Mention specific topics I need to understand in order to get understand this question and get it right. Go in the order of the questions.
1. x
x lim1 cos (2 )
(2 ) x 0
2
2
− = →
(A) 0
(B) 1
4
(C) 1
2
(D) 1
=
< −
− − ≤ ≤
− >







f x
x
x
x x
x x
( )
2 for 1
3 for 1 2
4 3 for 2
2
2. Let f be the function defined above. At what values of x, if any, is f not
differentiable?
(A) x = -1 only
(B) x = 2 only
(C) x = -1 and x = -2
(D) f is differentiable for all values of x.
x f(x) fʹ(x) g(x) gʹ(x)
1 2 -4 -5 3
2 -3 1 8 4
3. The table above gives values of the differentiable functions f and g and their
derivatives at selected values of x. If h is the function defined by
h(x) = f(x)g(x) + 2g(x), then hʹ (1) =
(A) 32
(B) 30
(C) -6
(D) -16
4. If x3 - 2xy + 3y2
 = 7, then dy
dx =
(A) x y +
x
3 4
2
2
(B) −
−
x y
x y
3 2
2 6
2
(C) −
x
x y
3
2 6
2
(D) −
x
y
3
2 6
2
5. The radius of a right circular cylinder is increasing at a rate of 2 units per
second. The height of the cylinder is decreasing at a rate of 5 units per second.
Which of the following expressions gives the rate at which the volume of the
cylinder is changing with respect to time in terms of the radius r and height h
of the cylinder?
(The volume V of a cylinder with radius r and height h is V = πr2
h.)
(A) -20πr
(B) -2πrh
(C) 4πrh - 5πr2
(D) 4πrh + 5πr2
6. Which of the following is equivalent to the definite integral ∫ x dx
6
2 ?
(A) →∞ ∑= n
k
n
lim
4 4
n k
n
1
(B) →∞ ∑= n
k
n
lim
6 6
n k
n
1
(C) ∑ + →∞ = n
k
n
lim
4
2
4
n k
n
1
(D) ∑ + →∞ = n
k
n
lim
6
2
6
n k
n
1
Graph of g
1 2 O 34567 8
7. The figure above shows the graph of the continuous function g on the interval
[0, 8]. Let h be the function defined by h(x) = ∫
x
g t dt
3 ( ) . On what intervals is h
increasing?
(A) [2, 5]only
(B) [1, 7]
(C) [0, 1] and [3, 7]
(D) [1, 3] and [7, 8]
8. ∫ − = x
x
dx
1 9 2
(A) − − x C+
1
9
1 9 2
(B) x C 1
18
ln 1 9 2 − − +
(C) x C+
1
3
arcsin(3 )
(D) +
x x C
9. Shown above is a slope field for which of the following differential equations?
(A) = dy −
dx
y 2
2
(B) = dy −
dx
y 4
4
2
(C) = dy −
dx
x 2
2
(D) = dy −
dx
x 4
4
2
10. Let R be the region bounded by the graph of x = ey
, the vertical line x = 10, and
the horizontal lines y = 1 and y = 2. Which of the following gives the area of R?
(A) ∫ e dy
2
1
y
(B)
e
e
ln x dx
2
∫
(C) ∫ − e dy
2
1
(10 )y
(D)
e
x dx
PART B (AB OR BC)
A graphing calculator is required on this part of the exam.
Graph of f
x
y
4
3
2
1
–1
–2
–1–2–3–4 1 2 3 4
–3
–4
O
11. The graph of the function f is shown in the figure above. The value
of → + lim (f x) x 1
 is
(A) -2
(B) -1
(C) 2
(D) nonexistent
12. The velocity of a particle moving along a straight line is given by v(t) = 1.3tln
(0.2t + 0.4) for time t ≥ 0. What is the acceleration of the particle at time t = 1.2?
(A) -0.580
(B) -0.548
(C) -0.093
(D) 0.660
x -1 0 2 4 5
fʹ(x) 11 9 8 5 2
13. Let f be a twice-differentiable function. Values of fʹ, the derivative of f, at
selected values of x are given in the table above. Which of the following
statements must be true?
(A) f is increasing for -1 ≤ x ≤ 5.
(B) The graph of f is concave down for -1 < x < 5.
(C) There exists c, where -1 < c < 5, such that f c ′( ) = − 3
2
.
(D) There exists c, where -1 < c < 5, such that f c ′′( ) = − 3
2
.
14. Let f be the function with derivative defined by f x ′( ) = +2 (2 8 x x − + )sin( 3).
How many points of inflection does the graph of f have on the interval 0 < x < 9 ?
(A) One
(B) Two
(C) Three
(D) Four
15. Honey is poured through a funnel at a rate of r(t) = 4e-0.35t
 ounces per minute,
where t is measured in minutes. How many ounces of honey are poured through
the funnel from time t = 0 to time t = 3?
(A) 0.910
(B) 1.400
(C) 2.600
(D) 7.429
PART A (BC ONLY)
Graphing calculators are not permitted on this part of the exam.
x 2 5
f(x) 4 7
fʹ(x) 2 3
16. The table above gives values of the differentiable function f and its derivative fʹ
at selected values of x.
If ∫ f x dx = 5
2 ( ) 14, what is the value of ∫ x f ⋅ ′ x dx
5
2 ( ) ?
(A) 13
(B) 27
(C) 63
2
(D) 41
17. The number of fish in a lake is modeled by the function F that satisfies the
logistic differential equation = − 

