Exam 2 - 5 Questions on Calculus I | MATH 180, Exams of Calculus

Download Exam 2 - 5 Questions on Calculus I | MATH 180 and more Exams Calculus in PDF only on Docsity! Calculus I (Math 180): Second Hour Exam UIC, April 10, 2009 Instructor: Martin Koerwien You may not use any documents or calculators. You must show all work in your exam booklet to receive full credit. Please return this exam sheet in your booklet. 1. (30 points) Differentiate the following functions. Do not simplify. f(x) = tan−1(ex) g(x) = sin(cos x) h(x) = x ln(x2) 2. (15 points) Evaluate the following limit: lim x→0 ln(1 + x2) sin 2x2 3. (15 points) Consider the curve in the x-y-plane defined by the equation x2 − y3 − 4xy = 1. (a) Show that the point (−1, 2) lies on the curve. (b) Find an equation for the line tangent to the curve at (−1, 2). 4. (20 points) Let f(x) = 2x3 − 3x2 − 12x. (a) Find the critical points and the inflection points of f . (b) On what interval(s) is f concave up/concave down? (c) Find the absolute maximum and minimum of f over the interval [−2, 1]. 5. (20 points) Find two numbers greater or equal to 1 whose sum equals 4 such that the sum of their squares is (a) maximal, (b) minimal.