ETHS: AP Calculus AB
Final Exam Review Problems
Semester 1
Part I. Calculators Allowed.
A. FUNCTIONS AND GRAPHS
𝑥 + 2,
𝑥≥1
1) Let 𝑓(𝑥) = & !
. How many values of 𝑐 satisfy 𝑓(𝑐) = 3.5 on [0, 5]?
−𝑥 + 4, 𝑥 < 1
(A) No values
(B) 1
(C) 2
(D) 3
(E) Infinitely many
B. LIMITS AND CONTINUITY
2) Let 𝑓 be defined as follows, where 𝑎 ≠ 0.
𝑥 ! − 𝑎!
𝑥 ≠ 𝑎.
𝑓(𝑥) = : 𝑥 − 𝑎 ,
0,
𝑥=𝑎
Which of the following are true about 𝑓?
I.
lim 𝑓(𝑥) exists.
II.
III.
(A) None
"→$
𝑓(𝑎) exists.
𝑓(𝑥) is continuous at 𝑥 = 𝑎.
(B) I only
(C) II only
(D) I and II only
(E) I, II, III
(D)∞
(E) DNE
C. DIFFERENTIAL CALCULUS
3) What is lim
"→%
,
&'( ")*+( "
"!
(A) !
?
(B) 1
(C) 2
4) If 𝑟 is positive and increasing, for what value of 𝑟 is the rate of increase of 𝑟 - twelve times that
of 𝑟?
(A) √3
(B) 2
(C) √12
(D) 2√3
(E) 6
5) If 𝑐 satisfies the conclusion of the Mean Value Theorem for 𝑓(𝑥) = 𝑥 . − 3𝑥 - + 2𝑥 − 3 on the
interval [−1, 2], then 𝑐 could be
(A) -0.59 only
(B) 0.00 only
(C) 0.84 only
(D) -0.59 and 0.84 (E) -0.43 and 0.54
6) An equation to the tangent line to the curve 𝑦 = sin(2𝑥 − 1) + 3 at 𝑥 =
(A) 𝑦 = 1.86𝑥 + 6.87
(D) 𝑦 = −1.86𝑥 − 6.87
(B) 𝑦 = 1.86𝑥 − 6.87
(E) None of these
-/
0
is approximately:
(C) 𝑦 = −1.86𝑥 + 6.87
D. INTEGRAL CALCULUS
7) If 𝑓 is continuous for all 𝑥, which of the following integrals necessarily have the same value?
1
I. H 𝑓(𝑥) 𝑑𝑥
$
(A) I and II only
(D) I, II, II
1)$
II. H 𝑓(𝑥 + 𝑎) 𝑑𝑥
%
123
III.
H 𝑓(𝑥 + 𝑐) 𝑑𝑥
$23
(B) I and III only
(C) II and III only
(E) No two necessarily have the same value.
Part I Calculators Allowed: Page 1
ETHS: AP Calculus AB
Final Exam Review Problems
Semester 1
Part II. No calculators Allowed.
A. FUNCTIONS AND GRAPHS
,
8) Let 𝑓(𝑥) = cos(𝑘𝑥). For what value of 𝑘 does 𝑓 have a period of 3?
"
!
(B)
(A) -
!/
9) The domain of 𝑓(𝑥) =
(A) All 𝑥 ≠ 0, 1
(C)
-
√!)"
" " )"
-/
!
(D) 6
(E) 6𝜋
(D) 𝑥 ≥ 1
(E) 𝑥 > 2
is
(B) 𝑥 ≤ 2, 𝑥 ≠ 0,1
(C) 𝑥 ≤ 2
10)Which of the following equations has a graph that is symmetric with respect to the origin?
"),
"
(A) 𝑦 = "
(B) 𝑦 = 2𝑥 . + 1
(C) 𝑦 = 𝑥 - + 2𝑥
(D) 𝑦 = 𝑥 - + 2
(E) 𝑦 = " ! 2,
11)Which of the following functions is not odd?
