3.1 Defining the Derivative
Vocabulary Examples

Difference Quotient

For a function f , the difference quotient Q is:

Q =

Alternately, for h * 0, Q =

Slope of a

Tangent Line

mtan =

Alternately, for h * 0, mtan =

Derivative of a

Function at a Point

The derivative of f (x) at a, denoted , is defined:

f ,(a) =

Or f ,(a) =

Instantaneous

Rate of Change

The instantaneous rate of change of a function f (x) at a is its

1. For each of the following functions, determine the slope of the secant line between x1 and x2. (a) f (x) = 4x + 7, x1 = 2, x2 = 5

Name:

Defining the Derivative

Section:

For use with OpenStax Calculus, free at https://openstax.org/details/books/calculus-volume-1

25

(b) f (x) =

x x+3

, x1 = 0, x2 = 3

2. For each of the following functions, determine the f ,(a)

(a) f (x) = 2×2 ? x, a = 4

(b) f (x) = ?x ? 7, a = 10

3. For each of the following functions f , write the equation for the line tangent to f at x = a

(a) f (x) = 1 x5 + 2x at a = 1

3

(b) f (x) =

x

?4 at a = 2

(c) f (x) = 54 + 5 at a = ?3

x3

4. Recall that the velocity of a moving object is instantaneous rate of change of its position. A projectile’s position d at time t is given by the function d(t) = ?4.9t2 + 20.1x + 24.3.

(a) Determine the velocity of the object after 2 seconds.

(b) Determine the velocity of the object after 3 seconds.

3.2 Derivative as a Function
Vocabulary Examples

Derivative

Function

For a function f , the derivative function, denoted , is the function whose domain consists of values of x such that the following limit exists:

f ,(x) =

Notations:

Theorem on

Differentiabil- ity and Continuity

If a function f is differentiable at a, then f is at a.

Higher-Order

Derivative

The of a

1. Use the definition of a derivative to determine the derivative of the following functions. (a) f (x) = 3×2 ? 2

(b) f (x) = x?2

(c) f (x) = ?3x ? 7

(d) f (x) = 3