Page 95 - Schaum's Outline of Theory and Problems of Advanced Calculus

86 DERIVATIVES [CHAP. 4

Supplementary Problems

DERIVATIVES
4.37. Use the definition to compute the derivatives of each of the following functions at the indicated point:
2
3
3
(a) ð3x 4Þ=ð2x þ 3Þ; x ¼ 1; ðbÞ x 3x þ 2x 5; x ¼ 2; ðcÞ p x; x ¼ 4; ðdÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6x 4; x ¼ 2:
ffiffiffi
1
Ans.(a) 17/25, (b)2, (c) ,(d) 1
4 2
d d 3 þ x 6
4
3
4.38. Show from definition that (a) x ¼ 4x ; ðbÞ ¼ ; x 6¼ 3
dx dx 3 x 2
ð3 xÞ
3
x sin 1=x; x 6¼ 0
4.39. Let f ðxÞ¼ .Prove that (a) f ðxÞ is continuous at x ¼ 0, (b) f ðxÞ has a derivative at
0; x ¼ 0
x ¼ 0, (c) f ðxÞ is continuous at x ¼ 0.
0
1=x 2
xe ; x 6¼ 0
4.40. Let f ðxÞ¼ . Determine whether f ðxÞ (a)iscontinuous at x ¼ 0, (b) has a derivative at
0; x ¼ 0
x ¼ 0:
Ans.(a)Yes; (b)Yes, 0
4.41. Give an alternative proof of the theorem in Problem 4.3, Page 76, using ‘‘ ; definitions’’.
h
x
4.42. If f ðxÞ¼ e ,show that f ðx 0 Þ¼ e x 0 depends on the result limðe 1Þ=h ¼ 1.
0
h!0
4.43. Use the results limðsin hÞ=h ¼ 1, limð1 cos hÞ=h ¼ 0toprove that if f ðxÞ¼ sin x, f ðx 0 Þ¼ cos x 0 .
0
h!0 h!0
RIGHT- AND LEFT-HAND DERIVATIVES
4.44. Let f ðxÞ¼ xjxj. (a) Calculate the right-hand derivative of f ðxÞ at x ¼ 0. (b) Calculate the left-hand
derivative of f ðxÞ at x ¼ 0. (c) Does f ðxÞ have a derivative at x ¼ 0? (d) Illustrate the conclusions in ðaÞ,
(b), and (c) from a graph.
Ans.(a)0; (b)0; (c)Yes, 0
p
4.45. Discuss the (a)continuity and (b)differentiability of f ðxÞ¼ x sin 1=x, f ð0Þ¼ 0, where p is any positive
number. What happens in case p is any real number?

2x 3; 0 @ x @ 2
4.46. Let f ðxÞ¼ 2 . Discuss the (a)continuity and (b)differentiability of f ðxÞ in
x 3; 2 < x @ 4
0 @ x @ 4.
4.47. Prove that the derivative of f ðxÞ at x ¼ x 0 exists if and only if f þ ðx 0 Þ¼ f ðx 0 Þ.
0
0
2
3
4.48. (a)Prove that f ðxÞ¼ x x þ 5x 6isdifferentiable in a @ x @ b, where a and b are any constants.
2
3
(b)Find equations for the tangent lines to the curve y ¼ x x þ 5x 6at x ¼ 0 and x ¼ 1. Illustrate
by means of a graph. (c)Determine the point of intersection of the tangent lines in (b). (d)Find
f ðxÞ; f ðxÞ; f ðxÞ; f ðIVÞ ðxÞ; .. . .
000
0
00
2
Ans.(b) y ¼ 5x 6; y ¼ 6x 7; ðcÞð1; 1Þ; ðdÞ 3x 2x þ 5; 6x 2; 6; 0; 0; 0; ...
2
4.49. If f ðxÞ¼ x jxj,discuss the existence of successive derivatives of f ðxÞ at x ¼ 0.
DIFFERENTIALS
4.50. If y ¼ f ðxÞ¼ x þ 1=x, find (a) y; ðbÞ dy; ðcÞ y dy; ðdÞð y dyÞ= x; ðeÞ dy=dx.
2
x 1 x 1
Ans: ðaÞ x ; ðbÞ 1 x; ðcÞ ð xÞ ; ðdÞ ; ðeÞ 1 :
x 2 2 2 x 2
xðx þ xÞ x ðx þ xÞ x ðx þ xÞ
Note: x ¼ dx.