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). Cover the mcq and the frq.

 1. If 3
y = sin , x then = dy
dx
(A) 3 cos x (B) 2 3cos x (C) 2 3sin x (D) 2 -3sin cos x x (E) 2 3sin cos x x
 2. The position of a particle moving in the xy-plane is given by the parametric equations ( ) 3 2 xt t t = - 3 and
( ) 2
y t tt = - 12 3 . At which of the following points ( ) x y , is the particle at rest?
(A) ( ) -4, 12 (B) ( ) -3, 6 (C) ( ) -2, 9 (D) ( ) 0, 0 (E) ( ) 3, 4
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 3. The graph of f is shown above for 0 4. £ £ x What is the value of ( )
4
0 f x dx ? Ú
(A) -1 (B) 0 (C) 2 (D) 6 (E) 12 
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 4. Which of the following integrals gives the length of the curve y = ln x from x = 1 to x = 2 ?
(A)
2
2
1
1 1 dx
x
+ ÛÙ
ı
(B)
2
2
1
1 1 dx
x
Ê ˆ + Ë ¯
ÛÙ
ı
(C)
2 2
1
1 x + e dx Ú
(D) ( )
2 2
1
1 ln + x dx Ú
(E) ( ) ( )
2 2
1
1 ln + x dx Ú
 5. The Maclaurin series for the function f is given by ( ) ( ) 0
. 4
n
n
x f x
=
•
= - Â What is the value of f ( ) 3 ?
(A) -3 (B) 3
7 - (C) 4
7 (D) 13
16 (E) 4 
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 6. Using the substitution 2
u x = - 3, ( ) 4 5 2
1
x x dx 3 - - Ú is equal to which of the following?
(A)
13 5
2
2 u du -Ú
(B)
13 5
2
u du -Ú
(C)
13 5
2
1
2 u du -Ú
(D)
4 5
1
u du -Ú
(E)
4 5
1
1
2 u du -Ú
 7. If arcsin ln , x y = then dy
dx =
(A) 2 1
y
- x
(B) 2 1
xy
- x
(C) 2 1
y
+ x
(D) arcsin x e
(E)
arcsin
2 1
x e
+ x
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-8-
t (hours) 4 7 12 15
R( )t
(liters/hour)
6.5 6.2 5.9 5.6
 8. A tank contains 50 liters of oil at time t = 4 hours. Oil is being pumped into the tank at a rate R( )t , where R( )t
is measured in liters per hour, and t is measured in hours. Selected values of R( )t are given in the table above.
Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the
number of liters of oil that are in the tank at time t = 15 hours?
(A) 64.9 (B) 68.2 (C) 114.9 (D) 116.6 (E) 118.2
 9. Which of the following series converge?
I.
1
8
!
n
n n =
•
Â II. 100
1
!
n
n
= n
•
Â III. ( )( )( ) 1
1
2 3 n
n
nn n =
• + Â + +
(A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III 
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10.
4 3 2
1
- = Ú t dt
(A) -1 (B) 7
8 - (C) 1
2 - (D) 1
2 (E) 1
11. Let f be the function defined by fx x ( ) = - 2 for all x. Which of the following statements is true?
(A) f is continuous but not differentiable at x = 2.
(B) f is differentiable at x = 2.
(C) f is not continuous at x = 2.
(D) ( ) 2
lim 0
x f x Æ π
(E) x = 2 is a vertical asymptote of the graph of f. 
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12. The points ( ) - - 1, 1 and ( ) 1, 5- are on the graph of a function y = f x( ) that satisfies the differential equation
2 . dy x y dx = + Which of the following must be true?
(A) ( ) 1, 5- is a local maximum of f.
(B) ( ) 1, 5- is a point of inflection of the graph of f.
(C) ( ) - - 1, 1 is a local maximum of f.
(D) ( ) - - 1, 1 is a local minimum of f.
(E) ( ) - - 1, 1 is a point of inflection of the graph of f.
13. What is the radius of convergence of the series ( )2
0
4 ?
3
n
n n
x
=
• - Â
(A) 2 3 (B) 3 (C) 3 (D) 3
2 (E) 0 
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14. Let k be a positive constant. Which of the following is a logistic differential equation?
