1. CALCULUS AB
SECTION I, Part A
Time – 60 minutes
Number of questions – 30
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
Directions: Solve each of the following problems, using the available space for scratchwork. After
examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe
correspondingoval onthe answersheet. No creditwill be givenfor anythingwritteninthe
test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f ( x ) is a real number.
1. If f ( x ) = 5𝑥
4
3 , then f´( 8 ) =
( A ) 10
( B )
40
3
( C ) 80
( D )
160
3
2. lim
𝑥→∞
5𝑥2−3𝑥+1
4𝑥2+2𝑥+5
is
( A ) 0
( B )
4
5
( C )
5
4
( D ) ∞
3. If f ( x ) =
3𝑥2+𝑥
3𝑥2−𝑥
, then f´( x ) is
( A ) 1
( B )
6𝑥2+1
6𝑥2−1
( C )
−6
(3𝑥−1)2
( D )
−2𝑥2
(𝑥2−𝑥)2
2. 4. lim
𝑥→0
𝑆𝑖𝑛𝑥2
𝑥
=
( A ) 1
( B ) 0
( C )
𝜋
2
( D ) The limit does not exist
5. If y = sin ( xy ) , find
𝑑𝑦
𝑑𝑥
( A ) Cos ( xy )
( B )
𝑦𝑐𝑜𝑠(𝑥𝑦)
1−𝑥𝑐𝑜𝑠(𝑥𝑦)
( C ) x cos ( xy )
( D )
𝑦𝑐𝑜𝑠(𝑥𝑦)
1−𝑥
6. Whichof the followingintegralscorrectlycorrespondstothe areaof the shadedregioninthe figure
above? f ( x ) = 5 ; g ( x ) = 1 + x2
( A ) ∫ (𝑥2 − 4)𝑑𝑥
2
1
( B ) ∫ (4 − 𝑥2)𝑑𝑥
2
1
( C ) ∫ (𝑥2 − 4)𝑑𝑥
5
1
( D ) ∫ (4 − 𝑥2)𝑑𝑥
5
1
7. If h ( x ) = f ( x ) g ( 4x ) , then h´( 2 ) =
( A ) f´( 2 ) g´( 8 )
( B ) f´( 2 ) g ( 8 ) + 4 f ( 2 ) g´( 8 )
3. ( C ) f´( 2 ) g ( 8 ) + f ( 2 ) g´( 8 )
( D ) 4 f´( 2 ) g´( 8 )
8. An equation of the line normal to the graph of y = √3𝑥2 + 2𝑥 at ( 2 , 4 ) is
( A ) 4x + 7y = 20
( B ) - 7x + 4y = 2
( C ) 7x + 4y = 30
( D ) 4x + 7y = 36
9. ∫ (3𝑥 + 1)2 =
2
0
( A ) 38
( B ) 49
( C )
343
3
( D ) 343
10. If f ( x ) = 𝑐𝑜𝑠2 𝑥,then f ´´( π ) =
( A ) - 2
( B ) 0
( C ) 1
( D ) 2
11. If f ( x ) =
5
𝑥2+1
and g ( x ) = 3x , then g ( f ( 2 ) ) =
( A )
5
37
( B ) 3
( C ) 5
( D )
37
5
12. ∫ 𝑥√5𝑥2 − 4dx =
4. ( A )
1
10
(5𝑥2 − 4)
3
2 + C
( B )
1
15
(5𝑥2 − 4)
3
2 + C
( C )
20
3
(5𝑥2 − 4)
3
2 + C
( D )
3
20
(5𝑥2 − 4)
3
2 + C
13. The slope of the line tangent to the graph of 3𝑥2 + 5𝑙𝑛𝑦 = 12 at ( 2 , 1 ) is
( A ) -
12
5
( B )
12
5
( C )
5
12
( D ) - 7
14. The equation y = 2 – 3 Sin
𝜋
4
( x – 1 ) has a fundamental period of
( A )
1
8
( B )
4
𝜋
( C ) 8
( D ) 2𝜋
15. If f ( x ) = {
𝑥2 + 5𝒊𝒇𝒙 < 𝟐
7𝑥 − 5𝒊𝒇𝒙 ≥ 𝟐
, for all real numbers x, which of the following must be true ?
I f ( x ) is continuous everywhere
II f ( x ) is differentiable everywhere
III f ( x ) has a local minimum at x = 2
( A ) I only
( B ) I and II only
( C ) II and III only
( D ) I , II and III
16. For what value of x does the function f ( x ) = 𝑥3 − 9𝑥2 − 120𝑥 + 6 have a local minimum ?
5. ( A ) 10
( B ) 4
( C ) - 4
( D ) - 10
17. The accelerationof a particle movingalongthe x – axisat time t is givenby a ( t ) = 4t – 12 . If the
velocity is 10 when t = 0 and the position is 4 when t = 0, then the particle is changing direction at
( A ) t = 1
( B ) t = 3
( C ) t = 5
( D ) t = 1 and t = 5
18. What is the area of the region between y = 2x2
and y = 12 – x2
?
