A® Calculus Midterm Practice Exam — 2022-2023
PART I: NON-Calculator. (15 questions - 30 minutes)
Multiple Choice. Please put all answers on your scantron sheet.
1. If y = xsinx, then &
(A)
=
Sy
J
\
\S
sinx + cosx
sinx + xcosx
(C)
sinx — xcosx
(D)
x(sinx + cosx)
(E)
x(sinx — cosx)
2. Let f be the function given by
f(x) = 300x - x°. On which of the following intervals is the
function f increasing?
- 400 -3y* = O
(A) (-00,-10] and [10,c0)
0
3( Ww-Ax*)=
.
(6)(C) 0.0
[0, 10] only
(D) [0,103] only
1Q,
=
Y
(E) [0, e)
B.
418
y= Bx
13x-y=8
wf S13
”
x + 13y = 66 9
-
710
—
‘
3. An equation of the line tangent to the graph ofy = =
A.
oO
/ te-xbl0+x)=)
?
2)
+y =18
E. x —13y= 64
A}
dn)
:
Gay
y
oO
“7
4.
7
Zlay
_
bx-4
dul.
2)— (24
2
ee
— by -4
Ape 7 k
-\s
(U5)
ne
xT
_
10
+
Y= Biel) |
. ~2x + 3y = 13
J
)
dy
at the point (1,5) is
.
YO
4
xe
The graph of fis shown in the figure above. Which of the following could be the graph of the derivative
ff
ca)
4 >
a
oO
/b
¥
(E)
\
(B)
v
—\7AN..,
:
a
oO
h
a
a
ic)
y
a
O
(D)
LO
_—
a
:
~~
a
NO
7
b
——~
i
oO
oo?
|
6
t
|
54--+
4}--4--4--
|
r
(
3°
t
2.
l
to
OF
1
4
ge
L
2
3
— L.
|
a
‘
4
5
6
+
Graph of f
5. The graph of the function f is shown above. Which of the following statements is false?
(A)
lim f(x)
exists. (/
(B) lim f(x)
exists. |
MH
(oe) lim f(x) exists.
lim f(x)
<—~
exists.
Y
(E) The function f is continuous at x = 3.
(x-2\PeF)
lim? 3242
6. Find“
4
x~-1
y
A) 2
B)
1
C) 0
E) Does not exist
VU FY -5
7.
If f(x)= -3x° +4
-4 4
-5, then there exists a number cin the interval te x <2 that satisfies the Mean Value
Theorem. Which of the following could be c? = f lx) 2
2
A) 3
8.
4
a
C) 5
Find an equation for the line tangent to the graph of f(x)
A)
9.
(Dy? 5
y=6x-2
i
B) y=—x-—
6
1
3
1
A) -2
B) -1
C) 0
Fiyj= Doel) sine + OelPcos x
Flo\=
LreetSid + 190
3
D) 1
—Qx?Y=
7S
we
E) i
D) -5
4
vx-7 at the point where x = 16. (Ho ,3)
;
. He,
\. Le
\
257)
ages
D) y=-6x-2
fo
pee
E)
4-32 4lel)
-
If f(x) = (x — 1)? sinx, then f’(0) =
“Ss
ko
C) \y=—x4+-—
6
F(2)-F(t)
Wlus
(D)
E) 2
y
-3-
ly—£
&
a
1 y
y=-—x+-
6
J
elx+¥lt
b
3
3
10. The top ofa 25-foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When
the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the
e a
x"
aK
oly 2-3 HY
26.
bottom of the ladder and the wall?
7
A.
.
— 3 ft/min
B.
Z
7
7
ia ft/min
.
C. a ft/min
4
=]
ye
7
2%
.
(0.3 ft/min
ao
+
zy
21
a
.
E. - ft/min
Vy
& 42.7.3 50
11. Find an equation for the tangent line to the graph of f(x) = 2x? — 2x + 3 at the point where x = 1.
Fiz 2-243=
A)
y=2x-2
B)
C)
y = 4x7 -6x+2
E)
none
of these
(»)
O
3
(13)
y= 4x*-6x+5
fy)
y=2x4+1
P(ijot
“2 -
qe4y 1 4
24x ~2
dee
r“~15
yee
12.Let f(x)=(2x+ 1)’ and let gbe the inverse function of £ Given that f(0)=1, what is the value of g'(1)?
gilyj=
o2 27
2. 3 (Hye
@t54
)
13. If fi
10
of27
(2H)
_
3
f (x)dx = 4and Siof
f°
(A)
3
4B)
3
(1!)6
-
worm
8
P\ (0) =
fo
/
=]
(Cc)
3
(D)
10
YS
?
|
_
3
_
ax = 7, then fi f(x)dx =
0
(1) < A
j
3
=
(2)
Le
{eo
—
11
~T
8
/
|
x
—
I
14. A particle moves along the x-axis so that at any time t its position is given by x(t) = 5 sine +cos(2t). What is the
acceleration of the particle at t=7?
