Interpolation. Error Theory

Assuming that the numbers written down are accurate only to the last significant digit, find approximate values of the following quantities:

An angle measured with a goniometer turned out to equal $32^{\circ }13'25'' \pm 10''$. What is the relative error?

The side of a square is approximately $1 \, \text{m}$. To what degree of precision must it be measured so that the error in the area is no more than $1 \, \text{cm}^{2}$?

A pendulum about $2 \, \text{m}$ long completed about 400 oscillations. Knowing that its length was measured to within $1 \, \text{mm}$ and the total duration of the oscillations was measured to within $0{,}5 \, s$, determine to what precision the value of the acceleration $g$ can be found from this data, and also find out which measurement error affects the result more strongly.

In measuring the span length of a bridge under construction, a baseline was laid out on one bank of the river, equal to $200 \, \text{m} \pm 0{,}01 \, \text{m}$. The angles between the baseline and the directions from its endpoints to a point across the river were measured. They turned out to be $90^{\circ }\pm 1'$ and $60^{\circ }\pm 1'$. To what degree of precision can the span length of the bridge be determined from this data?

For what value of the angle is the relative error in computing a leg of a right triangle from a given leg and the opposite angle especially small, if the error in measuring the given leg is neglected?

In the field of view of a level's telescope, two horizontal lines are visible. When viewing through this telescope a vertical staff with decimeter graduations, 10 divisions of the staff are visible between the crosshairs when the staff is at a distance of $50 \, \text{m}$. When the staff is moved to another point, $25 \pm 0{,}2$ divisions are visible between the same crosshairs. What is the distance to the new position of the staff?

In five-place logarithm tables, decimal logarithms of numbers are given to within $\frac{1}{2} \cdot 10^{-5}$. How large can the error be in finding a number from its logarithm, if the number lies between 300 and 400?

The same question applied to a table of logarithms of sines and logarithms of tangents, if the angle is about $45^{\circ }$.

The perimeter length of a regular inscribed 96-gon, which Archimedes used in computing $\pi$, is expressed for $r=1$ by the formula

If one computes directly from this formula, wishing to obtain $P$ to within $0{,}001$, to what precision must the quantities under the radicals be computed?

The hypsometric formula, by which one determines the height of a place above the ground, is as follows:

Here $p_{0}$ is the pressure at the earth's surface, $p$ is the pressure at the given place at height $h$, $a = \frac{1}{273}$, $t$ is the temperature in Celsius. Determine what the error is in determining the height, if $t=0^{\circ }$, and the value $p = 500 \, \text{mm}$ to within $1 \, \text{mm}$.

Find a polynomial in $x$, of as low a degree as possible, taking at $x=2, 4, 5, 10$ respectively the values $3, 7, 9, 19$.

Find a polynomial in $x$, of as low a degree as possible, taking at $x=2, 4, 5, 10, 15$ respectively the values $3, 7, 9, 19, 0$.

Find a polynomial in $x$, of as low a degree as possible, taking at $x=1,2,3,4,5$, respectively the values $2, 1, -1, 5, 0$.

Find a polynomial $f(x)$, of as low a degree as possible, taking at $x = -2, -1, 0, 1, 2$ the same values as the function $y = \sin \dfrac {\pi x}{2}$. Determine the value of the difference $f(x) - y$ at the values $x$ equal to $\dfrac {1}{2}, \dfrac {5}{3}, 10$.

Find a polynomial $f(x)$, of as low a degree as possible, taking at $x=-2, -\dfrac {4}{3}, 0, \dfrac {4}{3}, 2$ the same values as the function $y = \cos \dfrac {\pi x}{4}$. Determine the values of the difference $f(x) - y$ at the values $x$ equal to $-1, 3, 10$.

For the function $y = x^{3} - 3x^{2} + 5x - 7$ at $x=0$, the values of the differences $\Delta y_{0} = +3$, $\Delta^{2} y_{0} = 0$, $\Delta^{3} y_{0} = 5$ have been computed, corresponding to an increment $h_{0}=1$ of the argument. It is required to compute the differences $\overline{\Delta y_{0}}$, $\overline{\Delta^{2} y_{0}}$, $\overline{\Delta^{3} y_{0}}$, corresponding to an increment of the argument $h_{1} = \dfrac {1}{10}$.

For the function $y = 2x^{4} - x^{3} + 3x^{2} - 1$ at $x=0$, the differences $\Delta y_{0} = 4$, $\Delta^{2} y_{0} = 28$, $\Delta^{3} y_{0} = 66$, $\Delta^{4} y_{0} = 48$ have been computed, corresponding to an increment $h_{0}=1$ of the argument. It is required to compute the differences $\overline{\Delta y_{0}}$, $\overline{\Delta^{2} y_{0}}$, $\overline{\Delta^{3} y_{0}}$, $\overline{\Delta^{4} y_{0}}$, corresponding to an increment of the argument $h_{1} = \dfrac {1}{10}$.

Find the equation of the line for which the sum of the squares of the distances to the points $(3,2)$, $(7,5)$, $(12,9)$, $(-3,-2)$ would be smallest.

Find the equation of a second-order parabola such that the sum of the squares of the vertical distances to it from the points $(-3,4)$, $(-2,2)$, $(-1,1)$, $(0,0)$, $(1,1)$, $(2,2)$, $(3,3)$ would be smallest.

Using the table

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.

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

Using the table

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

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

Using the data of the table

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

Using the data of the table

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

Using the data of the table

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

Using the data of the table

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

Using the data of the table

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

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:

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

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?

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

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

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

How many people are in their 38th year?

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?