advertisement

ASSIGNMENT 15 for SECTION 001 This assignment is to be handed in. There are two parts: Part A and Part B. Part A will be graded for completeness. You will receive full marks only if every question has been completed. Part B will be graded for correctness, clarity and comprehensiveness. In both parts, you must show your work. Please submit Part A and Part B separately, with your name on each part. Part A From Calculus: Early Transcendentals: From section 4.3, complete questions: 2, 6, 8, 10, 12, 14, 16, 34, 36, 38, 40, 44, 46, 48, 50, 80 Part B 1. Determine if each of the following statements is true. If so, explain why; if not, provide a counterexample. 1. (a) The sum of two concave up functions is concave up. 1. (b) A function which is increasing and concave down everywhere must have a horizontal asymptote. 2. Find the critical points and inflection points for f (x) = x2 −x2 if x < 0 . if x ≥ 0 3. Find the number of inflection points for f (x) = x2 + cos(kx), where k is a real number. (There is more than one possibility.) 4. (a) Find a and b such that f (x) = ax3 + bx2 + 1 has an inflection point at (−1, 2). 4. (b) Using the constants a and b determined in part (a), sketch the graph of f (x). 5. Let f (x) have an inflection point at 0. Prove that the curve y = f (x) crosses the tangent line at 0. (Hint: let g(x) be the vertical distance between the function and the tangent line. What do the graphs of g 0 (x) and g(x) look like around 0?)