤2\left|x\right| \leq 2, ∣3x−x3∣≤2\left|3x-x^{3}\right| \leq 2;

?
Problem 1457

Find the "deviation from zero" of the polynomial

P(x)=x(x−1)2(x+2) P(x)=x(x-1)^{2}(x+2)

on the closed interval [−2,1]\left[-2,1\right], that is, find

EP=sup⁡−2≤x≤1∣P(x)∣. E_{P}=\sup _{-2 \leq x \leq 1}\left|P(x)\right| .
?
Problem 1458

What coefficient qq should be chosen so that the polynomial

P(x)=x2+q P(x)=x^{2}+q

has the smallest deviation from zero on the closed interval [−1,1]\left[-1,1\right], that is,

EP=sup⁡−1≤x≤1∣P(x)∣=min⁡. E_{P}=\sup _{-1 \leq x \leq 1}\left|P(x)\right|=\min .
?
Problem 1459

The number

Δ=sup⁡a≤x≤b∣f(x)−g(x)∣ \Delta =\sup _{a \leq x \leq b}\left|f(x)-g(x)\right|

is called the absolute difference of the functions f(x)f(x) and g(x)g(x) on the closed interval [a,b]\left[a,b\right].

Find the absolute difference of the functions f(x)=x2f(x)=x^{2} and g(x)=x3g(x)=x^{3} on the closed interval [0,1]\left[0,1\right].

?
Problem 1460

On the closed interval [x1,x2]\left[x_{1},x_{2}\right], approximate the function

f(x)=x2, f(x)=x^{2} ,

by the linear function

g(x)=(x1+x2)x+b g(x)=(x_{1}+x_{2})x+b

so that the absolute difference of f(x)f(x) and g(x)g(x) (see the problem above) is smallest, and find this smallest absolute difference.

?
Problem 1461

Find the minimum of the function

f(x)=max⁡{2∣x∣,∣1+x∣} f(x)=\max \left\{ 2\left|x\right|,\left|1+x\right|\right\}
?
Problem 1462

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

x3−6x2+9x−10=0. x^{3}-6x^{2}+9x-10=0 .
?
Problem 1463

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

x3−3x2−9x+h=0. x^{3}-3x^{2}-9x+h=0 .
?
Problem 1464

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

3x4−4x3−6x2+12x−20=0. 3x^{4}-4x^{3}-6x^{2}+12x-20=0 .
?
Problem 1465

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

x5−5x=a. x^{5}-5x=a .
?
Problem 1466

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

ln⁡x=kx. \ln x=kx .
?
Problem 1467

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

ex=ax2 (a>0). e^{x}=ax^{2} \ (a>0) .
?
Problem 1468

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

when 0≤x≤π,sin⁡3x⋅cos⁡x=a. \text{when } 0 \leq x \leq \pi , \sin ^{3}x \cdot \cos x=a .
?
Problem 1469

Determine the number of real roots of the following equation, and locate the intervals containing these roots:

ch⁡x=kx. \ch x=kx .
?
Problem 1470

Under what conditions does the equation

x3+px+q=0 x^{3}+px+q=0

have: (a) one real root; (б) three real roots. Sketch the corresponding regions in the (p,q)(p,q) plane.

?
(a)
(b)
Section 2.12Sketching the Graph of a Function from Its Characteristic Points[88/97%]
Problem 1471

Sketch the graph of the following function:

y=3x−x3. y=3x-x^{3} .
?
Problem 1472

Sketch the graph of the following function:

y=1+x2−x42. y=1+x^{2}-\dfrac {x^{4}}{2} .
?
Problem 1473

Sketch the graph of the following function:

y=(x+1)(x−2)2. y=(x+1)(x-2)^{2} .
?
Problem 1474

Sketch the graph of the following function:

y=2−x21+x4. y=\dfrac {2-x^{2}}{1+x^{4}} .
?
Problem 1475

Sketch the graph of the following function:

y=x2−1x2−5x+6. y=\dfrac {x^{2}-1}{x^{2}-5x+6} .
?
Problem 1476

Sketch the graph of the following function:

y=x(1+x)(1−x)2. y=\dfrac {x}{(1+x)(1-x)^{2}} .
?
Problem 1477

Sketch the graph of the following function:

y=x4(1+x)3. y=\dfrac {x^{4}}{(1+x)^{3}} .
?
Problem 1478

Sketch the graph of the following function:

