15 feet long with a probable error of 0.00012 inches. Is this sufficiently accurate for the computation of the distance to the sun?

Why is it that the distance to the sun cannot be determined with greater accuracy?

11. If y is a linear function of x, y = mx + b, is the percentage of error in the computed value of y ever less than the percentage of error in the measured value of x?

12. The radius and an arc of a circle were measured, and found to be 6 inches and 15 inches respectively, and the number of radians in the central angle subtended by the arc was computed (Theorem, page 171). Find the error and the relative error in the angle due to an error of onetenth of an inch in the radius; in the arc; assuming in each case that the other measurement is exact.

104. Approximate value of f(x + Ax). If y = f(x), it is frequently convenient to use f'(x) to denote the derivative, instead of Day. The latter notation is used when we are denoting a function by y, the former when we are talking of a function f(x), as in this section.

If the graph of f(x) is drawn, and if x = OM and Ax = MN, then the value of f(x + Ax) is represented by the ordinate NQ.

FIG. 172.

Hence we have the

Theorem. If Ax is small, an ap

proximate value of f(x + ▲x) is given by the relation f(x + Ax) = f(x) + ƒ′(x)▲x, where f'(x) denotes the derivative of f(x).

In more advanced work in mathematics the error in this approximation is considered, and also better approximations involving the second and higher powers of Ar. The utility of the approximation is seen in the examples following.

EXAMPLE 1. Find an approximate value of 3.013.

=

Since 3.01 may be put in the form 3 +0.01, we have 3.01 = x + Ax, where x = 3 and Ax 0.01. And if f(x) = x3, the required number is the value of f(x + Ax). By the Theorem we have approximately (x + Ax)3 3.013

Then

=

x3 + 3x2Ax.

=

33 + 3 x 32 x 0.01

=

27 +0.27

=

27.27.

As the exact value is 3.013

=

27.270901, the approximate value obtained

by the Theorem is correct to four significant figures.

EXAMPLE 2. Compute a table of squares, to two decimal places, for the values x = 2.01, 2.02, 2.03, . . . 2.10.

The square of 2.01 may be computed from the square of 2, the square of 2.02 from that of 2.01, etc., by means of the Theorem which shows that, approximately, (x + ▲x)2 = x2 + 2x^x.

The computation may be systematized by arranging it in tabular form. The last column in the table gives the squares of the numbers in the first column. The accuracy of the square of the last number in the table checks the accuracy of the entire computation.

The computation is carried to four decimal places, as these places have an accumulative effect which ultimately changes the value of the tabular difference (at what point in the table?), but very few of the digits in the third and fourth decimal places are accurate. The approximations in the last column are made by taking the nearest figure in the second decimal place.

The approximation given in the Theorem is much used in the computation of tables, as in Example 2. It cannot be used, however, to construct a complete table without making use of the limit of error of the approximation, Thus if the

table above was continued for 2.11, 2.12, etc., it would be found that the value obtained for 2.162 would be too small by 0.01. In the very simple function x2 the accumulative effect of the difference between (x + Ax)2 and the approximation x2 + 2x▲x may be seen very clearly (see Exercise 2 below). But for most functions the question of the error involved is much more complicated.

EXERCISES

1. Compute an approximate value of each of the functions below for the given value of the variable.

(a) x2, 6.2.

(b) x3, 4.1.

(c) Vx, 2.1.

(d) 1/x, 2.1. 2. Compute the squares of 1.01, 1.02, and 1.03 using (a) the approximation (x + ▲x)2 = x2 + 2x▲x; (b) the exact relation (x + ▲x)2 = x2 + 2x Ax + Ax2. Note the error in the approximation of 1.032 due to the accumulated effect of neglecting Ar2 in the approximation.

3. Compute a three-place table of

be carried before the accumulative effect of the error in the approximation becomes evident?

MISCELLANEOUS EXERCISES

=

1. The chord of the parabola y ax2 through the focus perpendicular to the axis of symmetry is called the latus rectum. Find its length, and show that the tangents at its extremities are perpendicular.

2. If P1 is any point on y = x", M its projection on the y-axis, and T the point of intersection of the y-axis and the line tangent at P1, show that TM nOM. Illustrate by figures for n>1, 0<n<1, n <0.

=

3. If the tangent and normal to the parabola y

=

ax2 at Pi cut the y-axis at T and N respectively, then the focus F is equidistant from P1, 7' and N.