(A) 𝑓(𝑥) = sin 𝑥
(B) 𝑓(𝑥) = sin 2𝑥
"
!
(D) 𝑓(𝑥) = " " 2,
(E) 𝑓(𝑥) = √2𝑥
(C) 𝑓(𝑥) = 𝑥 - + 1
12)The roots of the equation 𝑓(𝑥) = 0 are 1 and −2. The roots of 𝑓(2𝑥) = 0 are
(A) 1 and −2
(B) 0.5 and −1
(C) −0.5 and 1
(D) 2 and −4
(E) −2 and 4
13)The 𝑥-intercept(s) of the graph of 𝑓(𝑥) = 𝑥 - − 2𝑥 ! − 𝑥 + 2 is/are:
(C) 1, 2
(D) −1, 1, 2
(A) 1
(B)−1, 1
(E) −2, −1, 2
B. LIMITS AND CONTINUITY
14)Determine lim
" " ).
"→5 !2")." "
(A) −2
15)Find lim
(A) 1
"→%
*+( !"
"
.
,
,
(B) − .
(C) !
(D) 1
(B) 2
(C) 0.5
(D) 0
(E) DNE
.
(E) ∞
")-
16)Find lim " " )!")-.
(A) 0
"→%
(B) 1
(C) 0.25
(D) ∞
(E) None of these
Part III: Free Response (No Calculators): Page 2
ETHS: AP Calculus AB
Final Exam Review Problems
Semester 1
C. DIFFERENTIAL CALCULUS
17)What is lim
(A) −∞
#
"
#
"
78*9 26:)78*9 :
6→%
6
?
(B) −1
(C) 0
(D) 1
(E) ∞
18)At which of the five points on the graph in the figure at the right
;"<
;<
are ;" and ;" " both negative?
(A) A
(B) B
(C) C
(D) D
(E) E
19)The slope of the tangent line to the curve 𝑦 - 𝑥 + 𝑦 ! 𝑥 ! = 6 at (2,1) is:
-
(A) − !
(B) −1
0
-
(C) − ,.
(D) − ,.
(E) 0
20)Which of the following statements about the curve 𝑦 = 𝑥 . − 2𝑥 - is true?
(A) The curve has no relative extremum.
(B) The curve has one point of inflection and two relative extrema.
(C) The curve has two points of inflection and one relative extremum.
(D) The curve has two points of inflection and two relative extrema.
(E) The curve has two points of inflection and three relative extrema.
;
;
21)If ;" [𝑓(𝑥)] = 𝑔(𝑥) and if ℎ(𝑥) = 𝑥 ! , then ;" 𝑓Rℎ(𝑥)S =
(A) 𝑔(𝑥 ! )
(B) 2𝑥𝑔(𝑥)
(C) 𝑔′(𝑥)
(D) 2𝑥𝑔(𝑥 ! )
(E) 𝑥 ! 𝑔(𝑥 ! )
22)A point (𝑥, 𝑦) is moving along a curve 𝑦 = 𝑓(𝑥). At the instant when the slope of the curve is
,
− -, the 𝑥-coordinate of the point is increasing at the rate of 5 units per second. The rate of
change in units per second of the 𝑦-coordinate of the point is
0
,
(A) − -
(B) − !)"
=
-
0
(D) 0
(E) -
;<
23)If 𝑦 = -"2, then ;" =
(A) − (-"2,)"
,
(C) -
@")0
(B) (-"2,)"
24)If 𝑦 = cos ! 𝑥, then 𝑦 B =
(A) − sin! 𝑥
(B) 2 sin 𝑥 cos 𝑥
A
=
-)@"
(C) − (-"2,)"
(D) (-"2,)"
(E) (-"2,)"
(C) − sin 2𝑥
(D) 2 cos 𝑥
(E) −2 sin 𝑥
Part III: Free Response (No Calculators): Page 3
ETHS: AP Calculus AB
Final Exam Review Problems
Semester 1
/ /
25) An equation of the tangent to the curve 𝑦 = 𝑥 sin 𝑥 at the point U ! , ! V is
(A) 𝑦 = 𝑥 − 𝜋
(B) 𝑦 =
/
(C) 𝑦 = 𝜋 − 𝑥
!
