Approximate calculation by differentials (linearization)

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.

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

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?

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

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

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

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

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

Prove the approximate formula:

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:

Prove the approximate formula:

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

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?

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?

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 $g$?

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

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.

Prove the approximate formula

and use it to compute the roots: $\sqrt[5]{245}$, $\sqrt[7]{129}$, $\sqrt[9]{515}$, $\sqrt[10]{1027}$.

Prove that the tabular difference of ordinary five-digit logarithms of numbers, i.e. $\lg (n+1)-\lg n$, for $1000 < n < 10000$ is approximately expressed by the formula $\Delta = 10^{5} \cdot \dfrac {M}{n} = \dfrac {43429}{n}$; $M = 0{,}43429$ is the modulus of common logarithms, $\Delta$ being expressed in hundred-thousandths.

Prove that the tabular difference of logarithms of sines given one minute apart, expressed in hundred-thousandths, can be given by the approximate formula

Prove that formulas of the same kind for sines and for logarithms of tangents have the form:

The approximate equalities $\sin x \approx x$, $\tg x \approx x$ have high accuracy for small angles (in some cases the more accurate formulas $\sin x \approx x-\dfrac {x^{3}}{6}$, $\tg x \approx x+\dfrac {x^{3}}{3}$ are needed). At what visual angle is the diameter of a factory chimney of radius 2 m seen from a distance of 10 km?

The disk of the Sun is seen at an angle of $30'$. How many times greater is the distance to the Sun than the diameter of the Sun?

A railway embankment has a grade of $0{,}012$. What angle does it make with the horizontal?

Prove the approximate formula: $l = \sqrt{13h}$, where $h$ is the height of the observer above the horizon in meters, and $l$ is the distance to the horizon in kilometers.

Replacing the increment of the function by its differential, find the approximate value of the function $y = y(x)$ at the indicated points:

Prove that for all values of $\Delta x$ small compared with $x_{0}$ the approximate formula holds

Using this formula, approximately compute:

Determine by approximately how much the volume of a ball increases if its radius $R = 15$ cm is increased by 0.2 cm.

Determine approximately the relative error in computing the surface area of a sphere if the relative error in determining its radius was $1\%$.

By approximately how much (in percent) will the current in a conductor change if its resistance increases by $1\%$?

By approximately how much should the length of a pendulum $l = 20 \mathrm{~ cm}$ be changed so that the period of oscillation of the pendulum increases by $0.05 \mathrm{s}$? The period $T$ is given by the formula $T = 2 \pi \sqrt{l / g}$.