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) = x^{3} 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).