(A) = dy kt dt
(B) = dy ky dt
(C) = - ( ) 1 dy kt t dt
(D) = - ( ) 1 dy ky t dt
(E) = - ( ) 1 dy ky y dt
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15. The graph of a differentiable function f is shown above. If ( ) () 0 , x
h x f t dt = Ú which of the following is true?
(A) hh h () () () 66 6 < < ¢ ¢¢
(B) hh h () () () 6 66 < < ¢¢ ¢
(C) h hh ¢ ¢ () () () 66 6 < < ¢
(D) h hh ¢¢() () () 66 6 < < ¢
(E) h hh ¢¢ ¢ () () () 6 66 < <
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16. Let y = f x( ) be the solution to the differential equation = - dy x y dx with initial condition f ( ) 1 3. = What is
the approximation for f ( ) 2 obtained by using Euler’s method with two steps of equal length starting at x = 1 ?
(A) 5
4 - (B) 1 (C) 7
4 (D) 2 (E) 21
4
17. For x > 0, the power series ( ) ( )
246 2
1 1 3! 5! 7! 2 1!
n xxx n x
n
- + - + +- +
+ " " converges to which of the
following?
(A) cos x (B) sin x (C) sin x
x
 (D)
2 x x e e - (E)
2
1 x x + - e e
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18. The graph of f ¢, the derivative of a function f, consists of two line segments and a semicircle, as shown in the
figure above. If f ( ) 2 1, = then f ( ) - = 5
(A) 2 2 p -
(B) 2 3 p -
(C) 2 5 p -
(D) 6 2 - p
(E) 4 2 - p
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19. The function f is defined by ( ) . 2 = +
x f x
x What points ( ) x y, on the graph of f have the property that the
line tangent to f at ( ) x y, has slope 1 ? 2
(A) ( ) 0,0 only
(B) ( ) 1 1, 2 5 only
(C) ( ) 0,0 and ( ) -4,2
(D) ( ) 0,0 and ( ) 2 4, 3
(E) There are no such points.
20.
1
2
0
5 8
3 2
x dx
x x
+
+ +
ÛÙ
ı is
(A) ln 8( ) (B) ( ) 27 ln 2 (C) ln 18 ( ) (D) ln 288 ( ) (E) divergent 
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21. The line y = 5 is a horizontal asymptote to the graph of which of the following functions?
(A) sin 5( ) = x
y
x
 (B) y = 5x (C) 1
5 = - y
x
 (D) 5
1 = -
x
y
x
 (E)
2
2
20
1 4
- = +
x x
y
x
22. The power series ( )
0
3 n
n
n
a x
=
•
Â - converges at x = 5. Which of the following must be true?
(A) The series diverges at x = 0.
(B) The series diverges at x = 1.
(C) The series converges at x = 1.
(D) The series converges at x = 2.
(E) The series converges at x = 6.
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23. If P t( ) is the size of a population at time t, which of the following differential equations describes linear growth
in the size of the population?
(A) = 200 dP
dt
(B) = 200 dP t dt
(C) 2 = 100 dP t dt
(D) = 200 dP P dt
(E) 2 = 100 dP P dt
24. Let f be a differentiable function such that ( ) ( ) 3 f x x dx f x x x x dx sin =- + cos 4 cos . Ú Ú Which of the
following could be f x( ) ?
(A) cos x (B) sin x (C) 3 4x (D) 4 -x (E) 4
x
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25.
2
1
x xe dx
• -
Ú is
(A) 1
e - (B) 1
2e
 (C) 1
e
 (D) 2
e (E) divergent 
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26. What is the slope of the line tangent to the polar curve r = + 1 2sin q at q = 0 ?
(A) 2 (B) 1
2 (C) 0 (D) 1
2 - (E) -2
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27. For what values of p will both series 2
1
1
p n= n
•
Â and
1 2
n
n
p
=
• Ê ˆ ÂË ¯ converge?
(A) -< < 2 2 p only
(B) 1 1
2 2 -< < p only
(C) 1 2 2 < < p only
(D) 1
2 p < and p > 2
(E) There are no such values of p. 