( A ) 0
( B ) 16
( C ) 32
( D ) 48
19 ∫(𝑒3𝑙𝑛𝑥 + 𝑒3𝑥)𝑑𝑥 =
( A ) 3 +
𝑒3𝑥
3
+ C
( B )
𝑒 𝑥4
4
+ 3𝑒3𝑥+ C
( C )
𝑒 𝑥4
4
+
𝑒3𝑥
3
+ C
( D )
𝑥4
4
+
𝑒3𝑥
3
+ C
20. If f ( x ) = ( x2
+ x + 11 ) √𝑥3 + 5𝑥 + 121, then f´( 0 ) =
( A )
5
2
( B )
27
2
( C ) 22
6. ( D )
247
2
21. If f ( x ) = 53x
, then f´( x ) =
( A ) 53x
( ln 125 )
( B )
53𝑥
3𝑙𝑛5
( C ) 3 ( 52x
)
( D ) 3 ( 53x
)
22. A solidisgeneratedwhenthe regionin the first quadrant enclosed by the graph of y =
( x2
+ 1 )3
,the line x = 1 ,the x – axis,andthe y – axis. Itsvolume isfoundbyevaluatingwhichof the
following integrals?
( A ) 𝜋 ∫ (𝑥2 + 1)3𝑑𝑥
8
1
( B ) 𝜋 ∫ (𝑥2 + 1)6𝑑𝑥
8
1
( C ) 𝜋 ∫ (𝑥2 + 1)3𝑑𝑥
1
0
( D ) 𝜋 ∫ (𝑥2 + 1)6𝑑𝑥
1
0
23. lim
𝑥→0
4
sin𝑥cos𝑥−sin𝑥
𝑥2
=
( A ) 2
( B )
40
3
( C ) 0
( D ) undefined
24. If
𝑑𝑦
𝑑𝑥
=
3𝑥2+2
𝑦
and y = 4 when x = 2, then when x = 3 ; y = ?
( A ) 18
( B ) 58
( C ) ±√74
( D ) ±√58
7. 25. ∫
𝑑𝑥
9+𝑥2
=
( A ) 3 tan- 1
(
𝑥
3
) + 𝐶
( B )
1
3
tan- 1
(
𝑥
3
) + 𝐶
( C )
1
3
tan- 1 (𝑥) + 𝐶
( D )
1
9
tan- 1 (𝑥) + 𝐶
26. If f ( x ) = cos3
( x + 1 ) , then f´( 𝜋) =
( A ) - 3 cos2
( 𝜋 + 1) sin (𝜋 + 1)
( B ) 3 cos2
( 𝜋 + 1)
( C ) 3 cos2
( 𝜋 + 1) sin (𝜋 + 1)
( D ) 0
27. ∫ 𝑥√𝑥 + 3 𝑑𝑥
( A )
2(𝑥+3)
3
2
3
+ 𝐶
( B )
2
5
(𝑥 + 3)
5
2 − 2(𝑥 + 3)
3
2 + C
( C )
3(𝑥+3)
3
2
2
+ C
( D )
4𝑥2(𝑥+3)
3
2
3
+ C
28. If f ( x ) = ln ( ln ( 1 – x ) ) , then f´( x ) =
( A ) -
1
ln(1−𝑥)
( B )
1
(1−x)ln(1−𝑥)
( C )
1
(1−𝑥)2
( D ) -
1
(1−x)ln(1−𝑥)
29. lim
𝑥→0
𝑥𝑒 𝑥
1−𝑒 𝑥
=
8. ( A ) - ∞
( B ) - 1
( C ) 1
( D ) ∞
30. ∫ 𝑡𝑎𝑛6 𝑥𝑠𝑒𝑐2 𝑥𝑑𝑥=
( A )
𝑡𝑎𝑛7 𝑥
7
+ C
( B )
𝑡𝑎𝑛7 𝑥
7
+
𝑠𝑒𝑐3 𝑥
3
+ C
( C )
𝑡𝑎𝑛7 𝑥𝑠𝑒𝑐3 𝑥
21
+ 𝐶
( D ) 7 tan7
x + C
9. CALCULUS AB
SECTION I, Part B
Time – 45 Minutes
Number of questions – 15
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION
Directions: Solve each of the following problems, using the available space for scratchwork. After
examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe
correspondingoval onthe answersheet. No creditwill be givenfor anything writteninthe
test book. Do not spend too much time on any one problem.
