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p ! 1 1 2 ln 1 cos x dx  ln ; j j < 1. 2 & ln 1 2 cos x 2 dx dx 59 : 3 2048 5 3 cos x  ln 2 ; j j < 1 . 0; j j > 1 Discuss the case j j 1.
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INTEGRATION UNDER THE INTEGRAL SIGN ' ' 1 & 2 2 & 1 8.54. Verify that 2 x2 dx d 2 x2 d dx.
0 1 1 0
2 8.55. Starting with the result
sin x dx 2 , prove that for all constants a and b, 2 f b sin x 2 a sin x 2 g dx 2 b2 a2
2 8.56. Use the result
dx 2 p ; > 1 to prove that sin x 2 1    2  5 3 sin x 9 dx 2 ln ln 5 4 sin x 8 0 =2 dx cos 1 p ; 0 @ < 1 to show that for 0 @ a < 1; 0 @ b < 1 1 cos x 1 2   =2 1 b cos x sec x ln dx 1 f cos 1 a 2 cos 1 b 2 g 2 1 a cos x 0 52 . 72
(a) Use the result
=2 (b) Show that
sec x ln 1 1 cos x dx 2
APPLICATIONS OF PARTIAL DERIVATIVES
[CHAP. 8
MAXIMA AND MINIMA, LAGRANGE MULTIPLIERS 8.58. Find the maxima and minima of F x; y; z xy2 z3 subject to the conditions x y z 6, x > 0; y > 0, z > 0. Ans. maximum value 108 at x 1; y 2; z 3 What is the volume of the largest rectangular parallelepiped which can be inscribed in the ellipsoid p x2 =9 y2 =16 z2 =36 1 Ans. 64 3 (a) Find the maximum and minimum values of x2 y2 subject to the condition 3x2 4xy 6y2 140. (b) Give a geometrical interpretation of the results in (a). Ans. maximum value 70, minimum value 20 Solve Problem 8.23 using Lagrange multipliers. Prove that in any triangle ABC there is a point P such that PA PB PC is a minimum and that P is the intersection of the medians. (a) Prove that the maximum and minimum values of f x; y x2 xy y2 in the unit square 0 @ x @ 1, 0 @ y @ 1 are 3 and 0, respectively. (b) Can the result of (a) be obtained by setting the partial derivatives of f x; y with respect to x and y equal to zero. Explain. Find the extreme values of z on the surface 2x2 3y2 z2 12xy 4xz 35. Ans. maximum 5, minimum 5 Establish the method of Lagrange multipliers in the case where we wish to nd the extreme values of F x; y; z subject to the two constraint conditions G x; y; z 0, H x; y; z 0. Prove that the shortest distance p from the origin to the curve of intersection of the surfaces xyz a and y bx where a > 0; b > 0, is 3 a b2 1 =2b. Find the volume of the ellipsoid 11x2 9y2 15z2 4xy 10yz 20xz 80. Ans. p 64 2=3
2 2 2
8.61. 8.62.
APPLICATIONS TO ERRORS 8.68. The diameter of a right circular cylinder is measured as 6:0 0:03 inches, while its height is measured as 4:0 0:02 inches. What is the largest possible (a) error and (b) percent error made in computing the volume Ans. (a) 1.70 in3 , (b) 1.5% The sides of a triangle are measured to be 12.0 and 15.0 feet, and the included angle 60.08. If the lengths can be measured to within 1% accuracy, while the angle can be measured to within 2% accuracy, nd the maximum error and percent error in determining the (a) area and (b) opposite side of the triangle. Ans. (a) 2.501 ft2 , 3.21%; (b) 0.287 ft, 2.08%
MISCELLANEOUS PROBLEMS 8.70. If  and  are cylindrical coordinates, a and b are any positive constants, and n is a positive integer, prove that the surfaces n sin n a and n cos n b are mutually perpendicular along their curves of intersection. Find an equation for the (a) tangent plane and (b) normal line to the surface 8r 2 at the point where r 1,  =4;  =2; r; ;  being spherical coordinates. p p p x y 2=2 z 2=2 Ans: a 4x 2 4 y 4 2 z 2 2; b 2 2 4  4  4
CHAP. 8]
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