Barrons AP Calculus - David Bock

SOLUTION: is the derivative of f (x) = x^4 at the point where x = 2. Since f ′(x) =

4 x^3 the value of the given limit is f ′(2) = 4(2^3 ) = 32.

EXAMPLE 49

Find

SOLUTION: where The value of the limit is

EXAMPLE 50

Find

SOLUTION: where

Verify that and compare with Example 17.

EXAMPLE 51

Find

SOLUTION: where f (x) = ex. The limit has value e^0 or 1 (see also Example 41).

EXAMPLE 52

Find

SOLUTION: is f ′(0), where f (x) = sin x, because we can write

The answer is 1, since f ′(x) = cos x and f ′(0) = cos 0 = 1. Of course, we already know that the

given limit is the basic trigonometric limit with value 1. Also, L’Hôpital’s Rule yields 1 as the

answer immediately.

Chapter Summary

In this chapter we have reviewed differentiation. We’ve defined the derivative as the instantaneous
rate of change of a function, and looked at estimating derivatives using tables and graphs. We’ve
reviewed the formulas for derivatives of basic functions, as well as the product, quotient, and chain
rules. We’ve looked at derivatives of implicitly defined functions and inverse functions, and
reviewed two important theorems: Rolle’s Theorem and the Mean Value Theorem.
For BC Calculus students, we’ve reviewed derivatives of parametrically defined functions and
the use of L’Hôpital’s Rule for evaluating limits of indeterminate forms.