© 2014 Shmoop University, Inc. All rights reserved.
Derivatives

Derivatives

Example 1

Let f(x) = sin(x). This function is differentiable everywhere. Prove that there is some c in (0, 2π) with f'(c) = 0. By looking at the graph of f, determine how many such values of c there are in (0,2π).

Example 2

For the function f shown below, determine we're allowed to use Rolle's Theorem to guarantee the existence of some c in (a,b) with f'(c) = 0. If not, explain why not.

Example 3

For the function f shown below, determine we're allowed to use Rolle's Theorem to guarantee the existence of some c in (a,b) with f'(c) = 0. If not, explain why not. 

Example 4

For the function f shown below, determine we're allowed to use Rolle's Theorem to guarantee the existence of some c in (a,b) with f'(c) = 0. If not, explain why not.

Example 5

For the function f shown below, determine we're allowed to use Rolle's Theorem to guarantee the existence of some c in (a,b) with f'(c) = 0. If not, explain why not.

Example 6

For each given function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f'(c) = 0. If not, explain why not.

  •  f(x) = x3 on the interval (-2,2)

Example 7

For each given function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f'(c) = 0. If not, explain why not.

  • f(x) = cos(x) on the interval (-π,3π) (yes, cos(x) is differentiable!)

Example 8

For each given function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f'(c) = 0. If not, explain why not. 

  • f(x) = (x-2)2 + 4 on the interval (-2,2)

Example 9

For each given function f and interval, determine if Rolle's Theorem guarantees the existence of some c in that interval with f'(c) = 0. If not, explain why not.

  •  f(x) = \frac{1}{x + 1} on the interval (-1,1).
Advertisement
Noodle's College Search
Advertisement
Advertisement
Advertisement