y=(1+x1−x)4. y=\left(\dfrac {1+x}{1-x}\right)^{4} .
?
Problem 1479

Sketch the graph of the following function:

y=x2(x−1)(x+1)2. y=\dfrac {x^{2}(x-1)}{(x+1)^{2}} .
?
Problem 1480

Sketch the graph of the following function:

y=x(1−x2)2. y=\dfrac {x}{(1-x^{2})^{2}} .
?
Problem 1481

Sketch the graph of the following function:

y=(x+1)3(x−1)3. y=\dfrac {(x+1)^{3}}{(x-1)^{3}} .
?
Problem 1482

Sketch the graph of the following function:

y=x4+8x3+1. y=\dfrac {x^{4}+8}{x^{3}+1} .
?
Problem 1483

Sketch the graph of the following function:

y=11+x−103x2+11−x. y=\dfrac {1}{1+x}-\dfrac {10}{3x^{2}}+\dfrac {1}{1-x} .
?
Problem 1484

Sketch the graph of the following function:

y=(x−3)x. y=(x-3)\sqrt{x} .
?
Problem 1485

Sketch the graph of the following function:

y=±8x2−x4. y=\pm \sqrt{8x^{2}-x^{4}} .
?
Problem 1485.1
[NO TASK]
Problem 1485.1
[NO TASK]
Problem 1486

Sketch the graph of the following function:

y=±(x−1)(x−2)(x−3). y=\pm \sqrt{(x-1)(x-2)(x-3)} .
?
Problem 1487

Sketch the graph of the following function:

y=x3−x2−x+13. y=\sqrt[3]{x^{3}-x^{2}-x+1} .
?
Problem 1488

Sketch the graph of the following function:

y=x23−x2+13. y=\sqrt[3]{x^{2}}-\sqrt[3]{x^{2}+1} .
?
Problem 1489

Sketch the graph of the following function:

y=(x+2)2/3−(x−2)2/3. y=(x+2)^{2/3}-(x-2)^{2/3} .
?
Problem 1490

Sketch the graph of the following function:

y=(x+1)2/3+(x−1)2/3. y=(x+1)^{2/3}+(x-1)^{2/3} .
?
Problem 1491

Sketch the graph of the following function:

y=xx2−13. y=\dfrac {x}{\sqrt[3]{x^{2}-1}} .
?
Problem 1492

Sketch the graph of the following function:

y=x2x2−12x2−1. y=\dfrac {x^{2}\sqrt{x^{2}-1}}{2x^{2}-1} .
?
Problem 1493

Sketch the graph of the following function:

y=(1+x)3/2x. y=\dfrac {(1+x)^{3/2}}{\sqrt{x}} .
?
Problem 1494

Sketch the graph of the following function:

y=1−x+x33+x. y=1-x+\sqrt{\dfrac {x^{3}}{3+x}} .
?
Problem 1495

Sketch the graph of the following function:

y=x2x+13. y=\sqrt[3]{\dfrac {x^{2}}{x+1}} .
?
Problem 1496

Sketch the graph of the following function:

y=x4+3x2+1. y=\sqrt{\dfrac {x^{4}+3}{x^{2}+1}} .
?
Problem 1497

Sketch the graph of the following function:

y=sin⁡x+cos⁡2x. y=\sin x+\cos ^{2}x .
?
Problem 1498

Sketch the graph of the following function:

y=(7+2cos⁡x)sin⁡x. y=(7+2\cos x)\sin x .
?
Problem 1499

Sketch the graph of the following function:

y=sin⁡x+13sin⁡3x. y=\sin x+\dfrac {1}{3}\sin 3x .
?
Problem 1500

Sketch the graph of the following function:

y=cos⁡x−12cos⁡2x. y=\cos x-\dfrac {1}{2}\cos 2x .
?
Problem 1501

Sketch the graph of the following function:

y=sin⁡4x+cos⁡4x. y=\sin ^{4}x+\cos ^{4}x .
?
Problem 1502

Sketch the graph of the following function:

y=sin⁡x⋅sin⁡3x. y=\sin x \cdot \sin 3x .
?
Problem 1503

Sketch the graph of the following function:

y=sin⁡xsin⁡(x+π4). y=\dfrac {\sin x}{\sin \left(x+\dfrac {\pi }{4}\right)} .
?
Problem 1504

