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between the z axis and the normal to the surface F x; y; z 0 at any point is q 2 2 2 given by sec
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Fx Fy Fz =jFz j. The equation of a surface is given in cylindrical coordinates by F ; ; z 0, where F is continuously di erentiable. Prove that the equations of (a) the tangent plane and (b) the normal line at the point P 0 ; 0 ; z0 are given respectively by A x x0 B y y0 C z z0 0 and x x0 y y0 z z0 A B C
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where x0 0 cos 0 , y0 0 sin 0 and 1 A F jP cos 0 F jP sin 0 ;  8.35. 1 B F jP sin 0 F jP cos 0 ;  C Fz jP
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Use Problem 8.34 to nd the equation of the tangent plane to the surface z  at the point where  2,  =2, z 1. To check your answer work the problem using rectangular coordinates. Ans. 2x y 2z 0
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TANGENT LINE AND NORMAL PLANE TO A CURVE 8.36. Find the equations of the (a) tangent line and (b) normal plane to the space curve x 6 sin t, y 4 cos 3t, z 2 sin 5t at the point where t =4. p p p p x 3 2 y 2 2 z 2 Ans: a b 3x 6y 5z 26 2 3 6 5 The surfaces x y z 3 and x2 y2 2z2 2 intersect in a space curve. (a) tangent line (b) normal plane to this space curve at the point 1; 1; 1 . Ans: a x 1 y 1 z 1 ; 3 1 2 b 3x y 2z 0 Find the equations of the
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ENVELOPES 8.38. Find the envelope of each of the following families of curves in the xy plane. In each case construct a graph. x2 y2 (a) y x 2 ; b 1. 1 Ans. (a) x2 4y; 8.39. b x y 1; x y 1
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Find the envelope of a family of lines having the property that the length intercepted between the x and y axes is a constant a. Ans. x2=3 y2=3 a2=3 Find the envelope of the family of circles having centers on the parabola y x2 and passing through its vertex. [Hint: Let ; 2 be any point on the parabola.] Ans. x2 y3 = 2y 1 Find the envelope of the normals (called an evolute) to the parabola y 1 x2 and construct a graph. 2 Ans. 8 y 1 3 27x2 Find the envelope of the following families of surfaces: a x y 2 z 1; b x 2 y2 2 z Ans. a 4z x y 2 ; b y2 z2 2xz Prove that the envelope of the two parameter family of surfaces F x; y; z; ; 0, if it exists, is obtained by eliminating and in the equations F 0; F 0; F 0. Find the envelope of the two parameter families (a) z x y 2 2 and (b) x cos y cos z cos
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a where cos2 cos2 cos2
1 and a is a constant. Ans. a 4z x2 y2 ; b x2 y2 z2 a2
DIRECTIONAL DERIVATIVES 8.45. (a) Find the directional derivative of U 2xy z2 at 2; 1; 1 in a direction toward 3; 1; 1 . (b) In what direction is the directional derivative a maximum (c) What is the value of this maximum p Ans. a 10=3; b 2i 4j 2k; c 2 6
CHAP. 8]
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The temperature at any point x; y in the xy plane is given by T 100xy= x2 y2 . (a) Find the directional derivative at the point 2; 1 in a direction making an angle of 608 with the positive x-axis. (b) In what direction p from 2; 1 would the derivative be a maximum (c) What is the value of this maximum Ans. (a) 12 3 6; (b)p a direction making an angle of  tan 1 2 with the positive x-axis, or in the in direction i 2j; (c) 12 5 Prove that if F ; ; z is continuously di erentiable, the maximum directional derivative of F at any point is s      @F 2 1 @F 2 @F 2 2 . given by @ @z  @
DIFFERENTIATION UNDER THE INTEGRAL SIGN 1= 1= d 1 1 1 Ans. p x2 sin x2 dx 2 cos p cos 2 8.48. If  p cos x2 dx, nd . d 2 8.49. (a) If F Ans. x dF dx, nd by Leibnitz s rule. (b) Check the result in (a) by direct integration. d 1 2 1 a 2 tan 2 ln 1 tan 1
2
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