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

MA 3972-MA-Book April 11, 2018 16:32

AP Calculus AB Practice Exam 1 363

f ́ ́

f ́

f

• 
  

x = 1

Concave down

Increasing Increasing

Concave up

Point of inflection

0

0

+

+ +

Note that the graph offchanges concavity

from down to up. Sincef is a differentiable

function, it has a point of inflection atx=1.

16. The correct answer is (C).

Since

∫− 8

8

f(x)dx=−10,

∫ 8

− 8

f(x)dx=10.

Also,

∫− 2

− 8

f(x)dx+

∫ 8

− 2

f(x)dx=

∫ 8

− 8

f(x)dx,

so 4+

∫ 8

− 2

f(x)dx=10 or

∫ 8

− 2

f(x)dx=6.

17. The correct answer is (D).

Using implicit differentiation, you have

4 xy+ 2 x^2

dy

dx

− 3 y^2 − 3 x

(

2 y

dy

dx

)

=0.

Substitutex=2 andy=1, and obtain

8 + 8

dy

dx

− 3 − 12

dy

dx

=0, or− 4

dy

dx

=−5, or

dy

dx

=

5

4

.

18. The correct answer is (B).

x

y 4 x

x^2

(2, 8)

(2, 4)

f

Sincef(x) is discontinuous atx=2,f′(2)

does not exist. Statement III is false. The limit

in statement II represents the right-hand

derivative (f+)′(x), and (f+)′(2) does not exist

because the point (2, 8) is not part of the

function. Statement II is false. The limit in

statement I represents the left-hand derivative

(f−)′(x), and forx≤2, (f−)′(x)= 2 x. Thus,

(f−)′(2)=4. Statement I is true.

19. The correct answer is (B).

The volume of a sphere isV=

4

3

πr^3 ,so

dV

dt

= 4 πr^2

dr

dt

,or10= 4 πr^2

dr

dt

, which yields

dr

dt

=

5

2 πr^2

. The diameterDis twice the

radius;D= 2 r, and

dD

dt

= 2

dr

dt

. Therefore,
dD
dt

= 2

(

5

2 πr^2

)

=

5

πr^2

, and

dD

dt

∣∣

∣∣

r= 5

=

(

5

π( 52 )

)

=

1

5 π

centimeters per

second.

20. The correct answer is (B).
    Note that the definition of the derivative
    f′(x)=lim
    h→ 0

f(x+h)−f(x)

h

. Therefore, the

lim

h→ 0

sin

(

π

3

+h

)

−sin

(

π

3

)

h

is equivalent to

d

dx

(sinx)

∣

∣∣

∣x=π

3

or cos

(

π

3

)

=

1

2

. Note that

there are other approaches to solving this

problem, e.g., applyingL'Hôpital'sRule or

expanding sin

(

π

3

+h

)

.