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# Problems Sheet 2 For tutorials: 6), 8), 11) (ii), 14) i) 1. Show that f (x) as x for each of the following cases: (a) f (x) = x3 . (b) f (x) = x2 + x...

Mathematical Analysis
Problems Sheet 2 For tutorials: 6), 8), 11) (ii), 14) i) 1. Show that f ( x ) →∞ as x →∞ for each of the following cases: (a) f ( x )= x 3 . (b) f ( x )= x 2 + x cos x . (c) f ( x )= x log( x ). 2. For each of the following functions decide whether f ( x ) tends to a limit as x →∞ . When the limit exits ±nd it. (a) f ( x )= x 3 x 2 +1 . (b) f ( x )= x sin x x 2 +1 . (c) f ( x )= 1 - x 1+ x . 3. For each of the following functions and points a , decide whether f ( x ) tends to a limit as x a . When the limit exists ±nd it. (a) f ( x )= 2 x 2 +1 3 x 2 +3 x +1 ,a = 0. (b) f ( x )= ( x 2 +1) sin x x ,a = 0. (c) f ( x )= sin x x 2 ,a = 0. 4. Decide whether the following functions are continuous at a = 0: 1) f ( x )= x 2 if x< 0 and f ( x ) = sin x if x 0. 2) f ( x )= ± | x | sin 1 x if x 0 and f (0) = 0. 5. Discuss the continuity of the function f de±ned for all x [0 , 1] by f ( x )= x if x is rational, and f ( x )= x 2 if x is irrational. 6. Let f be continuous on [ a, b ], and suppose that f ( c ) ± = 0 for some c [ a, b ]. Show that there exists a δ > 0 with the property that f ( x ) ± =0 for all x such that | x - c | < δ . 7. Let f be a continuous function with domain and image [ a, b ]. By consid- ering the function g ( x )= f ( x ) - x , show that there exists c [ a, b ] such that f ( c )= c . Deduce that the equation x +1 3 = sin πx 2 has a solution 0 < x < 1. 8. Let f and g be continuous functions on [0 , 1], and suppose that f (0) < g (0), f (1) >g (1). Show that there exists 0 < x < 1 such that f ( x )= g ( x ). 9. a) Show that the equation 2 sin x = x 2 - 1 has a solution 1 < x < 2. b) Show that the equation xe x = 1 has a solution 0 < x < 1. Prove that this solution is unique. 10. Determine the inverse of the function f ( x )= 1 1 - x . What is its domain? 11. Find the derivatives of the following functions f . In each case state explicitely the values of x for which the formula for f ± is valid. (i) f ( x )= e x sin x . (ii) f ( x ) = cos[log(1 + x 2 )]. (iii) x log( x ) , x > 0. 12. Prove that the equation x 3 - 4 x 2 + cos x has one and only one solution between 0 and 1. 1
13. Prove that if 0 < a < b < π 2 , then ( b - a ) cos b< sin b - sin a< ( b - a ) cos a 14. Evaluate the following limits: i) lim x 0 e x 2 - 1 sin x 2 . ii) lim x 0 cos x - 1 x 2 . iii) lim x 1 ( x - 1) 3 log x . 15. Let f be continuous and diFerentialble on [ a, b ], and suppose that f attains its maximum and minimum at points c and d , respectively, where c, d [ a, b ]. (i) Show that f ± ( c ) = 0. (ii) Show that f ± ( d ) = 0 16. Compute the derivatives of the following functions: 1) f :[ - 1 , 1] [ - π 2 , π 2 ] given by f ( x ) = sin - 1 ( x ). 2) g :[ - 1 , 1] [0 , π ] given by g ( x )= cos - 1 ( x ). 3) h : R ( - π 2 , π 2 ) given by h ( x ) = tan - 1 ( x ). 17. Let f : R (0 , ) have the property that f ± ( x )= f ( x ) for all x . Show that f is an increasing function for all x . Show also that ( f - 1 ) ± ( x )= 1 x . 18. ±ind the radius of convergence, and the open interval of convergence for each of the following power series: 1) n =1 x n 3 n , 2) n =1 3 n x n n ! , 3) n =1 ( 3 n 1+2 n ) x 2 n 4) n =1 ( - 1) n x 2 n +1 (2 n +1)! . 5) n =1 ( x - 1) n n n . 6) n =1 n n x n n ! . 19. ±ind the interval of convergence of n =1 a n x n , where: 1) a n = 2 n n +1 , 2) a n = n ! ( n +1) n , 3) a n = ( n !) 2 (2 n )! . 2

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9b)
To prove: xex – 1 =0, has unique solution and it belongs to (0, 1).
Let, f(x) = xex – 1,
Case(i):
Let x&lt;0, then it is clear that f(x)&lt;-1, as ex &gt;0.
Hence f(x) cannot be zero if...

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