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Then using the constraint conditions, we nd x2 y2 z2 1 . For a generalization of this problem, see Problem 8.76.
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APPLICATIONS TO ERRORS p 8.28. The period T of a simple pendulum of length l is given by T 2 l=g. Find the (a) error and (b) percent error made in computing T by using l 2 m and g 9:75 m=sec2 , if the true values are l 19:5 m and g 9:81 m=sec2 .
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(a) T 2l 1=2 g 1=2 . Then 2g 1=2 1 l 1=2 dl 2 2l
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1 g 3=2 dg 2
s  l p dl  3 dg lg g
Error in g g dg 0:06; error in l l dl 0:5 The error in T is actually T, which is in this case approximately equal to dT. Thus, we have from (1), s  2 Error in T dT p 0:05  0:06 0:0444 sec (approx.) 9:75 3 2 9:75 The value of T for l 2; g 9:75 is T 2 r 2 2:846 sec (approx.) 9:75
Percent error (or relative error) in T Another method:
dT 0:0444 1:56%: T 2:846
Since ln T ln 2 1 ln l 1 ln g, 2 2     dT 1 dl 1 dg 1 0:05 1 0:06 1:56% T 2 l 2 g 2 2 2 9:75
as before.
Note that (2) can be written Percent error in T 1 Percent error in l 1 Percent error in g 2 2
MISCELLANEOUS PROBLEMS 1 x 1 dx. 8.29. Evaluate 0 ln x
In order to evaluate this integral, we resort to the following device. De ne 1 x 1  dx >0 0 ln x Then by Leibnitz s rule  0 1   1 1 @ x 1 x ln x 1 dx x dx dx ln x 1 0 @ 0 ln x 0
Integrating with respect to ,  ln 1 c. But since  0 0; c 0; and so  ln 1 . Then the value of the required integral is  1 ln 2. The applicability of Leibnitz s rule can be justi ed here, since if we de ne F x; x 1 = ln x, 0 < x < 1, F 0; 0; F 1; , then F x; is continuous in both x and for 0 @ x @ 1 and all nite > 0.
8.30. Find constants a and b for which F a; b is a minimum.
fsin x ax2 bx g2 dx
CHAP. 8]
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The necessary conditions for a minimum are @F=@a 0, @F=@b 0. Performing these di erentiations, we obtain   @F @ fsin x ax2 bx g2 dx 2 x2 fsin x ax2 bx g dx 0 @a @a 0 0   @F @ 2 2 fsin x ax bx g dx 2 xfsin x ax2 bx g dx 0 @b 0 @b 0 From these we nd   8  > a x4 dx b x3 dx x2 sin x dx > < 0 0 0   >  3 > a x dx b x2 dx x sin x dx :
0 0 0
or 8 5 4 > a  b > 2 4 < 5 4 > 4 a 3 b > :  4 3 Solving for a and b, we nd a 20 320 5 % 0:40065;  3 b 240 12 2 % 1:24798 4 
We can show that for these values, F a; b is indeed a minimum using the su ciency conditions on Page 188. The polynomial ax2 bx is said to be a least square approximation of sin x over the interval 0;  . The ideas involved here are of importance in many branches of mathematics and their applications.
Supplementary Problems
TANGENT PLANE AND NORMAL LINE TO A SURFACE 8.31. Find the equations of the (a) tangent plane and (b) normal line to the surface x2 y2 4z at 2; 4; 5 . x 2 y 4 z 5 : Ans. (a) x 2y z 5; b 1 2 1 If z f x; y , prove that the equations for the tangent plane and normal line at point P x0 ; y0 ; z0 are given respectively by a z z0 fx jP x x0 fy jP y y0 and b x x0 y y0 z z0 fx jP fy jP 1
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