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DERIVATIVES
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Verify Rolle s theorem for f x x2 1 x 2 , 0 @ x @ 1. Prove that between any two real roots of ex sin x 1 there is at least one real root of ex cos x 1. [Hint: Apply Rolle s theorem to the function e x sin x: (a) If 0 < a < b, prove that 1 a=b < ln b=a < b=a 1 (b) Use the result of (a) to show that 1 < ln 1:2 < 1. 6 5 p Prove that =6 3=15 < sin 1 :6 < =6 1=8 by using the mean value theorem. Show that the function F x in Problem 4.20(a) represents the di erence in ordinants of curve ACB and line AB at any point x in a; b . (a) If f 0 x @ 0 at all points of a; b , prove that f x is monotonic decreasing in a; b . (b) Under what conditions is f x strictly decreasing in a; b (a) Prove that sin x =x is strictly decreasing in 0; =2 . (b) Prove that 0 @ sin x @ 2x= for 0 @ x @ =2. sin b sin a cot , where  is between a and b. (a) Prove that cos a cos b (b) By placing a 0 and b x in (a), show that  x=2. Does the result hold if x < 0
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L HOSPITAL S RULE 4.75. Evaluate each of the following limits. x sin x (a) lim (e) lim x3 ln x x!0 x!0 x3 (b) (c) (d) lim e 2e 1 cos 3x 2 cos 2x cos x
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 (i) lim 1=x csc x
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(m) lim x ln
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 ( f ) lim 3x 2x =x
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( j) lim xsin x
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(n) lim
sin x x
 x 3 x 3 1=x2
x!1
lim x2 1 tan x=2 lim x3 e 2x
(g) lim 1 3=x 2x
k lim 1=x2 cot2 x (o) lim x ex e2x 1=x
x!0 x!1
(h) lim 1 2x 1=3x (l) lim
tan x sin x x 1 cos x g e ;
(p) lim sin x 1= ln x
x!0
Ans. (a) 1 ; b 1; 6 (k) 2 ; l 1 ; m 6; 3 3
c 4=; d 0; e 0; n e 1=6 ; o e2 ; p e
f ln 3=2;
h 1;
i 0;
j 1,
MISCELLANEOUS PROBLEMS r 1 x ln 1 x < < 1 if 0 < x < 1. 4.76. Prove that 1 x sin 1 x 4.77. (a) Prove that f f x g 2 f x f x 2 x 2f x x f x , n f x n (b) derive an expression for n f x where n is any positive integer, (c) show that lim n f x x!0 x if this limit exists. If f x f x x f x , Complete the analytic proof mentioned at the end of Problem 4.36. Find the relative maximum and minima of f x x2 , x > 0. Ans. f x has a relative minimum when x e 1 . A train moves according to the rule x 5t3 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 A stone thrown vertically upward has the law of motion x 16t2 96t. (Assume that the stone is at ground level at t 0, that t is measured in seconds, and that x is measured in feet.) (a) What is the height of the stone at t 2 seconds (b) To what height does the stone rise (c) What is the initial velocity, and what is the maximum speed attained
4.78. 4.79.
CHAP. 4]
DERIVATIVES
A particle travels with constant velocities v1 and v2 in mediums I and II, respectively (see adjoining Fig. 4-11). Show that in order to go from point P to point Q in the least time, it must follow path PAQ where A is such that sin 1 = sin 2 v1 =v2 Note: This is Snell s Law; a fundamental law of optics rst discovered experimentally and then derived mathematically.