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

(Marvins-Underground-K-12) #1
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.)
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