In this test: 1. The exactnumerical value of the correct answerdoesnot alwaysappearamongthe
choices given. When this happens, select from among the choices the number that best
approximates the exact numerical value.
2. Unlessotherwise specified,the domainof the functionf is assumedtobe the set of
all real numbers x for which f ( x ) is a real number.
31. ∫ sin 𝑥𝑑𝑥 +∫ cos 𝑥𝑑𝑥
0
−
𝜋
4
𝜋
4
0 =
( A ) - 1
( B ) 0
( C ) 1
( D ) √2
32. Boats A and B leave the same place at the same time. Boat A heads due north at 12 km/hr. Boat B
headsdue eastat 18 km/hr. After2.5 hours,how fast isthe distance betweenthe boatsincreasing(
in km/hr ) ?
( A ) 21.63
( B ) 31.20
( C ) 75.00
( D ) 9.84
33. If ∫ 𝑓(𝑥) 𝑑𝑥 = 𝐴𝑎𝑛𝑑 ∫ 𝑓(𝑥) 𝑑𝑥
100
50 = 𝐵, 𝑡ℎ𝑒𝑛
100
30 ∫ 𝑓(𝑥) 𝑑𝑥
50
30 =
( A ) A + B
( B ) A – B
( C ) B – A
10. ( D ) 20
34. If f ( x ) = 3x2
– x and g ( x ) = f – 1
( x ) , then g´( 10 ) could be
( A ) 59
( B )
1
59
( C )
1
10
( D )
1
11
35. lim
𝑥→0
3sin 𝑥−1
𝑥
=
( A ) 0
( B ) 1
( C ) ln 3
( D ) 3
36. The volume generatedbyrevolvingaboutthe y – axis the regionenclosedby the graphs y = 9 – x2
and y = 9 – 3x , for 0 ≤ 𝑥 ≤ 2 , is
( A ) - 8𝜋
( B ) 4𝜋
( C ) 8𝜋
( D ) 24𝜋
37. The average value of the function f ( x ) = ln 2
x on the interval [2,4] is
( A ) 1.204
( B ) 2.159
( C ) 2.408
( D ) 8.636
38.
𝑑
𝑑𝑥
∫ cos(𝑡)𝑑𝑡
3𝑥
0 =
11. ( A ) sin 3x
( B ) cos 3x
( C ) 3 sin 3x
( D ) 3 cos 3x
39. Find the average value of f ( x ) = 4 cos ( 2x ) on the interval from x = 0 to x = 𝜋
( A ) -
2
𝜋
( B ) 0
( C )
2
𝜋
( D )
4
𝜋
40. The radiusof a sphere isincreasingatarate proportional toitself. If the radiusis 4 initially,andthe
radius is 10 after two seconds, what will the radius be after three seconds?
( A ) 62.50
( B ) 15.81
( C ) 16.00
( D ) 25.00
41. Suppose f ( x ) = ∫ (𝑡3 + 𝑡)𝑑𝑡
𝑥
0 . Find f´( 5 )
( A ) 130
( B ) 120
( C ) 76
( D ) 74
42. ∫𝑙𝑛2𝑥𝑑𝑥=
( A )
ln 2𝑥
2𝑥
+ 𝐶
( B ) xlnx – x + C
( C ) xln2x – x + C
( D ) 2xln2x – 2x + C
12. 43. Given f ( x ) = {𝑎𝑥2 + 3𝑏𝑥 + 14;𝑥 ≤ 12
3𝑎𝑥 + 5𝑏; 𝑥 > 2
, findthe valuesof a and b that make f differentiable
for all x.
( A ) a = - 6 , b = 2
( B ) a = 6 ; b = - 2
( C ) a = 2 ; b = 6
( D ) a = 2 ; b = - 6
44. Two particles leave the origin at the same time and move along the y – axis with their respective
positions determinedby the functions y1 = cos 2t and y2 = 4 sin t for 0 < 𝑡 < 6. For how many
values of t do the particles have the same acceleration?
( A ) 0
( B ) 1
( C ) 2
( D ) 3
45. Find the distance traveled ( to three decimal places ) in the first four seconds, for a particle whose
velocity is given by v ( t ) = 7𝑒−𝑡2
, where t stands for time.
( A ) 0.976
( B ) 6.204
( C ) 6.359
( D ) 12.720
13. SECTION II
GENERAL INSTRUCTIONS
You may wish to look over the problems before starting to work on them, since it is not expected that
everyone will be able to complete all parts of all problems. All problems are given equal weight, but the
parts of a particular problem are not necessarily given equal weight.