4. If the tangent at P1 to the hyperbola xy: = a cuts the x-axis at T, show that OP1 = TP1.

5. If the normal at P1 to the hyperbola xy = a (a>0) cuts the bisector of the first and third quadrants at B, then OP1 = BP1.

6. The line y = mx + la passes through the focus (0, a) of the parabola y = ax2. Find the abscissas of the points at which it intersects the parabola. Show that the tangents at these points are perpendicular, and that they intersect on the directrix.

7. If V。 is the volume of a quantity of water at 0° Centigrade then the volume at a temperature 0° Centigrade is given by

V = Vo(1 -0.00005758 0 + 0.000007560 – 0.0000000351 03). Show that the volume is least, and hence the density greatest, when 0 = 3°.92C. (Let the decimals be denoted by a, b, c respectively until the end of the computation, and write 0.00005758 in the form 5.758 × 10-5, etc.)

8. A bandit is walking along a street at the rate of 3 miles an hour toward the intersection with a second street making an angle of 45° with the first. A tree stands 100 feet from the intersection and 25 feet toward the second street from the first. A timid citizen walks along the second street endeavoring to keep the tree between himself and the bandit. How fast does he walk?

=

9. There are two sources of light at the points A and B. At a distance x from the first the illumination is I 8/x2, and at a distance x from the second the illumination is I2 = 27/x2. Find the point on the segment AB where the sum of the illuminations is a minimum.

are the values of n measurements Then the errors in the measure

a1, x A2, x a3,

xan, some of which

10. Suppose that a1, a2, aз, . . . an, of a magnitude whose true value is x. ments are respectively x are positive, and some of which are negative. The theory of least squares asserts that the most probable value of x is such that the sum of the squares of the errors is a minimum. Show that the most probable value of x is the arithmetic mean (or average).

11. Find the dimensions of the strongest beam which can be sawed from a log 12 inches in diameter, assuming that the strength of the beam varies as the breadth and the square of the depth.

12. Find the dimensions of the stiffest beam which can be cut from a log 10 inches in diameter, if the stiffness is proportional to the breadth and the cube of the depth.

13. Find the dimensions of the rectangle with maximum perimeter which can be inscribed in a circle of radius r.

=

14. A point moves along the straight line y 2x+10. If its abscissa is increasing at the rate of 3 inches per second when x = 1, find the rate at which its ordinate changes.

15. Two railroad tracks intersect at right angles. A train on one track is 24 miles from the intersection and is approaching it at the rate of 30 miles an hour. A train on the other track is 7 miles from the intersection, and is receding from it at 45 miles an hour. Is the distance between the trains increasing or decreasing? How rapidly?

16. A 16 foot ladder resting against the side of a barn begins to slip. When the foot of the ladder is 10 feet from the barn it is moving 2 feet per second. How fast is the top of the ladder moving?

17. The side of a square is 39.51 inches, with an error of not more than 0.01. Find the error in the area of the square. Is the value of 39.512 given in Huntington's Tables sufficiently accurate for the area?

18. The intensity of heat at a distance of x feet from a source of heat is I = 100/x2. If a body moves directly away from the source at the uniform rate of 3 feet per second, how rapidly is the intensity of the heat changing when x = 5 feet? When x = 20 feet?

19. The side of an equilateral triangle is 5.4 ± 0.1 inches. What is the percentage of error in the area of the triangle?

20. If a body moves so that v2 = ks, show that the acceleration is constant. An automobile moving v miles per hour on a slippery pavement should be able to stop in s feet, where v2 17s. Find the acceleration in

feet per second per second.

=

21. A rectangular grain bin, without a cover, is to be divided into two equal parts by a partition parallel to the ends. The bin is to be 3 feet deep and to hold 98 cubic feet. Find its dimensions if the amount of lumber required is a minimum.

22. A rectangular pan is to have its width two-thirds of its length and its capacity is to be 1% of a cubic foot. Find its dimensions if the amount of tin used is a minimum.

23. The thrust of an aeroplane is given by the equation (see Exercise 11, page 157)

Find the value of 0 for which t is a minimum. Substitute this value of in the given equation, and determine how t varies as the fineness f changes.

24. The power required for oblique flight of an aeroplane up a slope m is

=

For gliding flight T = 0; what relation must connect & and m for gliding flight? If 0 01 makes m a minimum, mi, show that m1 = 201. What relation exists between m and the horizontal and vertical distances the aeroplane glides? How does the fineness ƒ affect the greatest horizontal distance the aeroplane can glide? Show that the aeroplane can glide along a given slope for two different values of the angle of incidence and hence for two different velocities (see Note, page 156).