5 Steps to a 5 AP Calculus BC 2019

90 STEP 4. Review the Knowledge You Need to Score High

6.4 Approximating a Derivative

Given a continuous and differentiable function, you can find the approximate value of a

derivative at a given point numerically. Here are two examples.

Example 1

The graph of a functionf on [0, 5] is shown in Figure 6.4-1. Find the approximate value

off′(3). (See Figure 6.4-1.)

0

1

2

3

4

5

6

7

8

123 4567

y

x

f

Figure 6.4-1

Sincef′(3) is equivalent to the slope of the tangent tof(x)atx=3, there are several ways

you can find its approximate value.

Method 1: Using the slope of the line segment joining the points atx=3 andx=4.

f(3)=3 and f(4)= 5

m=

f(4)− f(3)

4 − 3

=

5 − 3

4 − 3

= 2

Method 2: Using the slope of the line segment joining the points atx=2 andx=3.

f(2)=2 and f(3)= 3

m=

f(3)− f(2)

3 − 2

=

3 − 2

3 − 2

= 1