A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS
SECTION OF THE EXAMINATION
You should write all work for each part of each problem in the space provided for that part in the
booklet. Be sure to write clearlyandlegibly. If youmake an error, youmay save time by crossingit
out rather than trying to erase it. Erased or crossed-out work will not be graded.
Show all your work. You will be graded on the correctness and completeness or your methods as
well as your answers. Correct answers without supporting work may not receive credit.
Justifications require that you give mathematical ( noncalculator ) reasons and that you clearly
identify functions, graphs, tables, or other objects you use.
You are permittedtouse your calculator to solve an equation,findthe derivative of a functionat a
point, or calculate the value of a definite integral. However, you must clearlyindicate the setup of
your problem, namely the equation, function, or integral you are using. If you use other built-in
features or programs, you must show the mathematical steps necessary to produce your results.
Your work mustbe expressedinstandardmathematical notationratherthancalculatorsyntax. For
example, ∫ 𝑥2 𝑑𝑥
5
1 may not be written as fnInt ( X2
, X , 1 , 5 ).
Unlessotherwise specified,answers( numericor algebraic) neednot be simplified. If your answer
is given as a decimal approximation, it should be correct to three places after the decimal point.
Unlessotherwisespecified,the domainof afunction f isassumedtobe the set of all real numbers x
for which f ( x ) is a real number.
SECTION II, PART A
Time – 30 minutes
Number of problems – 2
A graphing calculator is required for some problems or parts or problems.
During the timed portion for Part A, you may work only on the problems in Part A.
On part A, youare permittedtouse your calculatorto solve anequation,findthe derivative of afunctionat
a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your
problem, namely the equation, function, or integral you are using. If you use other built-in features or
programs, you must show the mathematical steps necessary to produce your results.
14. 1. A particle movesalongthe x-axissothatitsacceleration at any time t > 0 is given by a ( t ) = 12t –
18 . At time t = 1 , the velocity of the particle is v ( 1 ) = 0 and the position is x ( 1 ) = 9
( a ) Write an expression for the velocity of the particle v ( t )
( b ) At what values of t does the particle change direction ?
( c ) Write an expression for the position x ( t )
( d ) Find the total distance traveled by the particle from t =
3
2
to t = 6
2. Let R be the region enclosed by the graphs of y = 2lnx and y =
𝒙
𝟐
, and the lines x = 2 and x = 8
( a ) Find the area of R
( b ) Set up, but do not integrate, an integral expression, in terms of a single variable,for the
volume of the solid generated when R is revolved about the x-axis.
( c ) Set up, but do not integrate, an integral expression, in terms of a single variable,for the
volume of the solid generated when R is revolved about the line x = - 1
15. SECTION II, PART B
Time – 1 hour
Number of problems – 4
No calculator is allowed for these problems.
Duringthe timed portionforPart B, youmay continue toworkon the problemsinPartA withoutthe use of
any calculator.
3. Consider the equation x2
– 2xy + 4y2
= 52
( a ) Write an expression for the slope of the curve at any point ( x , y ).
( b ) Find the equation of the tangent lines to the curve at the point x = 2
( c ) Find
𝑑2 𝑦
𝑑𝑥2
at ( 0 , √21 )
4. Waterisdrainingatthe rate of 48𝜋 ft3
/secondfromthe vertexatthe bottomof aconical tankwhose
diameter at its base is 40 feet and whose height is 60 feet.
( a ) Findan expressionforthe volume of water inthe tank, in termsof its radius,at the surface
of the water.
( b ) At what rate is the radius of the water in the tank shrinking when the radius is 16 feet?
( c ) How fast is the height of the water in the tank dropping at the instant that the radius is 16
feet?
5. Let f be the function given by f ( x ) = 2x4
– 4x2
+ 1.
( a ) Find an equation of the line tangent to the graph at ( - 2 , 17 )
( b ) Findthe x-andy-coordinatesofthe relativemaximaandrelative minima. Verifyyouranswer.
( c ) Find the x-and y-coordinates of the points of inflection. Verify your answer.
6. Let F( x ) = ∫ [𝑐𝑜𝑠 (
𝑡
2
) +(
3
2
)]
𝑥
0 dt on the closed interval [0, 2𝜋]
( a ) Approximate F ( 2𝜋 ) using four right hand rectangles.
( b ) Find F´( 2𝜋 )
( c ) Find the average value of F´( x ) on the interval [0,4𝜋].
16. CALCULUS AB
SECTION I, Part A
Time – 60 minutes
Number of questions – 30
A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION
Directions: Solve each of the following problems, using the available space for scratchwork. After
examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe
correspondingoval onthe answersheet. No creditwill be givenfor anythingwritteninthe
test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f ( x ) is a real number.