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28. Let g be a continuously differentiable function with g( ) 1 6 = and g ( ) 1 3. =¢ What is
( )
( )
1
1
lim ? 6
x
x
g t dt
Æ g x -
Ú
(A) 0 (B) 1
2 (C) 1 (D) 2 (E) The limit does not exist.

76. The function f, whose graph is shown above, is defined on the interval -£ £ 2 2. x Which of the following
statements about f is false?
(A) f is continuous at x = 0.
(B) f is differentiable at x = 0.
(C) f has a critical point at x = 0.
(D) f has an absolute minimum at x = 0.
(E) The concavity of the graph of f changes at x = 0.
77. Let f and g be the functions given by ( ) = x fx e and ( ) 4
gx x = . On what intervals is the rate of change
of f x( ) greater than the rate of change of g x( ) ?
(A) ( ) 0.831, 7.384 only
(B) ( ) -•, 0.831 and ( ) 7.384, •
(C) ( ) - - •, 0.816 and ( ) 1.430, 8.613
(D) ( ) -0.816, 1.430 and ( ) 8.613, •
(E) ( ) -• •,
B B B B B BB B B
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78. The graph of the piecewise linear function f is shown above. What is the value of ( ) ( ) 9
1
3 2? - + Ú f x dx
(A) 7.5 (B) 9.5 (C) 27.5 (D) 47 (E) 48.5
79. Let f be a function having derivatives of all orders for x > 0 such that f ( ) 3 2, = f ¢( ) 3 1, = - f ¢¢( ) 3 6, = and
f ¢¢¢( ) 3 12. = Which of the following is the third-degree Taylor polynomial for f about x = 3 ?
(A) 2 3 2 6 12 -+ + xx x
(B) 2 3 2 32 -+ + xx x
(C) ( )( ) ( ) 2 3 2 3 6 3 12 3 --+ - + - xx x
(D) ( )( ) ( ) 2 3 2 33 3 4 3 --+ - + - xx x
(E) ( )( ) ( ) 2 3 2 33 3 2 3 --+ - + - xx x
B B B B B BB B B
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80. The graph of f ¢, the derivative of the function f, is shown above. Which of the following statements must be
true?
I. f has a relative minimum at x = -3.
II. The graph of f has a point of inflection at x = -2.
III. The graph of f is concave down for 0 4. < < x
(A) I only (B) II only (C) III only (D) I and II only (E) I and III only 
B B B B B BB B B
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0 1 < < x 1 2 < < x
f x( ) Positive Negative
f x ¢( ) Negative Negative
f x ¢¢( ) Negative Positive
81. Let f be a function that is twice differentiable on -< < 2 2 x and satisfies the conditions in the table above. If
fx f x () ( ) = - , what are the x-coordinates of the points of inflection of the graph of f on -< < 2 2? x
(A) x = 0 only
(B) x = 1 only
(C) x = 0 and x = 1
(D) x = -1 and x = 1
(E) There are no points of inflection on -< < 2 2. x
82. What is the average value of y = cos x on the interval 0 ? 2 x
p £ £
(A) -0.637 (B) 0.500 (C) 0.763 (D) 1.198 (E) 1.882 
B B B B B BB B B
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-29-
83. If the function f is continuous at x = 3, which of the following must be true?
(A) () () 3
3 limx f fx Æ <
(B) () ()
3 3
lim lim
x x
fx fx Æ Æ - + π
(C) () () ()
3 3
3 lim lim x x
f fx fx Æ Æ - + = =
(D) The derivative of f at x = 3 exists.
(E) The derivative of f is positive for x < 3 and negative for x > 3.
84. For - << 1.5 1.5, x let f be a function with first derivative given by ( ) ( ) 4 2 2 1
2. - + ¢ = - x x fx e Which of the
following are all intervals on which the graph of f is concave down?
(A) ( ) -0.418, 0.418 only
(B) ( ) -1, 1
(C) ( ) - - 1.354, 0.409 and ( ) 0.409, 1.354
(D) ( ) - - 1.5, 1 and ( ) 0, 1
(E) ( ) - - 1.5, 1.354 , ( ) -0.409, 0 , and ( ) 1.354, 1.5
B B B B B BB B B
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85. The fuel consumption of a car, in miles per gallon (mpg), is modeled by ( )
2
20 2400 6 ,
Ê ˆ - Á ˜ Ë ¯ =
s s
Fs e where s is the
speed of the car, in miles per hour. If the car is traveling at 50 miles per hour and its speed is changing at the rate
of 2 20 miles / hour , what is the rate at which its fuel consumption is changing?
