Calculus AB and Calculus BC

CHAPTER 3 Differentiation

K. RECOGNIZING A GIVEN LIMIT AS A DERIVATIVE

It is often extremely useful to evaluate a limit by recognizing that it is merely an expression for the definition of the derivative of a specific function (often at a specific point). The relevant definition is the limit of the difference quotient:

Image

EXAMPLE 48

Find Image

SOLUTION: Image is the derivative of f (x) = x4 at the point where x = 2. Since f (x) = 4x3 the value of the given limit is f (2) = 4(23) = 32.

EXAMPLE 49

Find Image

SOLUTION: Image where Image The value of the limit is Image

EXAMPLE 50

Find Image

SOLUTION: Image where Image

Verify that Image and compare with Example 17.

EXAMPLE 51

Find Image

SOLUTION: Image where f (x) = ex. The limit has value e0 or 1 (see also Example 41).

EXAMPLE 52

Find Image

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

Image

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.