 

 dF
dt
F F
0.04 1
5000
, where t is the time in
months and F(0) = 2000. What is →∞
lim ( F t) t ?
(A) 10,000
(B) 5000
(C) 2500
(D) 2000
18. A curve is defined by the parametric equations x(t) = t
2 + 3 and y(t) = sin (t
2
).
Which of the following is an expression for d y
dx
2
2 in terms of t ?
(A) -sin (t
2
)
(B) -2tsin (t
2
)
(C) cos (t
2
) - 2t
2
 sin (t
2
)
(D) 2cos (t
2
) - 4t
2
 sin (t
2
)
19. Which of the following series is conditionally convergent?
(A) k ( 1) 5
1
k
k
3
1
∑ − = +
∞
(B) k ( 1) 5
1
k
k 1
∑ − = +
∞
(C) k
k ( 1) 5
1
k
k 1
∑ − = +
∞
(D) k
k ( 1) 5
1
k
k
2
1
∑ − = +
∞
20. Let f be the function defined by f(x) = e2x
. Which of the following is the
Maclaurin series for fʹ, the derivative of f ?
(A) + + x + +  + +
x x x
n
1
2! 3! !
2 3 n
(B) + + x + +  + +
x x x
n
2 2
2
2!
2
3!
2
!
2 3 n
(C) + + x + +  + +
x x x
n
1 2 (2 )
2!
(2 )
3!
(2 )
!
2 3 n
(D) + + x + +  + +
x x x
n
PART B (BC ONLY)
A graphing calculator is required on this part of the exam.
y
x
O
21. The figure above shows the graph of the polar curve r = 2 + 4sin θ. What is the
area of the shaded region?
(A) 2.174
(B) 2.739
(C) 13.660
(D) 37.699
22. The function f has derivatives of all orders for all real numbers. It is known that
f x( ) ≤
12
5
(4) and f x( ) ≤ ≤x ≤ 3
2
for0 2 (5) . Let P x 4 ( ) be the fourth-degree
Taylor polynomial for f about x = 0. The Taylor series for f about x = 0 converges
at x = 2. Of the following, which is the smallest value of k for which the Lagrange
error bound guarantees that f P (2) ( − ≤ 2) k 4 ?
(A) ⋅ 2
5!
3
2
5
(B) ⋅ 2
5!
12
5
5
(C) ⋅ 2
4!
3
2
4
(D) ⋅ 2
4!
12
5
Section II: Free-Response
The following are examples of the kinds of free-response questions found on the exam.
PART A (AB OR BC)
A graphing calculator is required on this part of the exam.
t (hours) 0 2 4 6 8 10 12
R(t) (vehicles per hour) 2935 3653 3442 3010 3604 1986 2201
1. On a certain weekday, the rate at which vehicles cross a bridge is modeled by
the differentiable function R for 0 ≤ t ≤ 12, where R(t) is measured in vehicles
per hour and t is the number of hours since 7:00 a.m. (t = 0). Values of R(t) for
selected values of t are given in the table above.
(a) Use the data in the table to approximate Rʹ(5). Show the computations that
lead to your answer. Using correct units, explain the meaning of Rʹ(5) in the
context of the problem.
(b) Use a midpoint sum with three subintervals of equal length indicated by
the data in the table to approximate the value of ∫ R t dt
12
0 ( ) . Indicate units
of measure.
(c) On a certain weekend day, the rate at which vehicles cross the bridge is
modeled by the function H defined by H(t) = -t
3 - 3t
2 + 288t + 1300 for
0 ≤ t ≤ 17, where H(t) is measured in vehicles per hour and t is the number
of hours since 7:00 a.m. (t = 0). According to this model, what is the average
