Math Problems Vault

Problem 1376

Express the polynomial

P(x)=1+3x+5x^{2}-2x^{3}

as a polynomial in positive integer powers of the binomial $x+1$ .

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Problem 1377

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\frac{1+x+x^{2}}{1-x+x^{2}} \text{ up to the term in } x^{4} \text{. What is } f^{(4)}(0) \text{?}

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Problem 1378

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\frac{(1+x)^{100}}{(1-2x)^{40}(1+2x)^{60}} \text{ up to the term in } x^{2} \text{.}

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Problem 1379

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\sqrt[m]{a^{m}+x} \ (a>0) \text{ up to the term in } x^{2} \text{.}

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Problem 1380

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\sqrt{1-2x+x^{3}}-\sqrt[3]{1-3x+x^{2}} \text{ up to the term in } x^{3} \text{.}

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Problem 1381

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

e^{2x-x^{2}} \text{ up to the term in } x^{5} \text{.}

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Problem 1382

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\frac{x}{e^{x}-1} \text{ up to the term in } x^{4} \text{.}

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Problem 1383

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\sqrt[3]{\sin x^{3}} \text{ up to the term in } x^{13} \text{.}

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Problem 1384

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\ln \cos x \text{ up to the term in } x^{6} \text{.}

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Problem 1385

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\sin (\sin x) \text{ up to the term in } x^{3} \text{.}

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Problem 1386

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\tg x \text{ up to the term in } x^{5} \text{.}

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Problem 1387

Write out the expansion of the following function in positive integer powers of the variable $x$ , up to the term of the indicated order:

\ln \frac{\sin x}{x} \text{ up to the term in } x^{6} \text{.}

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Problem 1388

Find the first three terms of the expansion of the function $f(x)=\sqrt{x}$ in positive integer powers of the difference $(x-1)$ .

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Problem 1389

Expand the function $f(x)=x^{x}-1$ in positive integer powers of $(x-1)$ up to the term containing $(x-1)^{3}$ .

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Problem 1390

In a neighborhood of the point $x=0$ , approximate the function

y=a\ch \frac{x}{a} \ (a>0)

by a second-order parabola.

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Problem 1391

Expand the function $f(x)=\sqrt{1+x^{2}}-x \ (x>0)$ in positive integer powers of $\dfrac {1}{x}$ , up to the term in $\dfrac {1}{x^{3}}$ .

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Problem 1392

Expand the function $f(h)=\ln (x+h) \ (x>0)$ in positive integer powers of the increment $h$ , up to the term in $h^{n}$ ( $n$ a natural number).

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Problem 1393

Let:

\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 $f^{(n+1)}(x) \neq 0$ . Prove that:

\lim _{h \to 0}\theta =\frac{1}{n+1} .

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Problem 1393.1

[NO TASK]

Problem 1393.2

[NO TASK]

Problem 1393.3

[NO TASK]

Problem 1394

(a)(b)(c)(d)

Estimate the absolute error of the following approximate formulas:

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(a)

$e^{x} \approx 1+x+\dfrac {x^{2}}{2!}+\cdots +\dfrac {x^{n}}{n!}$ for $0 \leqslant x \leqslant 1$ ;

(b)

$\sin x \approx x-\dfrac {x^{3}}{6}$ for $\left|x\right| \leqslant \dfrac {1}{2}$ ;

(c)

$\tg x \approx x+\dfrac {x^{3}}{3}$ for $\left|x\right| \leqslant 0.1$ ;

(d)

$\sqrt{1+x} \approx 1+\dfrac {x}{2}-\dfrac {x^{2}}{8}$ for $0 \leqslant x \leqslant 1$ .

Problem 1395

For what values of $x$ is the approximate formula

\cos x=1-\frac{x^{2}}{2}

accurate to $0.0001$ .

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Problem 1395.1

[NO TASK]

Problem 1396

Using Taylor's formula, compute approximately, and estimate the error:

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(a)

$\sqrt[3]{30}$ ;

(b)

$\sqrt[5]{250}$ ;

(c)

$\sqrt[15]{4000}$ ;

(d)

$\sqrt{e}$ ;

(e)

$\sin 18°$ ;

(f)

$\ln 1.2$ ;

(g)

$\arctg 0.8$ ;

(h)

$\arcsin 0.45$ ;

(i)

$(1.1)^{1.2}$ .