1. If g(x) =
1
32
𝑥4 − 5𝑥2 , find g´(4)
( A ) - 72
( B ) - 32
( C ) 24
( D ) 32
2. lim
𝑥→0
8𝑥2
cos𝑥−1
=
( A ) - 16
( B ) - 1
( C ) 8
( D ) 6
3. lim
𝑥→5
𝑥2−25
𝑥−5
is
( A ) 0
( B ) 10
( C ) 5
( D ) The limit does not exist.
4. If f ( x ) =
𝑥5−𝑥+2
𝑥3+7
, find f´(x)
17. ( A )
5𝑥4−1
3𝑥2
( B )
(𝑥3+7)(5𝑥4−1)−(𝑥5−𝑥)(3𝑥2)
𝑥3+7
( C )
(𝑥5−𝑥+2)(3𝑥2)−(𝑥3+7)(5𝑥4−1)
(𝑥3+7)2
( D )
−(𝑥5−𝑥+2)(3𝑥2)+(𝑥3+7)(5𝑥4−1)
(𝑥3+7)2
5. Evaluate lim
ℎ→0
tan( 𝜋
4
+ℎ)−1
ℎ
( A ) 0
( B ) 1
( C ) 2
( D ) This limit does not exist
6. ∫ 𝑥√3𝑥𝑑𝑥 =
( A )
2√3
5
𝑥
5
2 + 𝐶
( B )
5√3
2
𝑥
5
2 + 𝐶
( C )
√3
2
𝑥
1
2 + 𝐶
( D )
5√3
2
𝑥
3
2 + 𝐶
7. For what value of k is f continuous at x = 1 if f ( x ) = {
𝑥2 − 3𝑘𝑥 + 2;𝑥 ≤ 1
5𝑥 − 𝑘𝑥2;𝑥 > 1
( A ) - 1
( B ) -
1
2
( C ) 2
( D ) 8
18. 8. Which of the followingintegralscorrectlygivesthe area of the regionconsistingof all points above
the x-axis and below the curve y = 8 + 2x – x2
?
( A ) ∫ (𝑥2 − 2𝑥 − 8) 𝑑𝑥
4
−2
( B ) ∫ (−𝑥2 + 2𝑥 + 8) 𝑑𝑥
2
−4
( C ) ∫ (−𝑥2 + 2𝑥 + 8) 𝑑𝑥
4
−2
( D ) ∫ (𝑥2 − 2𝑥 − 8) 𝑑𝑥
2
−4
9. Find
𝒅𝒚
𝒅𝒙
if y = sec ( 𝜋𝑥2 )
( A ) tan ( 𝜋𝑥2 )
( B ) ( 2𝜋𝑥 ) tan ( 𝜋𝑥2 )
( C ) sec ( 𝜋𝑥2 ) tan ( 𝜋𝑥2 )
( D ) sec ( 𝜋𝑥2 ) tan ( 𝜋𝑥2 ) ( 2𝜋𝑥 )
10. Given the curve y = 5 - (𝑥 − 2)
2
3 , find
𝑑𝑦
𝑑𝑥
at x = 2
( A ) -
2
3
( B ) -
2
3 √23
( C ) 5
( D ) The limit does not exist.
11. ∫
2
√1−𝑥2
𝑑𝑥
1
2
0 =
( A )
𝜋
3
( B ) -
𝜋
3
( C )
2𝜋
3
( D ) -
2𝜋
3
𝑑𝑥
12 Finda positive value c,forx, the satisfiesthe conclusion of the MeanTheoremfor Derivativesforf (
x ) = 3x2
- 5x + 1 on the interval [2,5]
( A ) 1
19. ( B )
11
6
( C )
23
6
( D )
7
2
13. Given f ( x ) = 2x2
– 7x – 10 , find the absolute maximum of f ( x ) on [−1, 3]
( A ) - 1
( B )
7
4
( C ) -
129
8
( D ) 0
14. Find
𝑑𝑦
𝑑𝑥
at ( 1 , 2 ) for y3
= xy – 2x2
+ 8
( A ) -
11
2
( B ) -
2
11
( C )
2
11
( D )
11
2
15. lim
𝑥→0
𝑥.2 𝑥
2 𝑥−1
=
( A ) ln 2
( B ) 1
( C ) 2
( D )
1
𝑙𝑛2
16. ∫𝑥𝑠𝑒𝑐2( 1 + 𝑥2 ) dx =
( A )
1
2
tan ( 1 + x2
) + C
( B ) 2 tan ( 1 + x2
) + C
( C )
𝑥
2
tan ( 1 + x2
) + C
20. ( D ) 2x tan ( 1 + x2
) + C
17. Find the equation of the tangent line to 9x2
+ 16y2
= 52 through ( 2 , - 1 )
( A ) - 9x + 8y – 26 = 0
( B ) 9x – 8y – 26 = 0
( C ) 9x – 8y – 106 = 0
( D ) 8x + 9y – 17 = 0
18. A particle´s position is given by s = t3
– 6t2
+ 9t . What is its acceleration at time t = 4 ?
