Math E-15 Homework 3

Math E-15 Homework 3 1. Consider the function f ( x )=5 x 3 . Do not round o ff your answers to the following. (a) What is the average rate of change of f on the interval [2, 2.1]? (b) What is the average rate of change of f on the interval [2, 2.01]? (c) What is the average rate of change of f on the interval [2, 2.001]? (d) Use the limit definition of the derivative to compute f ' ( x ). (e) Use your answer to (d) to compute the instantaneous rate of change of f at x = 2. (f) Find the equation of the tangent line to f at x = 2. 2. Let f ( x )= 2 4 - x . (a) Use the limit definition of the derivative to find f ' ( x ). (b) Find the equation of the tangent line to f at x = 1. 3. Let f ( x )=1 - 2 x - 3 x 2 . (a) Use the limit definition of the derivative to find f ' ( x ). (b) At what x -value(s) does f have a horizontal tangent line? 4. Shown below is a graph of f ' on its entire domain. The graph is NOT f . At which labelled x -value(s) (that is, the letters a through k ) (a) is f greatest? (b) is f least? (c) is f ' greatest? (d) is f ' least? (e) is the slope of f ' greatest? (f) is the slope of f ' least? On what interval(s) is (a) f increasing? (b) f decreasing? (c) f ' increasing? (d) f ' decreasing? f ’ (x) d c b a f g h e i j k