Sketch the graph of the following function:

y=cos⁡xcos⁡2x. y=\dfrac {\cos x}{\cos 2x} .
?
Problem 1504.1
[NO TASK]
Problem 1504.1
[NO TASK]
Problem 1505

Sketch the graph of the following function:

y=2x−tg⁡x. y=2x-\tg x .
?
Problem 1506

Sketch the graph of the following function:

y=e2x−x2. y=e^{2x-x^{2}} .
?
Problem 1507

Sketch the graph of the following function:

y=(1+x2)e−x2. y=(1+x^{2})e^{-x^{2}} .
?
Problem 1508

Sketch the graph of the following function:

y=x+e−x. y=x+e^{-x} .
?
Problem 1509

Sketch the graph of the following function:

y=x2/3e−x. y=x^{2/3}e^{-x} .
?
Problem 1509.1
[NO TASK]
Problem 1509.1
[NO TASK]
Problem 1510

Sketch the graph of the following function:

y=ex1+x. y=\dfrac {e^{x}}{1+x} .
?
Problem 1511

Sketch the graph of the following function:

y=1−e−x2. y=\sqrt{1-e^{-x^{2}}} .
?
Problem 1512

Sketch the graph of the following function:

y=ln⁡xx. y=\dfrac {\ln x}{\sqrt{x}} .
?
Problem 1513

Sketch the graph of the following function:

y=ln⁡(x+x2+1). y=\ln (x+\sqrt{x^{2}+1}) .
?
Problem 1514

Sketch the graph of the following function:

y=x2+1ln⁡(x+x2+1). y=\sqrt{x^{2}+1}\ln (x+\sqrt{x^{2}+1}) .
?
Problem 1515

Sketch the graph of the following function:

y=arcsin⁡x1−x2. y=\dfrac {\arcsin x}{\sqrt{1-x^{2}}} .
?
Problem 1516

Sketch the graph of the following function:

y=x+arctg⁡x. y=x+\arctg x .
?
Problem 1517

Sketch the graph of the following function:

y=x2+arcctg⁡x. y=\dfrac {x}{2}+\arcctg x .
?
Problem 1518

Sketch the graph of the following function:

y=xarctg⁡x. y=x \arctg x .
?
Problem 1519

Sketch the graph of the following function:

y=arcsin⁡2x1+x2. y=\arcsin \dfrac {2x}{1+x^{2}} .
?
Problem 1520

Sketch the graph of the following function:

y=arccos⁡1−x21+x2. y=\arccos \dfrac {1-x^{2}}{1+x^{2}} .
?
Problem 1521

Sketch the graph of the following function:

y=(x+2)e1/x. y=(x+2)e^{1/x} .
?
Problem 1522

Sketch the graph of the following function:

y=2x2+1−x2−1. y=2^{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}} .
?
Problem 1523

Sketch the graph of the following function:

y=ln⁡x2−3x+2x2+1. y=\ln \dfrac {x^{2}-3x+2}{x^{2}+1} .
?
Problem 1524

Sketch the graph of the following function:

y=aarcsin⁡xa−a2−x2(a>0). y=a\arcsin \dfrac {x}{a}-\sqrt{a^{2}-x^{2}} \quad (a>0) .
?
Problem 1525

Sketch the graph of the following function:

y=arccos⁡1−x1−2x. y=\arccos \dfrac {1-x}{1-2x} .
?
Problem 1526

Sketch the graph of the following function:

y=xx. y=x^{x} .
?
Problem 1527

Sketch the graph of the following function:

y=x1/x. y=x^{1/x} .
?
Problem 1528

Sketch the graph of the following function:

y=(1+x)1/x. y=(1+x)^{1/x} .
?
Problem 1529

Sketch the graph of the following function:

y=x(1+1x)x(x>0). y=x\left(1+\dfrac {1}{x}\right)^{x} \quad (x>0) .
?
Problem 1530

Sketch the graph of the following function (do not investigate concavity/convexity):

y=1e1−x21+x2. y=\dfrac {\dfrac {1}{e^{1-x^{2}}}}{1+x^{2}} .
?
Problem 1531

Sketch the curve represented by the following parametric equations:

x=(t+1)24,y=(t−1)24. x=\dfrac {(t+1)^{2}}{4}, \quad y=\dfrac {(t-1)^{2}}{4} .
?
Problem 1532

