Approximate formulas from Taylor expansions and their errors

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Problem 1411 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.10 "Taylor's Formula"

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

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

(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 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.10 "Taylor's Formula"

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.

Problem 1413 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 2.10 "Taylor's Formula"

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.

Problem 1104.1 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Problem 1104 "[NO TITLE]"

$\sqrt{5}$ ;

Problem 1393.1 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Problem 1393 "[NO TITLE]"

[NO TASK]

Problem 1395.1 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Problem 1395 "[NO TITLE]"

[NO TASK]

Problem 1722 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

Prove that for small $\dfrac {b}{a^{2}}$ the approximate equality $\sqrt{a^{2}+b} \approx a+\dfrac {b}{2a}$ has an error approximately equal to $\dfrac {a}{8}\left(\dfrac {b}{a^{2}}\right)^{2}$ , and use this equality to compute the values of the radicals:

(a)

$\sqrt{2}$ ;

(b)

$\sqrt{24}$ ;

(c)

$\sqrt{84}$ ;

(d)

$\sqrt{235}$ ;

(e)

$\sqrt{240}$ .

Problem 1724 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

There is the following approximate formula for extracting roots:

\sqrt[n]{N} = \sqrt[n]{a^{n}+b} \approx a+\frac{2ab}{2nN-(n+1)b},

where $a^{n}$ is an exact power close to $N$ . Prove that the error of this formula is approximately equal to $\dfrac {n^{2}-1}{12n^{3}}\left(\dfrac {b}{a^{n}}\right)^{3}$ .

Problem 1725 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

Using the formula of problem 1724 ( $\sqrt[n]{N} = \sqrt[n]{a^{n}+b} \approx a+\dfrac {2ab}{2nN-(n+1)b}$ ), compute the following roots:

(a)

$\sqrt[3]{30}$ ;

(b)

$\sqrt[3]{70}$ ;

(c)

$\sqrt[3]{500}$ ;

(d)

$\sqrt[5]{250}$ ;

(e)

$\sqrt{60}$ ;

(f)

$\sqrt{84}$ .

Problem 1729 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

Prove that the quantities

\Delta \ln n = \ln (n+1)-\ln n,

\Delta ^{2} \ln n = \ln (n+2)-2\ln (n+1)+\ln n,

\Delta ^{3} \ln n = \ln (n+3)-3\ln (n+2)+3\ln (n+1)-\ln n

have, respectively, the following series expansions:

\Delta \ln n = 2\left(\frac{1}{2n+1}+\frac{1}{3(2n+1)^{3}}+\frac{1}{5(2n+1)^{5}}+\ldots \right),

\Delta ^{2} \ln n = -\left(\frac{1}{(n+1)^{2}}+\frac{1}{2(n+1)^{4}}+\frac{1}{3(n+1)^{6}}+\ldots \right),

\Delta ^{3} \ln n = 2\left(\frac{8}{(2n+3)^{3}}+\frac{48}{(2n+3)^{5}}+\frac{312}{(2n+3)^{7}}+\ldots \right).

Problem 1736 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

In a day the Earth traverses approximately an arc of $1^{\circ }$ of its orbit, whose radius is $150 \cdot 10^{6}$ km. Determine by how much the Earth's path deviates from a straight line in 1 second (27 km) and in 1 minute?

Problem 1737 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

An arc of $1^{\circ }$ of the Earth's meridian is about 112 km. By how much is it longer than its chord?

Problem 1739 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

In measuring the distance between two points, the reading on a stretched tape showed 12 m. Knowing that the sag of the tape was 20 cm, and taking the shape of the tape to be an arc of a circle, determine the distance between the points with a correction for the sag.

Problem 1745 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.7 "Series and Operations on Them"

One of Huygens's rules for computing the circumference of a circle is equivalent to the equality

2\pi = \frac{P_{n}+2p_{n}}{3r},

where $P_{n}$ and $p_{n}$ are the perimeters of the circumscribed and inscribed regular polygons, and $r$ is the radius of the circle. Prove that the error of this equality is approximately equal to $\dfrac {30}{n^{4}}$ .

Problem 1783 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.8 "Evaluation of Indeterminate Forms"

If $a$ , $b$ , $c$ are the sides of a spherical triangle, and $A$ , $B$ , $C$ the opposite angles, then

\cos c = \cos a \cos b + \sin a \sin b \cos C.

What does this formula become if $a$ , $b$ and $c$ are small?

Problem 1784 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 4.8 "Evaluation of Indeterminate Forms"

A body falls in a medium whose resistance is proportional to the first power of the speed. The distance traveled in $t$ seconds is expressed by the formula

s = \frac{mgt}{a}-\frac{m^{2}g}{a^{2}}\left(1-e^{-\frac{at}{m}}\right),

where $m$ is the mass, $a$ the coefficient of friction. Find approximate formulas for $s$ when $\dfrac {at}{m}$ is large, and also for the case when $\dfrac {at}{m}$ is very small.

Problem VI.4.7 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VI.4 "Asymptotics of Implicit Functions and Recurrent Sequences"

The problem of "the globe in a string bag." The Earth's globe was girdled with an inextensible thread. Then the thread was lengthened by 1 cm and pulled taut by tugging at the knot. Compute the height to which it rose, with a relative error of no more than one ten-thousandth of a percent (take the radius of the globe to be $6400\ \text{km}$ ).

References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.

[ГюКу03] Гюнтер Н.М., Кузьмин Р.О. Сборник задач по высшей математике. 13-е изд., Издательство «Лань», 2003.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.