(A) 0.215 mpg per hour
(B) 4.299 mpg per hour
(C) 19.793 mpg per hour
(D) 25.793 mpg per hour
(E) 515.855 mpg per hour 
B B B B B BB B B
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-31-
86. If f x( ) >¢ 0 for all real numbers x and ( ) 7
4 = 0, Ú f t dt which of the following could be a table of values for the
function f ?
(A) x f x( )
4 -4
5 -3
 7 0
(B) x f x( )
4 -4
5 -2
 7 5
(C) x f x( )
4 -4
 5 6
 7 3
(D) x f x( )
 4 0
 5 0
 7 0
(E) x f x( )
 4 0
 5 4
 7 6 
B B B B B BB B B
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87. Let R be the region in the first quadrant bounded above by the graph of y = - ln 3 , ( ) x for 0 2. £ £ x R is
the base of a solid for which each cross section perpendicular to the x-axis is a square. What is the volume of
the solid?
(A) 0.442 (B) 1.029 (C) 1.296 (D) 3.233 (E) 4.071 
B B B B B BB B B
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88. The derivative of a function f is increasing for x < 0 and decreasing for x > 0. Which of the following could
be the graph of f ?
(A) (B)
(C) (D)
(E) 
B B B B B BB B B
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89. A particle moves along a line so that its acceleration for t ≥ 0 is given by ( ) 3
3 .
1
+ = +
t a t
t
 If the particle’s
velocity at t = 0 is 5, what is the velocity of the particle at t = 3 ?
(A) 0.713 (B) 1.134 (C) 6.134 (D) 6.710 (E) 11.710
90. If the series
1
n
n
a
=
•
Â converges and 0 n a > for all n, which of the following must be true?
(A) 1 lim 0 n
n n
a
a
+
Æ• =
(B) 1 n a < for all n
(C)
1
0 n
n
a
=
•
Â =
(D)
1
n
n
na
=
•
Â diverges.
(E)
1
n
n
a
n =
•
Â converges. 
B B B B B BB B B
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91. The figure above shows the graphs of the polar curves r = 2cos 3( ) q and r = 2. What is the sum of the areas of
the shaded regions?
(A) 0.858 (B) 3.142 (C) 8.566 (D) 9.425 (E) 15.708 
B B B B B BB B B
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92. The function h is differentiable, and for all values of x, hx h x () ( ) = - 2 . Which of the following statements
must be true?
I. ( )
2
0
h x dx > 0 Ú
II. h ( ) 1 0 =¢
III. h h ¢ ¢ () () 0 21 = =
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III
t (minutes) 0 4 9 15 20
W t( ) (degrees Fahrenheit) 55.0 57.1 61.8 67.9 71.0
 1. The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W,
where W t( ) is measured in degrees Fahrenheit and t is measured in minutes. At time t = 0, the temperature of
the water is 55 F. ∞ The water is heated for 30 minutes, beginning at time t = 0. Values of W t( ) at selected
times t for the first 20 minutes are given in the table above.
(a) Use the data in the table to estimate W ¢( ) 12 . Show the computations that lead to your answer. Using correct
units, interpret the meaning of your answer in the context of this problem.
(b) Use the data in the table to evaluate ( ) 20
0
W t dt ¢ . Ú Using correct units, interpret the meaning of ( ) 20
0
W t dt ¢ Ú
in the context of this problem.
-5-
(c) For 0 20, £ £t the average temperature of the water in the tub is ( ) 20
0
1 . 20 W t dt Ú Use a left Riemann sum
with the four subintervals indicated by the data in the table to approximate ( ) 20
0
1 . 20 W t dt Ú Does this
approximation overestimate or underestimate the average temperature of the water over these 20 minutes?
Explain your reasoning.
(d) For 20 25, £ £t the function W that models the water temperature has first derivative given by
Wt t t ( ) =¢ 0.4 cos 0.06 . ( ) Based on the model, what is the temperature of the water at time t = 25 ?