number of vehicles crossing the bridge per hour on the weekend day
for 0 ≤ t ≤ 12?
(d) For 12 < t < 17, L(t), the local linear approximation to the function H given
in part (c) at t = 12, is a better model for the rate at which vehicles cross the
bridge on the weekend day. Use L(t) to find the time t, for 12 < t < 17, at
which the rate of vehicles crossing the bridge is 2000 vehicles per hour.
Show the work that leads to your answer.
PART B (AB OR BC)
Graphing calculators are not permitted on this part of the exam.
O 1 2 3 4
x
y
Graph of f´
2. The figure above shows the graph of fʹ, the derivative of a twice-differentiable
function f, on the closed interval [0, 4]. The areas of the regions bounded by the
graph of fʹ and the x-axis on the intervals [0, 1], [1, 2], [2, 3], and [3, 4] are 2, 6, 10,
and 14, respectively. The graph of fʹ has horizontal tangents at x = 0.6, x = 1.6,
x = 2.5, and x = 3.5. It is known that f(2) = 5.
(a) On what open intervals contained in (0, 4) is the graph of f both decreasing
and concave down? Give a reason for your answer.
(b) Find the absolute minimum value of f on the interval [0, 4]. Justify your
answer.
(c) Evaluate ∫ f x f x ′ dx
4
0 () () .
(d) The function g is defined by g(x) = x3 f(x). Find gʹ (2). Show the work that
leads to your answer.
PART A (BC ONLY)
A graphing calculator is required on this part of the exam.
3. For 0 ≤ t ≤ 5, a particle is moving along a curve so that its position at time t
is (x(t), y(t)). At time t = 1, the particle is at position (2, -7). It is known that
= +


 

 dx
dt
t
t
sin
3
 and = dy
dt
ecost
.
(a) Write an equation for the line tangent to the curve at the point (2, -7).
(b) Find the y-coordinate of the position of the particle at time t = 4.
(c) Find the total distance traveled by the particle from time t = 1 to time t = 4.
(d) Find the time at which the speed of the particle is 2.5. Find the acceleration
vector of the particle at this time.
PART B (BC ONLY)
Graphing calculators are not permitted on this part of the exam.
4. The Maclaurin series for the function f is given by
f (x) = x
k
x
( 1) x x
4 9
k k
k
1
2
2 3
1
∑  − = − + −
+
=
∞
on its interval of convergence.
(a) Use the ratio test to determine the interval of convergence of the Maclaurin
series for f. Show the work that leads to your answer.
(b) The Maclaurin series for f evaluated at x = 1
4
is an alternating series whose
terms decrease in absolute value to 0. The approximation for 

 
 f 
1
4 using
the first two nonzero terms of this series is 15
64
. Show that this
approximation differs from 

 
 f 
1
4 by less than 1
500 .
(c) Let h be the function defined by ∫ h x = x
( ) f t dt
0 ( ) . Write the first three
nonzero terms and the general term of the Maclaurin series for h.