( A ) 0
( B ) - 9
( C ) - 12
( D ) 12
19. If f ( x ) = 3 𝜋𝑥 , then f´(x) =
( A )
3 𝜋𝑥
ln3
( B )
3 𝜋𝑥
𝜋
( C ) 𝜋(3 𝜋𝑥)
( D ) 𝜋𝑙𝑛3(3 𝜋𝑥)
20 The average value of f ( x ) =
1
𝑥
from x = 1 to x = e is
( A )
1
𝑒+1
( B )
1
1−𝑒
( C ) e – 1
( D )
1
𝑒−1
21. If f ( x ) = 𝑠𝑖𝑛2 𝑥, find f´´´( x )
21. ( A ) - sin2
x
( B ) cos 2x
( C ) - 4 sin 2x
( D ) - sin 2x
22. Find the slope of the normal line to y = x + cos xy at ( 0 , 1 )
( A ) 1
( B ) - 1
( C ) 0
( D ) Undefined
23. ∫
𝑐𝑠𝑐2
√ 𝑥
√ 𝑥
𝑑𝑥 =
( A ) 2cot√ 𝑥 + C
( B ) - 2cot√ 𝑥 + C
( C )
𝑐𝑠𝑐2
√ 𝑥
3√ 𝑥
+ C
( D )
𝑐𝑠𝑐2
√ 𝑥
6√ 𝑥
+ C
24. lim
𝑥→0
𝑡𝑎𝑛3(2𝑥)
𝑥3
=
( A ) - 8
( B ) 2
( C ) 8
( D ) The limit does not exist.
25. A solid is generated when the region in the firstquadrant bounded by the graph of y = 1 + 𝑠𝑖𝑛2 𝑥,
the line x =
𝜋
2
, the x-axis, and the y-axis is revolvedabout the x-axis. Its volume is found by
evaluating which of the following integrals?
( A ) 𝜋 ∫ (1 + 𝑠𝑖𝑛4 𝑥) 𝑑𝑥
1
0
( B ) 𝜋 ∫ (1 + 𝑠𝑖𝑛2 𝑥)2 𝑑𝑥
1
0
22. ( C ) 𝜋 ∫ (1 + 𝑠𝑖𝑛4 𝑥) 𝑑𝑥
𝜋
2
0
( D ) 𝜋 ∫ (1 + 𝑠𝑖𝑛2 𝑥)2 𝑑𝑥
𝜋
2
0
26. If y = (
𝑥3−2
2𝑥5−1
)
4
, find
𝑑𝑦
𝑑𝑥
at x = 1
( A ) - 52
( B ) - 28
( C ) 13
( D ) 52
27. ∫ 𝑥√5 − 𝑥𝑑𝑥 =
( A ) -
10
3
(5 − 𝑥)
3
2
( B )
10
3
√
5𝑥2
2
−
𝑥3
3
+ C
( C ) 10 (5 − 𝑥)
1
2 +
2
3
(5 − 𝑥)
3
2 + C
( D ) -
10
3
(5 − 𝑥)
3
2 +
2
5
(5 − 𝑥)
5
2 + C
28. If
𝑑𝑦
𝑑𝑥
=
𝑥3+1
𝑦
and y = 2 when x = 2 , y =
( A ) √
27
2
( B ) √
27
8
( C ) ± √
27
8
( D ) ± √
27
2
29. The graph of y = 5x4
– x5
has an inflection point ( or points ) at
( A ) x = 3 only
( B ) x = 0 , 3
23. ( C ) x = - 3 only
( D ) x = 0 , - 3
30 ∫ tan 𝑥𝑑𝑥
1
0
( A ) 0
( B ) ln ( cos ( 1 ) )
( C ) ln ( sec ( 1 ) )
( D ) ln ( sec ( 1 ) ) – 1
24. CALCULUS AB
SECTION I, Part B
Time – 45 Minutes
Number of questions – 15
A GRAPHING CALCULATOR IS REQUIRED FOR SOME QUESTIONS ON THIS PART OF THE EXAMINATION
Directions: Solve each of the following problems, using the available space for scratchwork. After
examiningthe formof the choices,decidewhich isthe bestof the choicesgivenandfill inthe
correspondingoval onthe answersheet. No creditwill be givenfor anythingwritten inthe
test book. Do not spend too much time on any one problem.