Sketch the curve represented by the following parametric equations:

x=2t−t2,y=3t−t3. x=2t-t^{2}, \quad y=3t-t^{3} .
?
Problem 1533

Sketch the curve represented by the following parametric equations:

x=t2t−1,y=tt2−1. x=\dfrac {t^{2}}{t-1}, \quad y=\dfrac {t}{t^{2}-1} .
?
Problem 1534

Sketch the curve given by the following parametric equations:

x=t21−t2,y=11+t2. x=\dfrac {t^{2}}{1-t^{2}}, \quad y=\dfrac {1}{1+t^{2}} .
?
Problem 1535

Sketch the curve given by the following parametric equations:

x=t+e−t,y=2t+e−2t. x=t+e^{-t}, \quad y=2t+e^{-2t} .
?
Problem 1536

Sketch the curve given by the following parametric equations:

x=acos⁡2t,y=acos⁡3t(a>0). x=a\cos 2t, \quad y=a\cos 3t \quad (a>0) .
?
Problem 1537

Sketch the curve given by the following parametric equations:

x=cos⁡4t,y=sin⁡4t. x=\cos ^{4}t, \quad y=\sin ^{4}t .
?
Problem 1538

Sketch the curve given by the following parametric equations:

x=tln⁡t,y=ln⁡tt. x=t\ln t, \quad y=\dfrac {\ln t}{t} .
?
Problem 1539

Sketch the curve given by the following parametric equations:

x=acos⁡3t,y=atan⁡3t(a>0). x=\dfrac {a}{\cos ^{3}t}, \quad y=a\tan ^{3}t \quad (a>0) .
?
Problem 1540

Sketch the curve given by the following parametric equations:

x=a(sh t−t),y=a(ch t−1)(a>0). x=a(\text{sh}\, t-t), \quad y=a(\text{ch}\, t-1) \quad (a>0) .
?
Problem 1541

Convert the following curve equation into parametric form, and then sketch this curve:

x3+y3−3axy=0(a>0). x^{3}+y^{3}-3axy=0 \quad (a>0) .
?
Problem 1542

Convert the following curve equation to parametric form, and then sketch this curve:

x2+y2=x4+y4. x^{2}+y^{2}=x^{4}+y^{4} .
?
Problem 1543

Convert the following curve equation to parametric form, and then sketch this curve:

x2y2=x3−y3. x^{2}y^{2}=x^{3}-y^{3} .
?
Problem 1544

Sketch the graph of the following curve:

xy=yx(x>0,y>0). x^{y}=y^{x} \quad (x>0, y>0) .
?
Problem 1545

Sketch the curve ch2x−ch2y=1\text{ch}^{2}x-\text{ch}^{2}y=1.

?
Problem 1546

Sketch the graphs of the following functions given in polar coordinates (φ,r)(\varphi ,r) (r≥0r \geq 0):

r=a+bcos⁡φ(0<a≤b). r=a+b\cos \varphi \quad (0<a \leq b) .
?
Problem 1547

Sketch the graphs of the following functions given in polar coordinates (φ,r)(\varphi ,r) (r≥0r \geq 0):

r=asin⁡3φ(a>0). r=a\sin 3\varphi \quad (a>0) .
?
Problem 1548

Sketch the graphs of the following functions given in polar coordinates (φ,r)(\varphi ,r) (r≥0r \geq 0):

r=acos⁡3φ(a>0). r=\dfrac {a}{\sqrt{\cos 3\varphi }} \quad (a>0) .
?
Problem 1549

Sketch the graph of the following function given in polar coordinates (φ,r)(\varphi ,r) (with r≥0r \geq 0):

r=ath φφ−1, where φ>1(a>0). r=a\dfrac {\text{th}\, \varphi }{\varphi -1} , \text{ where } \varphi >1 \quad (a>0) .
?
Problem 1550

Sketch the graph of the following function given in polar coordinates (φ,r)(\varphi ,r) (with r≥0r \geq 0):

cos⁡φ=r−1r2. \cos \varphi =\dfrac {r-1}{r^{2}} .
?
Problem 1551

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x2−2x+a. y=x^{2}-2x+a .
?
Problem 1552

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x+a2x. y=x+\dfrac {a^{2}}{x} .
?
Problem 1553

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x±a(1−x2). y=x \pm \sqrt{a(1-x^{2})} .
?
Problem 1554

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=x2+e−ax. y=\dfrac {x}{2}+e^{-ax} .
?
Problem 1555

Sketch the family of graphs of the following curves (aa denotes the parameter):

y=xe−xa. y=xe^{-\frac{x}{a}} .
?
Section 2.13Problems on Maxima and Minima of Functions[35/100%]
Problem 1556

Prove: if the function f(x)f(x) is nonnegative, then the function

F(x)=Cf2(x)(C>0) F(x)=Cf^{2}(x) \quad (C>0)

has the same extremum points as the function f(x)f(x).

