5 Steps to a 5 AP Calculus BC 2019

Graphs of Functions and Derivatives 121

Step 4: Critical numbers:

f′(x)= 0 ⇒− 42 x=0orx= 0

f′(x) is undefined atx=±5, which are not in the domain.

Possible points of inflection:

f′′(x)/=0 andf′′(x) is undefined atx=±5, which are not in the domain.

Step 5: Determine intervals:

Intervals are (−∞,−5), (−5, 0), (0, 5) and (5,∞).

Step 6: Set up a table:

INTERVALS (−∞,−5) x=−5(−5, 0) x= 0 (0, 5) x= 5 (5,∞)

f(x) undefined 4/25 undefined
f′(x) + undefined + 0 − undefined −
f′′(x) + undefined −−−undefined +
incr incr decr decr
concave concave concave concave
conclusion upward downward rel max downward upward

Step 7: Vertical asymptote:x=5 andx=− 5

Horizontal asymptote:y= 1

Step 8: y-intercept:

(

0,

4

25

)

x-intercept: (−2, 0) and (2, 0)

(See Figure 7.3-1.)

[–8,8] by [–4,4]

Figure 7.3-1

Graphing with Calculators

Example 1

Using a calculator, sketch the graph off(x)=−x^5 /^3 + 3 x^2 /^3 indicating all relative extrema,

points of inflection, horizontal and vertical asymptotes, intervals wheref(x) is increasing

or decreasing, and intervals wheref(x) is concave upward or downward.