In this test:
1. The exact numerical value of the correct answer does not always appear among the choices given.
When this happens, select from among the choices the number that best approximates the exact
numerical value.
2. Unlessotherwise specified,the domainof afunction f is assumedto be the set of all real numbersx
for which f ( x ) is a real number.
31. The average value of f ( x ) = 𝑒4𝑥2
on the interval [−
1
4
,
1
4
] is
( A ) 0.272
( B ) 0.545
( C ) 1.090
( D ) 2.180
32.
𝑑
𝑑𝑥
∫ 𝑠𝑖𝑛2 𝑡𝑑𝑡
𝑥2
0 =
( A ) x2
sin2
(x2
)
( B ) 2xsin2
(x2
)
( C ) sin2
(x2
)
( D ) x2
cos2
(x2
)
33. Find the value (s) of
𝑑𝑦
𝑑𝑥
of x2
y + y2
= 5 at y = 1
( A ) -
2
3
only
( B )
2
3
only
25. ( C ) ±
2
3
( D ) ±
3
2
34. The graph of y = x3
– 2x2
– 5x + 2 has a local maximum at
( A ) ( 2.120 , 0 )
( B ) ( 2.120 , - 8.061 )
( C ) ( - 0.786 , 0 )
( D ) ( - 0.786 , 4.209 )
35 Approximate ∫ 𝑠𝑖𝑛2 𝑥𝑑𝑥
1
0 using the Trapezoid Rule with n = 4 , to three decimal places.
( A ) 0.277
( B ) 0.555
( C ) 1.109
( D ) 2.219
36. The volume generated by revolving about the x-axis the regionabove the curve y = x3
, below the
line y = 1 , and between x = 0 and x = 1 is
( A )
𝜋
42
( B ) 0.143 𝜋
( C ) 0.643 𝜋
( D )
6𝜋
7
37. A sphere is increasing in volume at the rate of 20
𝑖𝑛.3
𝑠
. At what rate is the radius of the sphere
increasing when the radius is 4 in.?
( A ) 0.025
𝑖𝑛.
𝑠
( B ) 0.424
𝑖𝑛.
𝑠
( C ) 0.995
𝑖𝑛.
𝑠
( D ) 0.982
𝑖𝑛.
𝑠
26. 38. ∫
𝑙𝑛𝑥
3𝑥
𝑑𝑥 =
( A ) 6ln2 |𝑥| + C
( B )
1
3
𝑙𝑛2|𝑥| + C
( C )
1
6
𝑙𝑛2|𝑥| + C
( D )
1
3
𝑙𝑛|𝑥| + C
39. Find two non-negative numbers x and y whose sum is 100 and for which x2
y is a maximum.
( A ) x = 50 and y = 50
( B ) x = 33.333 and y = 66.667
( C ) x = 100 and y = 0
( D ) x = 66.667 and y = 33.333
40. Find the distance traveled ( to three decimal places ) from t = 1 to t = 5 seconds, for a particle
whose velocity is given by v ( t ) = t + ln t
( A ) 6.000
( B ) 1.609
( C ) 16.047
( D ) 148.413
41. ∫𝑠𝑖𝑛4(𝜋𝑥)cos(𝜋𝑥)𝑑𝑥 =
( A )
𝑠𝑖𝑛5(𝜋𝑥)
5𝜋
+ C
( B )
𝑠𝑖𝑛5(𝜋𝑥)
2𝜋
+ C
( C ) −
𝑐𝑜𝑠5(𝜋𝑥)
5𝜋
+ C
( D ) −
𝑐𝑜𝑠5(𝜋𝑥)
2𝜋
+ C
42. The volume of a cube isincreasing ata rate proportional toitsvolume atanytime t. If the volume is
8 ft3
originally, and 12 ft3
after 5 seconds, what is its volume at t = 12 seconds?
27. ( A ) 21.169
( B ) 22.941
( C ) 28.800
( D ) 17.600
43. If f ( x ) = (1 +
𝑥
20
)
5
, find f´´ ( 40 )
( A ) 0.068
( B ) 1.350
( C ) 5.400
( D ) 6.750
44. Find the equation of the line tangent to y = x tan x at x = 1
( A ) y = 4.983 x + 3.426
( B ) y = 4.983 x – 3.426
( C ) y = 4.983 x + 6.540
( D ) y = 4.983 x – 6.540
45. If f ( x ) is continuous and differentiable and f ( x ) = {
𝑎𝑥4 + 5𝑥; 𝑥 ≤ 2
𝑏𝑥2 − 3𝑥; 𝑥 > 2
, then b =
( A ) 0
( B ) 2
( C ) 6
( D ) There is no value of b.