?
Problem 1557

Prove: if, for −∞<x<+∞-\infty <x<+\infty, the function φ(x)\varphi (x) is monotonically increasing, then the functions

f(x) and φ(f(x)) f(x) \text{ and } \varphi (f(x))

have the same extremum points.

?
Problem 1558

The sum of two positive numbers equals the constant aa. Find the maximum value of the product of the mm-th power and nn-th power (m>0,n>0m>0, n>0) of these two positive numbers.

?
Problem 1559

The product of two positive numbers equals the constant aa. Find the minimum value of the sum of the mm-th power and nn-th power (m>0,n>0m>0, n>0) of these two numbers.

?
Problem 1560

For what base should logarithms be taken so that there is a number equal to its own logarithm?

?
Problem 1561

Among all rectangles of area SS, find the one with the smallest perimeter.

?
Problem 1562

If the sum of one leg of a right triangle and the hypotenuse is a constant, find the right triangle with the maximum area.

?
Problem 1563

What dimensions give a closed cylindrical can of volume VV the minimum surface area?

?
Problem 1564

Inscribe a rectangle of maximum area in a given circular segment not exceeding a semicircle.

?
Problem 1565

Inscribe, in the ellipse

x2a2+y2b2=1 \dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1

a rectangle of maximum area with sides parallel to the axes of the ellipse.

?
Problem 1566

Inscribe, in a triangle with base bb and height hh, a rectangle of maximum perimeter, and investigate the possibility of a solution to this problem.

?
Problem 1567

A beam with a rectangular cross-section is cut from a circular log of diameter dd; the base of this rectangle equals bb, and the height equals hh. If the strength of the beam is proportional to bh2bh^{2}, what dimensions of the beam give the maximum strength?

?
Problem 1568

Inscribe, in a hemisphere of radius RR, a right rectangular parallelepiped with a square base and of maximum volume.

?
Problem 1569

Inscribe, in a sphere of radius RR, a cylinder of maximum volume.

?
Problem 1570

Inscribe, in a sphere of radius RR, a cylinder of maximum surface area.

?
Problem 1571

Construct, about a given sphere, a circumscribed cone of minimum volume.

?
Problem 1572

Find the maximum volume of a cone with slant height ll.

?
Problem 1573

Inscribe, in a right circular cone with vertex angle 2α2\alpha and base radius RR, a cylinder of maximum surface area.

?
Problem 1574

Find the shortest distance from the point M(p,p)M(p,p) to the parabola y2=2pxy^{2}=2px.

?
Problem 1575

Find the shortest and longest distances from the point A(2,0)A(2,0) to the circle x2+y2=1x^{2}+y^{2}=1.

?
Problem 1576

Find the longest chord of the ellipse x2a2+y2b2=1\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1 (0<b<a0<b<a) passing through the vertex (0,−b)(0,-b).

?
Problem 1577

Draw a tangent line at the point M(x,y)M(x,y) on the ellipse x2a2+y2b2=1\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1; this tangent line, together with the coordinate axes, forms a triangle. Find the point MM for which the area of this triangle is minimum.

?
Problem 1578

A body has the shape of a right circular cylinder with a hemispherical top. If the volume of this body equals VV, what dimensions of the body give the minimum surface area?

?
Problem 1579

The cross-section of an open water channel is an isosceles trapezoid. If the cross-sectional area of the flowing water equals SS, and the height of the water surface equals hh, what should the inclination angle φ\varphi of the channel's sides be so that the wetted perimeter of the cross-section is minimum?

?
Problem 1580

Let the area enclosed by a closed curve be SS, and let a circle also enclose the same area SS; then the ratio of the length of the closed curve to the circumference of the circle is called the "curvature" of the curve.

Let the isosceles trapezoid ABCDABCD (AD∥BCAD \parallel BC) have base AD=2aAD=2a and acute angle BAD=αBAD=\alpha. What shape should the isosceles trapezoid have for the minimum curvature?

