Introduction to Mathematical Analysis I - Second Edition

106 4.1 DEFINITION AND BASIC PROPERTIES OF THE DERIVATIVE (a) Determine the values of α for which f is continuous on R . (b) Determine the values of α for which f is differentiable on R . In this case, find f 0 . 4.1.5 Use Theorems 4.1.3 and 4.1.5 to compute the derivatives of the following functions at the indicated points (see also Example 4.1.4 ) . (Assume known that the function sin x is differentiable at all points and that its derivative is cos x .) (a) f ( x ) = 3 x 4 + 7 x 2 x 2 + 3 at a = − 1. (b) f ( x ) = sin 5 ( 3 x 2 + π 2 x ) at a = π 8 4.1.6 Determine the values of x at which each function is differentiable. (a) f ( x ) =   x sin 1 x , if x 6 = 0; 0 , if x = 0 . (b) f ( x ) =   x 2 sin 1 x , if x 6 = 0; 0 , if x = 0 . 4.1.7 Determine if each of the following functions is differentiable at 0. Justify your answer. (a) f ( x ) = ( x 2 , if x ∈ Q ; x 3 , if x / ∈ Q . (b) f ( x ) = [ x ] sin 2 ( π x ) . (c) f ( x ) = cos ( p | x | ) . (d) f ( x ) = x | x | . 4.1.8 Let f , g be differentiable at a . Find the following limits: (a) lim x → a x f ( a ) − a f ( x ) x − a . (b) lim x → a f ( x ) g ( a ) − f ( a ) g ( x ) x − a . 4.1.9 Let G be an open subset of R and a ∈ G . Prove that if f : G → R is Lipschitz continuous, then g ( x ) = ( f ( x ) − f ( a )) 2 is differentiable at a . 4.1.10 B Let f be differentiable at a and f ( a ) > 0. Find the following limit: lim n → ∞ f ( a + 1 n ) f ( a ) ! n . 4.1.11 I Consider the function f ( x ) =   x 2 sin 1 x + cx , if x 6 = 0; 0 , if x = 0 ,