Page 97 - Schaum's Outline of Theory and Problems of Advanced Calculus

88 DERIVATIVES [CHAP. 4

2
2
4.67. Verify Rolle’s theorem for f ðxÞ¼ x ð1 xÞ ,0 @ x @ 1.
x
x
4.68. Prove that between any two real roots of e sin x ¼ 1there is at least one real root of e cos x ¼ 1. [Hint:
Apply Rolle’s theorem to the function e x sin x:Š
4.69. (a)If 0 < a < b,prove that ð1 a=bÞ < ln b=a < ðb=a 1Þ
1
1
(b)Use the result of (a)toshow that < ln 1:2 < .
6
5
p ffiffiffi 1
4.70. Prove that ð =6 þ 3=15Þ < sin :6 < ð =6 þ 1=8Þ by using the mean value theorem.
4.71. Show that the function FðxÞ in Problem 4.20(a) represents the difference in ordinants of curve ACB and line
AB at any point x in ða; bÞ.
4.72. (a)If f ðxÞ @ 0atall points of ða; bÞ,prove that f ðxÞ is monotonic decreasing in ða; bÞ.
0
(b) Under what conditions is f ðxÞ strictly decreasing in ða; bÞ?
4.73. (a)Prove that ðsin xÞ=x is strictly decreasing in ð0; =2Þ. (b)Prove that 0 @ sin x @ 2x= for
0 @ x @ =2.
sin b sin a
4.74. (a)Prove that ¼ cot , where is between a and b.
cos a cos b
(b)By placing a ¼ 0 and b ¼ x in (a), show that ¼ x=2. Does the result hold if x < 0?
L’HOSPITAL’S RULE
4.75. Evaluate each of the following limits.
x sin x x þ 3
3
(a) lim (e) lim x ln x (i) limð1=x csc xÞ (m) lim x ln
x!0 x 3 x!0þ x!0 x!1 x 3
2
2x
1=x
x
e 2e þ 1 sin x

x
x
(b) lim ( f ) limð3 2 Þ=x ( j) lim x sin x (n) lim
x!0 cos 3x 2cos 2x þ cos x x!0 x!0 x!0 x
2
2x 1=x
x
2
2
(c) lim ðx 1Þ tan x=2 (g) lim ð1 3=xÞ 2x ðkÞ limð1=x cot xÞ (o) lim ðx þ e þ e Þ
x!0
x!1þ x!1 x!1
tan 1 x sin 1 x
3 2x
(d) lim x e (h) lim ð1 þ 2xÞ 1=3x (l) lim (p) lim ðsin xÞ 1= ln x
x!0
x!1 x!1 xð1 cos xÞ x!0þ
6
Ans.(a) 1 6 ; ðbÞ 1; ðcÞ 4= ; ðdÞ 0; ðeÞ 0; ð f Þ ln 3=2; ðgÞ e ; ðhÞ 1; ðiÞ 0; ð jÞ 1,
2
(k) 2 ; 1 ; ðmÞ 6; ðnÞ e 1=6 ; ðoÞ e ; ð pÞ e
3 ðlÞ 3
MISCELLANEOUS PROBLEMS
r ffiffiffiffiffiffiffiffiffiffiffi
1 x
4.76. Prove that < lnð1 þ xÞ < 1if0 < x < 1.
1 þ x sin 1 x
2
4.77. If f ðxÞ¼ f ðx þ xÞ f ðxÞ, (a)Prove that f f ðxÞg ¼ f ðxÞ¼ f ðx þ 2 xÞ 2f ðx þ xÞþ f ðxÞ,
n
n
(b)derive an expression for f ðxÞ where n is any positive integer, (c)show that lim f ðxÞ ¼ f ðnÞ ðxÞ
n
x!0 ð xÞ
if this limit exists.
4.78. Complete the analytic proof mentioned at the end of Problem 4.36.
2
4.79. Find the relative maximum and minima of f ðxÞ¼ x , x > 0.
1
Ans. f ðxÞ has a relative minimum when x ¼ e .
3
4.80. A train moves according to the rule x ¼ 5t þ 30t, where t and x are measured in hours and miles,
respectively. (a) What is the acceleration after 1 minute? (b) What is the speed after 2 hours?
2
4.81. A stone thrown vertically upward has the law of motion x ¼ 16t þ 96t. (Assume that the stone is at
ground level at t ¼ 0, that t is measured in seconds, and that x is measured in feet.) (a) What is the height of
the stone at t ¼ 2 seconds? (b)To what height does the stone rise? (c) What is the initial velocity, and
what is the maximum speed attained?