- D Rolle’s Theorem states that if y = f(x) is continuous on the interval [a, b], and is differentiable
everywhere on the interval (a, b), and if f(a) = f(b) = 0, then there is at least one number c
between a and b such that f′(c) = 0. f(x) = 0 at both x = 0 and x = 2. Then, solve f′(c) = 8x^3 − 16
= 0. c = 2.
- A This may appear to be a limit problem, but it is actually testing to see whether you know the
definition of the derivative.
Step 1: You should recall that the definition of the derivative says
Thus, if we replace f (x) with tan (x), we can rewrite the problem as
Step 2: The derivative of tan x is sec^2 x. Thus,
Step 3: Because sec = , sec^2 = .
Note: If you had trouble with this problem, you should review the units on the definition of the
derivative and derivatives of trigonometric functions.
- D Using the Power and Addition Rules, take the derivative of f(x) and you get (D). Remember
that π and e are constants.
- D An absolute maximum or minimum occurs when the derivative of a function is zero or where
the derivative fails to exist or at an endpoint. First, find the derivative of y, set it equal to zero
and solve for x: = 5x^2 − 2x − 7 = 0, then x = −1 and x = . Determine the y-values
corresponding to each of these x-values and at the endpoints, x = −2 and x = 2. The resulting
points are , , , and . The maximum is occurs at
