MA 3972-MA-Book April 11, 2018 16:20
AP Calculus AB Practice Exam 2 391
=
1
3
u^3 /^2
3 / 2
+C
=
2
9
(x^3 +1)^3 /^2 +C
f(2)= 0 ⇒
2
9
(2^3 +1)^3 /^2 +C= 0
⇒
2
9
(9)^3 /^2 +C= 0
⇒ 6 +C=0orC=− 6
f(x)=
2
9
(x^3 +1)^3 /^2 − 6
f(0)=
2
9
− 6 =
− 52
9
- The correct answer is (C).
Statement I isnot true, e.g.,
2
∫ 4
0
xdx= 2
∫ 8
0
xdx.
Statement II is always∫ truesince
b
a
f(x)dx=−
∫a
b
f(x)dxby properties of
definite integrals.
Statement III is∣ not true, e.g.,
∣∣
∣
∫ 2
− 2
xdx
∣∣
∣∣=
∫ 2
− 2
|x|dx.
- The correct answer is (A).
f(0)=0; f′(0)<0 sincef is decreasing and
f′′<0,f is concave downward. Thus,f(0)
has the largest value.
- The correct answer is (D).
∫
x^4 − 1
x^2
dx=
∫(
x^2 −
1
x^2
)
dx
=
∫
(x^2 −x−^2 )dx
=
x^3
3
−
x−^1
− 1
+C=
x^3
3
+
1
x
+C
- The correct answer is (C).
Letu= 4 x; du= 4 dxor
du
4
=dx.
∫
q(4x)dx=
∫
q(u)
du
4
=
1
4
p(u)+c=
1
4
p(4x)+C
Thus,
∫ 0
− 1
q(4x)dx=
1
4
p(4x)
] 0
− 1
=
1
4
p(0)−
1
4
p(−4).
- The correct answer is (C).
The total amount of water leaked from the
tank for
1 ≤t≤ 3 =
∫ 3
1
f(t)dt
≈ 100 + 25 + 50 ≈175 gallons.
- The correct answer is (A).
f′(x)=10[cos(π−x)]−sin(π−x)
=10 cos(π−x) sin(π−x)
f′
(
π
2
)
=10 cos
(
π−
π
2
)
sin
(
π−
π
2
)
=10 cos
(
π
2
)
sin
(
π
2
)
=10(0)(1)= 0
- The correct answer is (C).
g′(x)=
3 x
x^3 + 1
;g′(2)=
3(2)
23 + 1
=
6
9
=
2
3
- The correct answer is (D).
∫ 2
k
(2x−2)dx=x^2 − 2 x
] 2
k
=
(
22 −2(2)
)
−
(
k^2 − 2 k
)
= 0 −
(
k^2 − 2 k
)
=−k^2 + 2 k
Set−k^2 + 2 k=− 3 ⇒ 0 =k^2 − 2 k− 3
0 =(k−3)(k+1)
⇒k=3ork=− 1.
- The correct answer is (A).
∫a
0
f(x)dx=−
∫ 0
−a
f(x)dx⇒ f(x)isan
odd function. The function whose graph is
shown in (A) is the only odd function.