>l
®B
0
=
©C
3
_
N
(A)A
V(t)
o)D
=5
~
(YE
~ 5
—7Sin
st
2 Sin
)
alt)- “Esint ~4. cus (2
fou
27 y(f)2 458.
we
ee Se
[a
~-
=
15. Suppose that / is a continuous function and is differentiable everywhere. Suppose also that /(0)=1,
f6)=—4,and f(-5) =-3.Which statement about / must be true?
I
J has exactly two zeros
| has at least two zeros
f must have a zero between 0 and -5
IV
(A)
Tl only
There is not enough information to determine anything about
the zeros of f
(B)
Il only
(C)
land Ill only
(2
( et)
II and III only
(E) IV only
AP Calculus Midterm Practice Exam
PART II: Calculator. (8 questions - 24 minutes)
Multiple Choice.
{
16.
p+
h
_
If f’(x) = Vx4 +14 x3 — 3x, then f has a local maximum at x =
(A) -2.314
(B) -1.332
(D) 0.829
(E) 1.234
17. Water runs into a conical tank at the rate of 3 ft?/min. The tank stands pointing down and has a height
of 15 ft and a base radius of 4 ft. How fast is the water level rising when the water is 8 ft deep?
t
7, ~
(A)
18.
).210 ft/min
y
o
Kg
Ne
Ni
(C) 0.070 ft/min
I yo ah
is
(B) 0.021 ft/min
ve
_
~
re
Vz
.
aL} sy
av
Gs
V=sar'h)
(Volume of a cone:
5
lo
a TT le We
lh
3
(pas
le
ie” nes
(D) 0.220 ft/min
tT -tte4
ar
Nas
aes
(E) 0.041 ft/min
The first derivative of the function f is given by f’(x) = x — 4e~“"'2")_ How many points of inflection does the
graph of f have on the interval 0 < x < 24?
C' Gividuhts dUAL CH
(A) Three
(B)
Four
(C)
Five
(D)
Six
(E)
Seven
19.
Graph of /”
ot
4 Iy
_
. The graph of f’, the derivative of f, is shown in the figure above. The function f has a local maximum at x =
(A) -3
20.
<1
(B) -1
(1)
\
(D) 3
(E)4
For how many values of x will the tangent lines to y =4sinx and
a) 0
b) 1
c)3
2
y=
Pe
parallel?
d) 4
Wo
4 css v
e) infinite
ys
x
Gd
21.
1
Graph of f”
The graph of f”. the second derivative off, is shown above for -2 < x < 4. What are all intervals on which
the graph of the function f is concave down?
(A)
-l<x<]
(B)
O<x<2
(C)
I<x <3 only
(D)
-2 <x <~—I only
(ey
filg
9
—2<x<-landil<x<3
22.
x
fl | £70) | eG) | eg’)
l
3
~2
~3
4
The table above gives values of the differentiable functions f and g and their derivatives at x
A(x)
(2f(x) + 3)(1 + g(x). then h’(1)
(A) -28
x
23.
(E) 47
(C) 40
(B) -16
3
f(x) | 20 |
4
17: |
5
12: |
6
16 |
7
20
Wy)
2 (x
art
_
1. If
[+ g(x) \)+ (othe) +3) qt
| +i) Heh
‘Cs L)
The function f is continuous and differentiable on the closed interval [3, 7]. The table above gives sel ed
values pf f on this interval. Which of the following statements must be true?
“ The minimum value off on [3, 7] is 12.
here exists ¢, for 3<¢ <7, such that f’(c) - 0. {AW
L f(x) >0 for 5 <x <7.
(A) 1 only
I] only
(C)
Ul only
(D) Land III only
(E) ITE, and Il
)4'')
aay
tf
°g
be
|
AP Calculus Midterm Practice Exam
Free Response: (Non calculator - 2 questions, 30 minutes)
oF
t
(minutes)
V7
01
C(t)
o
(ounces)
153)
|
2/3
a8
|
7
rt
|
4° 5
li2!128/
|'6
1381
|
145
Hot water is dripping through a coffeemaker, filling a large cup with coffee. The amount of coffee in the cup at
time #, 0 < ¢ < 6, is given by a differentiable function C, where ¢ is measured in minutes. Selected values of
C(t), measured in ounces, are given in the table above.
