MATH 135 – Calculus 1 The Derivative Function – Practice Problems October 4, 2013

1. Consider the function f(x) = \frac{x^{3}}{3} − x. a. Compute f′(x) = lim_{h→0} \frac{f(x+h)−f(x)}{h}.

b. Here is a plot of y = f(x). On the empty set of axes below plot y = f′(x).

c. Circle the correct answers: On the interval (−∞, −1), the derivative f′(x) is positive/negative and f(x) is increasing/decreasing. On the interval (−1, 1), the derivative f′(x) is positive/negative and f(x) is increasing/decreasing. On the interval (1, +∞), the derivative f′(x) is positive/negative and f(x) is increasing/decreasing.

d. What happens on y = f(x) when f′(x) = 0? e. (Harder – we have not discussed this!) What happens on the graph y = f(x) then the

tangent line to y = f′(x) is horizontal?

2. Given the graph y = f(x) below, make a qualitative sketch of y = f′(x) on the empty set of axes below. Show any places where f′(x) changes sign and places where f′(x) fails to exist.