Page 211 - Schaum's Outline of Theory and Problems of Advanced Calculus

202 APPLICATIONS OF PARTIAL DERIVATIVES [CHAP. 8

where x 0 ¼ 0 cos 0 , y 0 ¼ 0 sin 0 and
1 1
A ¼ F j P cos 0 F j P sin 0 ; B ¼ F j P sin 0 þ F j P cos 0 ; C ¼ F z j P

8.35. Use Problem 8.34 to find the equation of the tangent plane to the surface z ¼ at the point where ¼ 2,
¼ =2, z ¼ 1. To check your answer work the problem using rectangular coordinates.
Ans.2x y þ 2 z ¼ 0
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 ¼ 6 sin t, y ¼ 4cos 3t,
z ¼ 2 sin 5t at the point where t ¼ =4.
p ffiffiffi p ffiffiffi p ffiffiffi
x 3 2 y þ 2 2 2 p ffiffiffi
Ans: z þ 3x 6y 5z ¼ 26 2
3 ¼ 6 ¼ 5 ðbÞ
ðaÞ
2
2
2
8.37. The surfaces x þ y þ z ¼ 3 and x y þ 2z ¼ 2intersect in a space curve. Find the equations of the
(a)tangent line (b) normal plane to this space curve at the point ð1; 1; 1Þ.
x 1 y 1 z 1
Ans: ðaÞ ¼ ¼ ; ðbÞ 3x y 2z ¼ 0
3 1 2
ENVELOPES
8.38. Find the envelope of each of the following families of curves in the xy plane. In each case construct a graph.
x 2 y 2
2
(a) y ¼ x ; ðbÞ þ ¼ 1.
1
2
Ans.(a) x ¼ 4y; ðbÞ x þ y ¼ 1; x y ¼ 1
8.39. 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. x 2=3 þ y 2=3 ¼ a 2=3

2
8.40. Find the envelope of the family of circles having centers on the parabola y ¼ x and passing through its
3
2
2
vertex. [Hint: Let ð ; Þ be any point on the parabola.] Ans. x ¼ y =ð2y þ 1Þ
2
1
8.41. Find the envelope of the normals (called an evolute)tothe parabola y ¼ x and construct a graph.
2
3
Ans.8ðy 1Þ ¼ 27x 2
8.42. Find the envelope of the following families of surfaces:
2
2
2
ðx yÞ z ¼ 1; ðx Þ þ y ¼ 2 z
ðaÞ ðbÞ
2
2
2
Ans. ðaÞ 4z ¼ðx yÞ ; ðbÞ y ¼ z þ 2xz
8.43. Prove that the envelope of the two parameter family of surfaces Fðx; y; z; ; Þ¼ 0, if it exists, is obtained by
eliminating and in the equations F ¼ 0; F ¼ 0; F ¼ 0.
2
2
8.44. Find the envelope of the two parameter families (a) z ¼ x þ y and (b) x cos þ y cos þ
2
2
2
z cos
¼ a where cos þ cos þ cos
¼ 1 and a is a constant.
2
2
2
2
2
Ans. ðaÞ 4z ¼ x þ y ; ðbÞ x þ y þ z ¼ a 2
DIRECTIONAL DERIVATIVES
2
8.45. (a)Find the directional derivative of U ¼ 2xy z at ð2; 1; 1Þ in a direction toward ð3; 1; 1Þ.(b)In what
direction is the directional derivative a maximum? (c) What is the value of this maximum?
p
Ans. ðaÞ 10=3; ðbÞ 2i þ 4j 2k; ðcÞ 2 6 ffiffiffi