Problem 1397

Compute:

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(a)

$e$ to within $10^{-9}$ ;

(b)

$\sin 1°$ to within $10^{-8}$ ;

(c)

$\cos 9°$ to within $10^{-5}$ ;

(d)

$\sqrt{5}$ to within $10^{-4}$ ;

(e)

$\log_{10}11$ to within $10^{-5}$ .

Problem 1398

Using expansions I--V, find the following limits:

\lim _{x \to 0}\frac{\cos x-e^{-\frac{x^{2}}{2}}}{x^{4}} .

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Problem 1399

Using expansions I--V, find the following limits:

\lim _{x \to 0}\frac{e^{x}\sin x-x(1+x)}{x^{3}} .

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Problem 1400

Using expansions I--V, find the following limits:

\lim _{x \to +\infty }x^{\frac{3}{2}}(\sqrt{x+1}+\sqrt{x-1}-2\sqrt{x}) .

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Problem 1401

Using expansions I--V, find the following limits:

\lim _{x \to +\infty }(\sqrt[6]{x^{5}+x^{5}}-\sqrt[6]{x^{6}-x^{5}}) .

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Problem 1402

Using expansions I--V, find the following limits:

\lim _{x \to +\infty }\left[\left(x^{3}-x^{2}+\frac{x}{2}\right)e^{\frac{1}{x}}-\sqrt{x^{6}+1}\right] .

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Problem 1403

Using expansions I--V, find the following limits:

\lim _{x \to +0}\frac{a^{x}+a^{-x}-2}{x^{2}} \ (a>0) .

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Problem 1404

Using expansions I--V, find the following limits:

\lim _{x \to \infty }\left[x-x^{2}\ln \left(1+\frac{1}{x}\right)\right] .

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Problem 1405

Using expansions I--V, find the following limits:

\lim _{x \to 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right) .

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Problem 1406

Using expansions I--V, find the following limits:

\lim _{x \to 0}\frac{1}{x}\left(\frac{1}{x}-\ctg x\right) .

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Problem 1406.1

[NO TASK]

Problem 1406.2

[NO TASK]

Problem 1406.3

[NO TASK]

Problem 1407

As $x \to 0$ , find the leading term of the form $Cx^{n}$ ( $C$ a constant) of the infinitesimal $y$ , given:

y=\tg (\sin x)-\sin (\tg x) .

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Problem 1408

As $x \to 0$ , find the leading term of the form $Cx^{n}$ ( $C$ a constant) of the infinitesimal $y$ , given:

y=(1+x)^{x}-1 .

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Problem 1409

As $x \to 0$ , find the leading term of the form $Cx^{n}$ ( $C$ a constant) of the infinitesimal $y$ , given:

y=1-\frac{(1+x)^{\frac{1}{x}}}{e} .

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Problem 1410

For what choice of coefficients $a$ and $b$ is the quantity

x-(a+b\cos x)\sin x

an infinitesimal of order $5$ with respect to $x$ ?

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Problem 1410.1

[NO TASK]

Problem 1410.2

[NO TASK]

Problem 1411

(a)(b)(c)(d)

For small $\left|x\right|$ , derive simple approximate formulas for the following expressions:

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(a)

$\dfrac {1}{R^{2}}-\dfrac {1}{(R+x)^{2}} \ (R>0)$ ;

(b)

$\sqrt[3]{\dfrac {1+x}{1-x}}-\sqrt[3]{\dfrac {1-x}{1+x}}$ ;

(c)

$\dfrac {A}{x}\left[1-\left(1+\dfrac {x}{100}\right)^{-n}\right]$ ;

(d)

$\dfrac {\ln 2}{\ln \left(1+\dfrac {x}{100}\right)}$ .

Problem 1412

For small absolute value of $x$ , derive an approximate formula of the form

x=\alpha \sin x+\beta \tg x

accurate to the $x^{5}$ term.

Apply this formula to the approximate computation of the arc length for a small angle.

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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 $\sqrt{\dfrac {4}{3}}$ times the sagitta of the circular segment.

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