 Continue problem 2 on page 7.
-6-
 2. For t ≥ 0, a particle is moving along a curve so that its position at time t is ( ) xt yt () () , . At time t = 2, the
particle is at position ( ) 1, 5 . It is known that 2
t
dx t
dt e
+ = and 2 sin . dy t dt =
(a) Is the horizontal movement of the particle to the left or to the right at time t = 2 ? Explain your answer.
Find the slope of the path of the particle at time t = 2.
(b) Find the x-coordinate of the particle’s position at time t = 4.
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-7-
(c) Find the speed of the particle at time t = 4. Find the acceleration vector of the particle at time t = 4.
(d) Find the distance traveled by the particle from time t = 2 to t = 4.
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 3. Let f be the continuous function defined on [-4, 3] whose graph, consisting of three line segments and a
semicircle centered at the origin, is given above. Let g be the function given by ( ) () 1 . x
g x f t dt = Ú
(a) Find the values of g( ) 2 and g( ) -2 .
(b) For each of g ( ) -¢ 3 and g¢¢( ) -3 , find the value or state that it does not exist.
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3 33 3 3 3 3 3 3 3
NO CALCULATOR ALLOWED
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-15-
(c) Find the x-coordinate of each point at which the graph of g has a horizontal tangent line. For each of these
points, determine whether g has a relative minimum, relative maximum, or neither a minimum nor a
maximum at the point. Justify your answers.
(d) For -< < 4 3, x find all values of x for which the graph of g has a point of inflection. Explain your
reasoning.
NO CALCULATOR ALLOWED
 Continue problem 4 on page 17.
-16-
x 1 1.1 1.2 1.3 1.4
f x ¢( ) 8 10 12 13 14.5
 4. The function f is twice differentiable for x > 0 with f ( ) 1 15 = and f ¢¢( ) 1 20. = Values of f ¢, the derivative of
f, are given for selected values of x in the table above.
(a) Write an equation for the line tangent to the graph of f at x = 1. Use this line to approximate f ( ) 1.4 .

(b) Use a midpoint Riemann sum with two subintervals of equal length and values from the table to
approximate ( ) 1.4
1 f x dx ¢ . Ú Use the approximation for ( ) 1.4
1 f x dx ¢ Ú to estimate the value of f ( ) 1.4 . Show
the computations that lead to your answer.
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(c) Use Euler’s method, starting at x = 1 with two steps of equal size, to approximate f ( ) 1.4 . Show the
computations that lead to your answer.
(d) Write the second-degree Taylor polynomial for f about x = 1. Use the Taylor polynomial to approximate
f ( ) 1.4 .
-18-
 5. The rate at which a baby bird gains weight is proportional to the difference between its adult weight and its
current weight. At time t = 0, when the bird is first weighed, its weight is 20 grams. If B t( ) is the weight of the
bird, in grams, at time t days after it is first weighed, then
( ) 1 100 . 5
dB B dt = -
 Let y = B t( ) be the solution to the differential equation above with initial condition B( ) 0 20. =
(a) Is the bird gaining weight faster when it weighs 40 grams or when it weighs 70 grams? Explain your
reasoning.
(b) Find
2
2
d B
dt
 in terms of B. Use
2
2
d B
dt to explain why the graph of B cannot resemble the following graph.
-19-
(c) Use separation of variables to find y = B t( ), the particular solution to the differential equation with initial
condition B( ) 0 20. =
 Continue problem 6 on page 21.
-20-
 6. The function g has derivatives of all orders, and the Maclaurin series for g is
( )
21 3 5
0
1 . 2 33 5 7
n n
n
x xx x
n
+
=
•
- =- + - Â + "
(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for g.
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6 66 6 6 6 6 6 6 6
-21-
(b) The Maclaurin series for g evaluated at 1
2 x = is an alternating series whose terms decrease in absolute
value to 0. The approximation for ( ) 1
2 g using the first two nonzero terms of this series is 17 . 120 Show that
this approximation differs from ( ) 1
2 g by less than 1 . 200
(c) Write the first three nonzero terms and the general term of the Maclaurin series for g x ¢( ).