5 Steps to a 5 AP Calculus BC 2019

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

Step 3: Determine intervals.

–1 01

The intervals are (−∞,−1), (−1, 0), (0, 1), and (1,∞).

Step 4: Set up a table.

INTERVALS (−∞,−1) x=−1(−1, 0) x= 0 (0, 1) x= 1 (1,∞)

Test Point − 2 − 1 / 2 1/2 2

f′(x) − undefined + 0 − undefined +

f(x) decr rel min incr rel max decr rel min incr

Step 5: Write a conclusion.

Using the First Derivative Test, note that f(x) has a relative maximum atx= 0

and relative minimums atx=−1 andx=1.

Note thatf(−1)=0,f(0)=1, andf(1)=0. Therefore, 1 is a relative maximum value and

0 is a relative minimum value. (See Figure 7.2-13.)

[–3,3] by [–2,5]

Figure 7.2-13

TIP • Do not forget the constant,C, when you write the antiderivative after evaluating an

indefinite integral, e.g.,

∫

cosxdx=sinx+C.

Test for Concavity and Points of Inflection

Test for Concavity

Letf be a differentiable function.

1. If f′′>0 on an interval I, thenf is concave upward on I.


2. If f′′<0 on an interval I, thenf is concave downward on I.

(See Figures 7.2-14 and 7.2-15.)