Math Problems Vault

Problem 5208 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the table

$x$	$\ln x$	$10^{10} \Delta \ln x$	$10^{10} \Delta ^{2} \ln x$
1, 005	3{,}002166059	4, 319, 184{,}7	$-4\, 291{,}3$
1, 006	3{,}002597978	4, 314, 893{,}4	$-4\, 282{,}8$

compute $\ln 1005{,}237$ , determining the precision of the computation. The numbers in the table are given to within $\frac{1}{2}$ of the last digit.

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Problem 5209 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the table of problem 5208, find $x$ given $\ln x = 3{,}002268651$ .

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Problem 5210 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the table

$u$	$\ln \operatorname {sh} u$	$10^{5} \Delta \ln \operatorname {sh} u$	$\ln \operatorname {ch} u$	$10^{5} \Delta \ln \operatorname {ch} u$
0{,}500	9{,}71692	94{,}0	0{,}05217	20{,}1
0{,}501	9{,}71786	93{,}8	0{,}05237	20{,}1
0{,}502	9{,}71879	93{,}7	0{,}05257	20{,}1
0{,}503	9{,}71973	93{,}5	0{,}05277	20{,}2
0{,}504	9{,}72066	93{,}3	0{,}05297	20{,}2

find $\operatorname {sh} 0{,}50235$ .

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Problem 5211 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Find the root of the equation $\operatorname {ch} u = 1{,}1287$ . See the table of problem 5210.

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Problem 5212 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the data of the table

$x$	$\sin x$
$41^{\circ }$	0{,}6560590290
$42^{\circ }$	0{,}6691306064
$43^{\circ }$	0{,}6819983601
$44^{\circ }$	0{,}6946583705

compute, with the greatest possible precision, $\sin 41^{\circ }20'$ .

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Problem 5213 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the data of the table

$\varphi$	$F(\varphi ) = \int _{0}^{\varphi } \dfrac {d\varphi }{\sqrt{1 - \frac{1}{2}\sin ^{2}\varphi }}$
$21{,}0^{\circ }$	0{,}3706344
$21{,}5^{\circ }$	0{,}3796626
$22{,}0^{\circ }$	0{,}3887052
$22{,}5^{\circ }$	0{,}3977625

(table 13.1) compute $F(\varphi )$ at $\varphi = 21^{\circ }15'$ .

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Problem 5214 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the data of the table

$\varphi$	$F(\varphi ) = \int _{0}^{\varphi } \dfrac {d\varphi }{\sqrt{1 - \frac{1}{2}\sin ^{2}\varphi }}$
$28{,}0^{\circ }$	0{,}49846361
$28{,}5^{\circ }$	0{,}50772435
$29{,}0^{\circ }$	0{,}51700422
$29{,}5^{\circ }$	0{,}52630357

(table 13.2) compute the root of the equation $F(\varphi ) = 0{,}5$ .

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Problem 5215 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the data of the table

$x$	$\psi (x) = \dfrac {d \ln \Gamma (1+x)}{dx}$
0{,}71	0{,}2164501462
0{,}72	0{,}2242928871
0{,}73	0{,}2320769593
0{,}74	0{,}2398032061

(table 13.3) compute $\psi (0{,}715)$ . To estimate the error one should take into account that $\psi '(x) = \dfrac {1}{(x+1)^{2}} + \dfrac {1}{(x+2)^{2}} + \ldots$

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Problem 5216 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Using the data of the table

$x$	$\psi (x) = \dfrac {d \ln \Gamma (1+x)}{dx}$
0{,}83	0{,}3068612
0{,}84	0{,}3140589
0{,}85	0{,}3212000
0{,}86	0{,}3282922

(table 13.4) compute the root of the equation $\psi (x) = 0{,}31$ .

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Problem 5217 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

(a)(b)(c)

Using a table of values of reciprocals of integers and an interpolation formula taken up through the fourth differences inclusive, find the value of the fraction $\dfrac {1}{12{,}5}$ by means of the formulas:

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(a)

Newton--Gregory, with $y_{0} = \dfrac {1}{12}$ , $y_{1} = \dfrac {1}{13}$ , $y_{2} = \dfrac {1}{14}, \ldots$ ;

(b)

Newton--Gauss, and

(c)

Newton--Bessel, comparing the results obtained with the exact value.

Remark.

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Newton--Gregory formula: $y_{x} = y_{0} + x \Delta y_{0} + \dfrac {x(x-1)}{2!}\Delta^{2} y_{0} + \dfrac {x(x-1)(x-2)}{3!}\Delta^{3} y_{0} + \ldots$

Newton--Gauss formula: $y_{x} = y_{0} + x \Delta y_{0} + \dfrac {x(x-1)}{2!}\Delta^{2} y_{-1} + \dfrac {x(x^{2}-1)}{3!}\Delta^{3} y_{-1} + \ldots$

Newton--Bessel formula: $y_{x} = \dfrac {1}{2}(y_{0}+y_{1}) + \left(x - \dfrac {1}{2}\right)\Delta y_{0} + \dfrac {x(x-1)}{2!} \cdot \dfrac {\Delta^{2} y_{-1} + \Delta^{2} y_{0}}{2} + \ldots$

Here $y_{0}, y_{1}, \ldots , y_{n}$ are the values of $y$ at $x=a, a+h, \ldots , a+nh$ ; the quantities $\Delta y_{0}, \Delta^{2} y_{0}, \ldots$ are the successive differences.

Problem 5218 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

A similar question for the value $\sqrt{12{,}25}$ ; $y_{0} = \sqrt{12}$ , $y_{1} = \sqrt{13}$ , $y_{2} = \sqrt{14}, \ldots$

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Problem 5219 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

Of 1000 born, the number surviving to age 20, 30, 40, 50, 60 years is respectively 560, 530, 480, 400, and 300 people. How many people out of 1000 survive to age 35 and 45?

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Problem 5220 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

In a certain locality, the numbers of people whose ages fall into three consecutive five-year periods are as follows:

37\, 237, \quad 40\, 118, \quad 42\, 652.

How many people are at the age of the middle year of the second five-year period?

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Problem 5221 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

The numbers of people in the ages 25--30, 30--35, 35--40, 40--45 years are respectively as follows:

458\, 572, \quad 441\, 424, \quad 423\, 123, \quad 402\, 918.

How many people are in their 38th year?

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Problem 5222 from Гюнтер Н.М., Кузьмин Р.О. "Сборник задач по высшей математике" [ГюКу03]More problems in Section 13.1 "Interpolation. Error Theory"

A table contains, in one column, 20 consecutive values of a function. The sums of three consecutive columns are as follows: $630\, 611$ , $1\, 946\, 935$ , $3\, 246\, 319$ . With what number does the fourth column begin?

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Finite-difference interpolation formulas (Newton-Gregory, Gauss, Bessel)

References