(a)
Use the data in the table to approximate C’(3.5). Show the computations that lead to your answer, and
indicate units of measure.
(b)
Isthereatime t, 2 <1 < 4, at which C’(t) = 2? Justify your answer.
(c)
Use a midpoint sum with three subintervals of equal length indicated by the data in the table to approximate
6
6
the value of rf. C(t) dt. Using correct units, explain the meaning of Ff C(t) dt in the context of the
problem.
(d)
The amount of coffee in the cup, in ounces, is modeled by B(t) = 16 -— 16e°°“". Using this model, find the
rate at which the amount of coffee in the cup is changing when ¢ = 5.
-4t
a) CGS) ~ “=
ENE Me our
c(4)-c(3)
}
3) BlE)= Ne- Ibe
at (e)= wt GHC
=
~
B(s)= y+ bMe »
cL
ly + 6.4
(WVT Oy. Lub
Ft
C42
ue
— 4-5
2 AP fem ded fread
Ks cope
|
ya in
Hee nytt a
&é
gtr
citi
hot
oO
ClUdkLE a sae atte Qi]
ob
e
WY
+ LI
{0.6lo
1 (¢
.
’
io
paceeL f= 5
0) ely)
re
Vine CLUS
2
b = 10, | oma
a
.
oY.
Consider the function y = f(x) whose curve is given by the equation 2y” —6=ysinx
(a) Show
a
that2 =
OW
for y>0.
2225+
See"
‘ dx
4y-— sinx
(b) Write an equation for the line tangent to the curve at the point (0, V3).
(c) ForO <x <@ and y > 0, find the coordinates of the point where the line tangent to the curve is
horizontal.
(d) Determine whether f has a relative minimum, a relative maximum, or neither at the point found in
part (c). Justify your answer.
Uy 4 Avy - oyS cays YJ cosy
on
)|
yo
|
,
sl
4y
—
fink
C80)
c)
y COX
juse,0d
j
|
ox
coe’
i
=
ee
ayp—be Sj OCSin’s
© 9)
ynw
byttuys
—
Sim)
ely”
}
_
BBeaed)
oo
(-
1
;
—
2s (4.0
ui)
(3 sin 8)”
a
a
~
O
at aye
(aoncase
chown
cay
<=
O
}
; any
O
=
(24 ¥3)(4-2)
y sine
dy (ae) = <M
YR ANS
\I 2O
Yo
By
Lae
\
\V
of
Novitwn
,
int )*
(4
ye
(0/83)
é
-
yaw(¥g or)
S
4 COS
dy
(wecosy.
dy.
Kfy-sinwyta — yeoux
)
(a
d)
ay -b= Y sin y
a)
6
2)
Ct
f tf
aa ls
Uh
5
“dy
Ts
ui oy
- fp
yg
meres
fhe 4
Graph of fi’
The figure above shows the graph of h’, the derivative of a twice-differentiable function /7, on the
closed interval —1 < x < 7. The graph of h’ has horizontal tangent lines at 2 = 1, 2 = 3, and
= 5. The areas of the regions bounded by the x-axis and the graph of h! on the intervals
ely. The function /i is defined for
pect
[—1, 1], [1, 4], [4,6], and [6, 7] are 6.8, 5.6. and 6.
all real numbers and satisfiés A(5) =. 5a
h(7) =
(a) Find the x-coordinate of the critical point of h that nesponds
reason for your answer.
Minimun
telahve
Gwitehas
ln
WAL.
toarelative minimum of h. Givea
Cys
- +o
+
Y= b
(b) On what open intervals is h both increasing and concave up? Give a reason for your answer.
(S
A
Wi
is
(c) Find jim.
u(a)=8
lian
wer
Ir3
>
h'
-
ard
> O
(u,1)
S . Give a reason for your answer.
uly) -f
U Hupels
When
im concoe MUpealwhenteg SeIl onis A(1M,3I)NEaod
MALES
_- &
AyD
\
Wh
q
7
(9
\
3 Co
Mr
%
(d) The function a is defined by a(x) = (h(x))?. Find a'(5). Show the computations that lead to your
answer.
alk):
ln'{x)-
D{ hte)
a'(si= b{s)>2a(hi)
2(5),
“4
10
=
40)