Page 331 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes n
d
5. e x sin x is equal to:
dx n
x
n/2
x
(a) 2 n/2 .e cos(x +n /4) (b) 2 .e cos(x n /4)
n/2
x
x
n/2
(c) 2 .e sin (x + n /4) (d) 2 .e sin (x n / 4)
12.6 Review Questions
1. If y = sin (m sin x)
-1
Then prove, (1 x )y (2n + 1)xy + (m n )y = 0
2
2
2
n+2 n+1 n
-1
2. If y = cot x,
Then prove, (1 + x )y + 2(n + 1)xy + n(n + 1)y = 0
2
n+2 n+1 n
3. If y 1/m + y -1/m = 2x
2
2
2
Then prove, (x 1)y + (2n + 1)xy + (n m )y = 0
n+2 n+1 n
3
4. Let p and q be two real numbers with p > 0. Show that the cubic x + px + q has exactly one
real root.
5. Let a > 0 and f be continuous on [ a, a]. Suppose that f’(x) exists and f’(x) 1 for all
x ( a, a). If f(a) = a and f( a) = a, show that f(0) = 0.
2
6. Let f(x) = 1 + 12|x| 3x . Find the global maximum and the global minimum of f on [ 2, 5].
Verify it from the sketch of the curve y = f(x) on [ 2, 5].
Answers: Self Assessment
1. (c) 2. (d) 3. (c)
4. (d) 5. (c)

12.7 Further Readings

Books Husch, Lawrence S. Visual Calculus, University of Tennessee, 2001.
Smith and Minton, Calculus Early Trancendental, Third Edition, McGraw Hill 2008.

Online links http://www.suitcaseofdreams.net/Trigonometric_Functions.htm
http://library.thinkquest.org/20991/alg2/trigi.html
http://www.intmath.com/trigonometric functions/5 signs of trigonometric
functions.php

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