?
Problem 1581

What sector must be cut away from a circle of radius RR so that the remaining part, rolled up, forms a funnel of maximum volume?

?
Problem 1582

A railway running from south to north passes through town BB. A factory AA is at a shortest distance aa kilometers from this railway, and bb kilometers from town BB to the north. In order to transport goods from AA to BB most economically, a spur track is to be built from the factory. If the cost of transporting one ton of goods along the spur is pp rubles per kilometer, and along the railway is qq rubles per kilometer (p>qp>q), at what angle φ\varphi to the railway should the spur be laid?

?
Problem 1583

Two ships travel in straight lines with constant speeds uu and vv respectively, the angle between their two directions of travel being θ\theta. If at a certain moment their distances from the point where their paths cross are aa and bb respectively, find the minimum distance between the two ships.

?
Problem 1584

There are light sources at points AA and BB, with intensities S1S_{1} candlepower and S2S_{2} candlepower respectively. Find the point MM of minimum illumination on the segment AB=aAB=a.

?
Problem 1585

A luminous point lies on the line joining the centers of two non-intersecting spheres of radii RR and rr (R>rR>r), outside both spheres. Where should this luminous point be located so that the sum of the illuminated portions of the surfaces of the two spheres is maximal?

?
Problem 1586

Let the radius of a round table top be aa. At what height above the center of the table top should a lamp be placed so that the illuminance at the edge of the table is maximal?

?
Problem 1587

A canal bb meters wide is to be built connecting to a river aa meters wide, meeting it at a right angle. What is the maximum length of a boat that can enter this canal?

?
Problem 1588

The cost of a ship's voyage over a day and night consists of two parts: a fixed part equal to aa rubles, and a variable part that increases in proportion to the cube of the speed. At what speed vv is the voyage most economical?

?
Problem 1589

A body of weight PP lies on a rough horizontal surface and must be moved from its position by an applied force. If the coefficient of friction of the body is kk, at what angle of inclination of the applied force to the horizontal surface is the required force minimal?

?
Problem 1590

A teacup has the shape of a hemisphere of radius aa. A rod of length l>2al>2a is placed in the cup. Find the equilibrium position of the rod.

?
Section 2.14Tangency of Curves. Circle of Curvature. Evolute[26/100%]
Problem 1591

Choose the parameters kk and bb of the line

y=kx+b y=kx+b

so that it has tangency of order higher than one with the curve

y=x3−3x2+2 y=x^{3}-3x^{2}+2
?
Problem 1592

How should the coefficients a,ba, b and cc be chosen so that the parabola

y=ax2+bx+c y=ax^{2}+bx+c

has second-order tangency with the curve y=exy=e^{x} at the point x=x0x=x_{0}?

?
Problem 1593

What is the order of tangency of each of the following curves with the OxOx axis at the point x=0x=0:

?
(a)

y=1−cos⁡xy=1-\cos x；

(b)

y=tan⁡x−sin⁡xy=\tan x-\sin x；

(c)

y=ex−(1+x+x22)y=e^{x}-\left(1+x+\dfrac {x^{2}}{2}\right).

Problem 1594

Prove that the curve: y=e−1x2y=e^{-\frac{1}{x^{2}}} for x≠0x \neq 0, and y=0y=0 for x=0x=0, has tangency of infinite order with the OxOx axis at the point x=0x=0.

?
Problem 1595

Find the radius of curvature and center of curvature of the hyperbola

xy=1 xy=1

at each of the following points:

?
(a)

M(1,1)M(1,1)；

(b)

N(100,0.01)N(100,0.01).

Problem 1596

Find the radius of curvature of the following curve: the parabola y2=2pxy^{2}=2px.

?
Problem 1597

Find the radius of curvature of the following curve: the ellipse x2a2+y2b2=1\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1.

?
Problem 1598

Find the radius of curvature of the following curve: the hyperbola x2a2−y2b2=1\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}=1.

?
Problem 1599

Find the radius of curvature of the following curve: the astroid x23+y23=a23x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}.

?
Problem 1600

Find the radius of curvature of the following curve: the ellipse x=acos⁡t,y=bsin⁡tx=a\cos t, y=b\sin t.

?
Problem 1601

Find the radius of curvature of the following curve: the cycloid x=a(t−sin⁡t),y=a(1−cos⁡t)x=a(t-\sin t), y=a(1-\cos t).

