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

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

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

Steps:

1. Domain: all real numbers; Range: all real numbers


2. No symmetry


3. Relative maximum: (1.2, 2.03)
   Relative minimum: (0, 0)
   Points of inflection: (−0.6, 2.56)


4. No asymptote


5. f(x) is decreasing on (−∞, 0], [1.2,∞) and increasing on (0, 1.2).


6. Evaluatingf′′(x) on either side of the point of inflection (−0.6, 2.56)

d

(

−x∧

(

5

3

)

+ 3 ∗x∧

(

2

3

)

,x,2

)

x=− 2 → 0. 19

d

(

−x∧

(

5

3

)

+ 3 ∗x∧

(

2

3

)

,x,2

)

x=− 1 →− 4. 66

⇒ f(x) is concave upward on (−∞,− 0 .6) and concave downward on (− 0 .6,∞).

(See Figure 8.3-2.)

[–2, 4] by [–4, 4]

Figure 8.3-2

Example 2

Using a calculator, sketch the graph of f(x)=e−x^2 /^2 , indicating all relative minimum and

maximum points; points of inflection; vertical and horizontal asymptotes; and intervals on

whichf(x) is increasing, decreasing, concave upward, or concave downward.

Steps:

1. Domain: all real numbers; Range (0, 1]


2. Symmetry:f(x) is an even function, and thus is symmetrical with respect to they-axis.


3. Relative maximum: (0, 1)
   No relative minimum
   Points of inflection: (−1, 0.6) and (1, 0.6)


4. y=0 is a horizontal asymptote; no vertical asymptote.


5. f(x) is increasing on (−∞, 0] and decreasing on [0,∞).


6. f(x) is concave upward on (−∞,−1) and (1, ∞); and concave downward on
   (−1, 1).

(See Figure 8.3-3.)