/
(D) 𝑦 = 𝑥 + !
(E) 𝑦 = 𝑥
26)Given 𝑠(𝑡) = 𝑡 - − 6𝑡 ! + 12𝑡 − 8. The velocity of the particle is increasing:
(A) When 𝑡 > 2
(B) for all 𝑡, 𝑡 ≠ 2
(C) when 𝑡 < 2
(D) for 1 < 𝑡 < 3 (E) for 1 < 𝑡 < 2
27)What is the smallest perimeter possible for a rectangle whose area is to be 100 square inches?
(A) 20
(B) 30
(C) 40
(D) 50
D. INTEGRAL CALCULUS
;<
,
28)If ;" = 9𝑥 ! and 𝑦 = 1 when 𝑥 = 0, what is the value of 𝑦 when 𝑥 = - ?
,
(A) −2
,
(B) − -
(C) -
(D)
,%
A
(E) 8
29)∫(𝑥 − 1)√𝑥 𝑑𝑥 =
-
,
!
√"
! !
(A) ! √𝑥 −
$
!
+𝐶
$
,
%
,
(C) ! 𝑥 ! − 𝑥 + 𝐶
(B) - 𝑥 " + ! 𝑥 " + 𝐶
(D) 𝑥 " − 𝑥 " + 𝐶
0
-
!
,
(E) ! 𝑥 ! + 2𝑥 " − 𝑥 + 𝐶
/
!
30) H 2 sin 𝑥 𝑑𝑥 =
%
(A) 2
(B) 0
/
(C) !
(D) −1
(E) −2
Part III: Free Response (No Calculators): Page 4
ETHS: AP Calculus AB
Final Exam Review Problems
Semester 1
Part III: Free Response (No Calculators)
(1) Consider the curve given by 𝑥𝑦 ! − 𝑥 - 𝑦 = 6.
;<
a. Show that ;" =
-" " <)< "
!"<)" !
.
b. Find all points on the curve whose 𝑥-coordinate is 1 and write an equation for the tangent
line at each of these points.
c. Find the 𝑥-coordinate of each point on the curve where the tangent line is vertical.
Part III: Free Response (No Calculators): Page 5
ETHS: AP Calculus AB
Final Exam Review Problems
Semester 1
(2) Use the limit definition of the derivative to find 𝑓′(𝑥) if 𝑓(𝑥) = 2𝑥 ! + 3𝑥.
(3) Use the definition of the derivative to find 𝑓 B (𝑥) if 𝑓(𝑥) = √2𝑥 + 3.
-
(4) Use the definition of derivative to find 𝑓′(𝑥) if 𝑓(𝑥) = !"2,.
Part III: Free Response (No Calculators): Page 6
ETHS: AP Calculus AB
Final Exam Review Problems
Semester 1
Part IV: Free Response (Calculators Allowed)
(5) A container has the shape of an open right circular cone, as shown in
the figure to the right. The height of the container is 10 cm and the
diameter of the opening is 10 cm and the diameter of the opening is
10 cm. Water in the container is evaporating so that its depth ℎ is
changing at the constant rate of − ,%cm/hr. (The volume of a cone of
,
height ℎ and radius 𝑟 is given by 𝑉 = - 𝜋𝑟 ! ℎ. )
a. Find the volume 𝑉 of water in the container when ℎ = 5 𝑐𝑚.
Indicate units of measure.
b. Find the rate of change of the volume of water in the container, with respect to time, when
ℎ = 5 𝑐𝑚. Indicate units of measure.
c. Show that the rate of change of the volume of water in the container due to evaporation is
directly proportional to the exposed surface area of the water. What is the constant of
proportionality?
Part III: Free Response (Calculators): Page 7