5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 17:21

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

The graph indicates that (1) f(5)=0; (2) f′(5) < 0, since f is decreasing; and

(3)f′′(5)>0, since fis concave upward. Thus, f′(5)< f(5)<f′′(5), choice (C).

TIP • Move on. Do not linger on a problem too long. Make an educated guess. You can earn

many more points from other problems.

8.3 Sketching the Graphs of Functions

Main Concepts:Graphing without Calculators, Graphing with Calculators

Graphing without Calculators

General Procedure for Sketching the Graph of a Function

STRATEGY Steps:

1. Determine the domain and if possible the range of the functionf(x).


2. Determine if the function has any symmetry, i.e., if the function is even (f(x)=f(−x)),
   odd (f(x)=−f(−x)), or periodic (f(x+p)=f(x)).


3. Findf′(x) andf′′(x).


4. Find all critical numbers (f′(x)=0orf′(x) is undefined) and possible points of
   inflection (f′′(x)=0orf′′(x) is undefined).


5. Using the numbers in Step 4, determine the intervals on which to analyzef(x).


6. Set up a table using the intervals, to
   (a) determine where f(x) is increasing or decreasing.
   (b) find relative and absolute extrema.
   (c) find points of inflection.
   (d) determine the concavity of f(x) on each interval.


7. Find any horizontal, vertical, or slant asymptotes.


8. If necessary, find thex-intercepts, they-intercepts, and a few selected points.


9. Sketch the graph.

Example 1

Sketch the graph off(x)=

x^2 − 4

x^2 − 25

.

Step 1: Domain: all real numbersx=±5.

Step 2: Symmetry: f(x) is an even function (f(x)=f(−x)); symmetrical with respect to

they-axis.

Step 3: f′(x)=

(2x)(x^2 −25)−(2x)(x^2 −4)

(x^2 −25)^2

=

− 42 x

(x^2 −25)^2

f′′(x)=

−42(x^2 −25)^2 −2(x^2 −25)(2x)(− 42 x)

(x^2 −25)^4

=

42(3x^2 +25)

(x^2 −25)^3