Review Problems | First Year Interest Group of Seminar | N 1, Study notes of Health sciences

Download Review Problems | First Year Interest Group of Seminar | N 1 and more Study notes Health sciences in PDF only on Docsity! MATH 361K – REVIEW 3 Will not be graded, but will be discussed on Thu, May 7, in class. 1. Problem Assume that f : I = (a, b) → J = (c, d) is differentiable, and f ′(x) 6= 0 for x ∈ (a, b). Furthermore, assume that f is bijective and thus has an inverse, f−1 : J → I. This means that (f ◦ f−1)(y) = y for all y ∈ J , and (f−1 ◦ f)(x) = x for all x ∈ I. Prove that (f−1)′(y) = 1/f ′(f−1(y)) for all y ∈ J , using the chain rule. 2. Problems Assume that f : [a, b] → R is continuous, and differentiable on (a, b). Assume that there exists exactly one point c ∈ (a, b) where f ′(c) = 0, and that f(c) > 0. Moreover, assume that f(a) = 0 = f(b). Determine the absolute maximum and absolute minimum of f on [a, b]. 3. Problem Assume that f : (a, b) → R is differentiable on (a, b). Assume that there exists a continuous function g : (a, b)→ R such that f(x)−f(y) = g(x)(x−y) and g(y) = 2− f ′(y), for some y ∈ (a, b). What can you say about f ′(y) ? 4. Problem (a) Prove that 0 ≤ ln(1 + x) ≤ x for 0 < x < 1. (b) Determine the Taylor remainder term Rn(x) in ln(1 + x) = x− x 2 2 + · · ·+ (−1)n+1 x n n + Rn(x) . Find an upper bound on |Rn(x)| for 0 < x < 0.1 and n = 5. (c) Prove that x− x22 ≤ ln(1 + x) ≤ x− x2 2 + x3 3 for 0 < x < 1. 5. Problems (a) Find ( sin xex ) ′ and sin(ex 2 )′. (b) Find ddx ∫ x 2 arctan(e sin t)dt. 1