?
Problem 1602

Find the radius of curvature of the following curve: the involute of a circle x=a(cos⁡t+tsin⁡t),y=a(sin⁡t−tcos⁡t)x=a(\cos t+t\sin t), y=a(\sin t-t\cos t).

?
Problem 1603

Prove that for the quadratic curve

y2=2px−qx2 y^{2}=2px-qx^{2}

the radius of curvature is proportional to the cube of the length of the normal segment.

?
Problem 1604

Write down the formula for the radius of curvature of a curve given in polar coordinates.

?
Problem 1605

Find the radius of curvature of the curve given by the following polar equation: the Archimedean spiral r=aφr=a\varphi.

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Problem 1606

Find the radius of curvature of the curve given by the following polar equation: the logarithmic spiral r=aemφr=ae^{m\varphi }.

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Problem 1607

Find the radius of curvature of the curve given by the following polar equation: the cardioid r=a(1+cos⁡φ)r=a(1+\cos \varphi ).

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Problem 1608

Find the radius of curvature of the curve given by the following polar equation: the lemniscate r2=a2cos⁡2φr^{2}=a^{2}\cos 2\varphi.

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Problem 1609

Find the point of maximum curvature on the curve y=ln⁡xy=\ln x.

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Problem 1610

The cubic parabola y=kx36y=\dfrac {kx^{3}}{6} (0≤x<+∞,k>00 \leq x<+\infty , k>0) has maximum curvature equal to 11000\dfrac {1}{1000}; find the point xx at which this maximum curvature is attained.

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Problem 1611

Find the equation of the evolute of each of the following curves: the evolute of the parabola y2=2pxy^{2}=2px.

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Problem 1612

Find the equation of the evolute of each of the following curves: the evolute of the ellipse x2a2+y2b2=1\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1.

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Problem 1613

Find the equation of the evolute of each of the following curves: the evolute of the astroid x23+y23=a23x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}.

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Problem 1614

Find the equation of the evolute of each of the following curves: the evolute of the tractrix x=aln⁡a+a2−y2y−a2−y2x=a\ln \dfrac {a+\sqrt{a^{2}-y^{2}}}{y}-\sqrt{a^{2}-y^{2}}.

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Problem 1615

Find the equation of the evolute of each of the following curves: the evolute of the logarithmic spiral r=aemφr=ae^{m\varphi }.

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Problem 1616

Prove that the evolute of the cycloid

x=a(t−sin⁡t),y=a(1−cos⁡t) x=a(t-\sin t), \quad y=a(1-\cos t)

is again a cycloid, differing from the given cycloid only in its position.

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Section 2.15Approximate Solution of Equations[11/100%]
Problem 1617

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): x3−6x+2=0x^{3}-6x+2=0.

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Problem 1618

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): x4−x−1=0x^{4}-x-1=0.

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Problem 1619

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): x−0.1sin⁡x=2x-0.1\sin x=2.

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Problem 1620

Using the method of proportional parts (regula falsi), find the root of the following equation (accurate to 0.0010.001): cos⁡x=x2\cos x=x^{2}.

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Problem 1621

Using Newton's method, find the root of the following equation (accurate to the specified precision): x2+1x2=10xx^{2}+\dfrac {1}{x^{2}}=10x (accurate to 10−310^{-3}).

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Problem 1622

Using Newton's method, find the root of the following equation (accurate to the specified precision): xlg⁡x=1x\lg x=1 (accurate to 10−410^{-4}).

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Problem 1623

Using Newton's method, find the root of the following equation (accurate to the specified precision): cos⁡x⋅ch x=1\cos x \cdot \text{ch}\, x=1 (accurate to 10−310^{-3}, find two positive roots).

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Problem 1624

Using Newton's method, find the root of the following equation (accurate to the specified precision): x+ex=0x+e^{x}=0 (accurate to 10−510^{-5}).

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Problem 1625

Using Newton's method, find the root of the following equation (accurate to the specified precision): x th x=1x\, \text{th}\, x=1 (accurate to 10−610^{-6}).

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Problem 1626

Find the smallest three positive roots of the equation

tan⁡x=x \tan x=x

(accurate to 0.0010.001).

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Problem 1627

Find the two positive roots of the equation

cot⁡x=1x−x2 \cot x=\dfrac {1}{x}-\dfrac {x}{2}

(accurate to 10−310^{-3}).

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