94 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 ′. 4.1.5 Use Theorems 4.1.2 and 4.1.3 to compute the derivatives of the following functions at the indicated points (see also Example 4.1.4). (Assume known that the function sinx is differentiable at all points and that its derivative is cosx.) (a) f (x) = 3x4 +7x 2x2 +3 at x0 =−1. (b) f (x) =sin5(3x2 +π 2x) at x0 = π 8. 4.1.6 Determine the values of x at which each function is differentiable. (a) f (x) = xsin 1 x , if x̸ =0; 0, if x =0. (b) f (x) = x2sin 1 x , if x̸ =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) =( x2, if x ∈Q; x3, if x / ∈Q. (b) f (x) = [x]sin2(πx). (c) f (x) =cos(p|x|). (d) f (x) =x|x|. 4.1.8 Let f,gbe differentiable at x0. Find the following limits: (a) limx→x0 x f (x0)−x0 f (x) x−x0 . (b) limx→x0 f (x)g(x0)−f (x0)g(x) x−x0 . 4.1.9 Let I be an open interval of Randx0 ∈I. Prove that if f : I →Ris Lipschitz continuous, then g(x) = (f (x)−f (x0)) 2 is differentiable at x 0. 4.1.10 ▷Let f be differentiable at x0 and f (x0) >0. Find the following limit: lim n→∞ f (x0 + 1 n) f (x0) ! n . 4.1.11 ▶Consider the function f (x) = x2sin 1 x +cx, if x̸ =0; 0, if x =0,
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