169 Exercise 4.2.6. 1. Apply Rolle’s theorem to the function f (x) =a1x+a2 x2 2 +· · ·+an xn n on the interval [0,1]. 2. Apply Rolle’s theorem to the function f (x) = n ∑ k=0 sin(2k+1)x 2k+1 on the interval [0,π/2]. Exercise 4.2.8. (a) Given ε >0, first find x0 large enough so that a−ε/2< f ′(x) <a+ε/2 for x >x0. Then use the identity f (x) x = f (x)−f (x0)+f (x0) x−x0 +x0 = f (x)−f (x0) x−x0 +f (x0) x−x0 1+ x0 x−x0 , and the mean value theorem to show that, for x large, a−ε < f (x) x <a+ε. (b) Use the method in part (a). (c) Consider f (x) =sin(x). SECTION 4.3 Exercise 4.3.2. (a) We can prove that f is uniformly continuous onRby definition. Given anyε >0, choose δ = ε ℓ+1 1 α and get | f (u)−f (v)| ≤ℓ|u−v|α < ℓδα =ℓ ε ℓ+1 <ε whenever |u−v| <δ. Note that we use ℓ+1 here instead of ℓ to avoid the case where ℓ =0. (b) We will prove that f is a constant function by showing that it is differentiable onRand f ′(a) =0 for all a∈R. Fix anya∈R. Then, for x̸ =a, f (x)−f (a) x−a ≤ ℓ|x−a|α |x−a| =ℓ|x−a|α−1. Since α>1, by the squeeze theorem, lim x→a f (x)−f (a) x−a =0.
RkJQdWJsaXNoZXIy NTc4NTAz