Derivatives Exercises

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(x) = 2^x, determine whether we're allowed to use Rolle's Theorem to guarantee the existence of some c in (0, 1) with f ' (c) = 0. If not, explain why not.

Example 3

For the function f shown below, determine whether we're allowed to use Rolle's Theorem to guarantee the existence of some c in (-1 ,1) with f ' (c) = 0. If not, explain why not.

(Insert graph of f(x) = 3 for x ≤ -1, f(x) = x² for -1 < x < 1 and f(x) = 3 for x ≥ 1)

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.

(Insert graph of the function f(x) = -2(x-a) for x ≤ a, f(x) = 0 for a < x < b and f(x) = 2(x-b) for x ≥ b)

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.

(Insert graph of f(x) = sin(x) on the interval (0, 2π) On the x-axis, label the origin as a, and then label x = 3π/2 as b.)

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 the 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 the 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