Math Problems Vault

Problem 1083

(a)(b)(c)

Let:

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(a)

$\Delta x=1$ ;

(b)

$\Delta x=0.1$ ;

(c)

$\Delta x=0.01$ ,

for the function

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

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

Problem 1084

(a)(b)(c)

The equation of motion is

x=5t^{2},

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

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(a)

(b)

(c)

Problem 1085

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

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Problem 1086

Find the differential of the function $y$ , given: $y=\dfrac {1}{a}\arctg \dfrac {x}{a} \ (a \neq 0)$ .

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Problem 1087

Find the differential of the function $y$ , given: $y=\dfrac {1}{2a}\ln \left|\dfrac {x-a}{x+a}\right|$ .

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Problem 1088

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

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Problem 1089

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

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Problem 1090

Find:

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(a)

$d(xe^{x})$ ;

(b)

$d(\sin x-x\cos x)$ ;

(c)

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

(d)

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

(e)

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

(f)

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

(g)

$d\ln (1-x^{2})$ ;

(h)

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

(i)

$d\left(\dfrac {\sin x}{2\cos^{2}x}+\dfrac {1}{2}\ln \left|\tg \left(\dfrac {x}{2}+\dfrac {\pi }{4}\right)\right|\right)$ .

Problem 1091

Let $u$ , $v$ , $w$ be differentiable functions of $x$ . Find the differential of the function $y$ , given: $y=uvw$ .

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Problem 1092

Let $u$ , $v$ , $w$ be differentiable functions of $x$ . Find the differential of the function $y$ , given: $y=\dfrac {u}{v^{2}}$ .

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Problem 1093

Let $u$ , $v$ , $w$ be differentiable functions of $x$ . Find the differential of the function $y$ , given: $y=\dfrac {1}{\sqrt{u^{2}+v^{2}}}$ .

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Problem 1094

Let $u$ , $v$ , $w$ be differentiable functions of $x$ . Find the differential of the function $y$ , given: $y=\arctg \dfrac {u}{v}$ .

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Problem 1095

Let $u$ , $v$ , $w$ be differentiable functions of $x$ . Find the differential of the function $y$ , given: $y=\ln \sqrt{u^{2}+v^{2}}$ .

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Problem 1096

Find

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(a)

$\dfrac {d}{dx^{3}}(x^{3}-2x^{6}-x^{9})$ ;

(b)

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

(c)

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

(d)

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

(e)

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

Problem 1097

(a)(b)

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

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(a)

(b)

Problem 1098

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

T=2\pi \sqrt{\frac{l}{g}},

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

In order to increase the period $T$ by $0.05$ seconds, how much modification is needed to the pendulum length $l=20$ cm?

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Problem 1099

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

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Problem 1100

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

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Problem 1101

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

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Problem 1102

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

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Problem 1103

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

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Problem 1104

(a)(b)(c)

Prove the approximate formula:

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

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

Use this formula to approximately compute:

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(a)

(b)

$\sqrt{34}$ ;

(c)

$\sqrt{120}$

and compare with the values in the table.

Problem 1104.1

$\sqrt{5}$ ;

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Problem 1105

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

Prove the approximate formula:

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

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

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(a)

$\sqrt[3]{9}$ ;

(b)

$\sqrt[4]{80}$ ;

(c)

$\sqrt[7]{100}$ ;

(d)

$\sqrt[10]{1000}$ .

Problem 1106

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

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Problem 1107

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

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Problem 1108

(a)(b)

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

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(a)

(b)

Problem 1109

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

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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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