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

MA 3972-MA-Book April 11, 2018 14:46

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

23. (a) Summarize the information off′on
    a number line.

.5 2.5

decr. incr.

rel. min.

l′ – 0 +

l

Sincefhas only one relative

extremum, it is the absolute

extremum. Thus, atx=3, it is an

absolute minimum.

(b) The functionfis decreasing on the

interval (−∞, 3) and increasing on

(3,∞).

(c)

f′

f′′

f

++ 3

incr.

concave

upward

concave

upward

0 incr.

No change of concavity⇒No point

of inflection.

(d) The functionfis concave upward for

the entire domain (−∞,∞).

(e) Possible sketch of the graph forf(x).

(See Figure 9.7-1.)

3

(3, –2)

x

0

y

f

Figure 9.7-1

24. (Calculator) (See Figure 9.7-2.)

Figure 9.7-2

Step 1: Differentiate:

2 x+ 2 y

dy

dx

= 0 ⇒

dy

dx

=−

x

y

.

Step 2: Set

dy

dx

=− 1 ⇒

−x

y

=− 1 ⇒

y=x.

Step 3: Solve fory:x^2 +y^2 = 36 ⇒

y^2 = 36 −x^2 ;

y=±

√

36 −x^2.

Step 4: Thus,y=x⇒±

√

36 −x^2 =

x⇒ 36 −x^2 =x^2 ⇒

36 = 2 x^2 orx=± 3

√

2.

25. (Calculator) (See Figure 9.7-3.)

Step 1: Distance formula:

z=

√

√ (x−1)^2 +(x^3 )^2 =

(x−1)^2 +x^6.

Figure 9.7-3