28. SECTION II
GENERAL INSTRUCTIONS
You may wish to look over the problems before starting to work on them, since it is not expected that
everyone will be able to complete all parts of all problems. All problems are given equal weight, but the
parts of a particular problem are not necessarily given equal weight.
A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS
SECTION OF THE EXAMINATION
You should write all work for each part of each problem in the space provided for that part in the
booklet. Be sure to write clearlyandlegibly. If youmake an error, youmay save time by crossingit
out rather than trying to erase it. Erased or crossed-out work will not be graded.
Show all your work. You will be graded on the correctness and completeness or your methods as
well as your answers. Correct answers without supporting work may not receive credit.
Justifications require that you give mathematical ( noncalculator ) reasons and that you clearly
identify functions, graphs, tables, or other objects you use.
You are permittedtouse your calculator to solve an equation,findthe derivative of a functionat a
point, or calculate the value of a definite integral. However, you must clearlyindicate the setup of
your problem, namely the equation, function, or integral you are using. If you use other built-in
features or programs, you must show the mathematical steps necessary to produce your results.
Your work mustbe expressedinstandardmathematical notationratherthancalculatorsyntax. For
example, ∫ 𝑥2 𝑑𝑥
5
1 may not be written as fnInt ( X2
, X , 1 , 5 ).
Unlessotherwise specified,answers( numericor algebraic) neednot be simplified. If your answer
is given as a decimal approximation, it should be correct to three places after the decimal point.
Unlessotherwisespecified,the domainof afunction f isassumedtobe the set of all real numbers x
for which f ( x ) is a real number.
SECTION II, PART A
Time – 30 minutes
Number of problems – 2
A graphing calculator is required for some problems or parts or problems.
During the timed portion for Part A, you may work only on the problems in Part A.
On part A, youare permittedtouse your calculatorto solve anequation,findthe derivative of afunctionat
a point, or calculate the value of a definite integral. However, you must clearly indicate the setup of your
problem, namely the equation, function, or integral you are using. If you use other built-in features or
programs, you must show the mathematical steps necessary to produce your results.
1. The temperature onNewYear´s Day in Hinterlandwas givenby T(H) = - A – B cos (
𝜋𝐻
12
) , where T
esthe temperature indegreesFahrenheitandHisthe numberof hoursfrommidnight ( 0≤ 𝐻 < 24)
( a ) The initial temperature atmidnightwas - 15°F and at noonof New Year´sDay was 5°F. Find
A and B
( b ) Find the average temperature for the first 10 hours
( c ) Use the Trapezoid Rule with 4 equal subdivisions to estimate ∫ 𝑇(𝐻)𝑑𝐻
8
6
( d ) Find an expression for the rate that the temperature is changing with respect to H
29. 2. Sea grass grows on a lake. The rate of growth of the grass is
𝑑𝐺
𝑑𝑡
= kG, where k is a constant.
( a ) FindanexpressionforG,the amountof grassinthe lake ( in tons),intermsof t, the number
of years, if the amount of grass is 100 tons initially and 120 tons after one year.
( b ) In how many years will the amount of grass available be 300 tons ?
( c ) If fishare now introducedintothe lake andconsume a consistent80 tons/yearof seagrass,
how long will it take for the lake to be completely free of sea grass ?
30. SECTION II, PART B
Time – 1 hour
Number of problems – 4
No calculator is allowed for these problems.
During the timedportionfor Part B, you may continue to work on the problemsinPart A withoutto use of
any calculator.
3. Consider the curve defined by y = x4
+ 4x3
.
( a ) Find the equation of the tangent line to the curve at x = - 1
( b ) Find the coordinates of the absolute minimum.
( c ) Find the coordinates of the point ( s ) of inflection.
4. Water is being poured into a hemispherical bowl of radius 6 inches at the rate of 4 in.3
/ sec.
( a ) Giventhatthe volume ofthe waterinthesphericalsegmentshownaboveis V = 𝜋ℎ2 (𝑅 −
ℎ
3
),
where Risthe radiusof the sphere,findthe rate thatthe water levelisrisingwhen the water
is 2 inches deep.
( b ) Finan expressionforr,the radiusof the surface of the spherical segment of water,in terms
of h.
( C ) How fast is the circular area of the surface of the spherical segment of water growing ( in
in.2
/sec ) when the water is 2 inches deep?
5. Let R be the region in the first quadrant bounded by y2
= x and x2
= y.
( a ) Fin the area of region R
( b ) Find the volume of the solid generated when R is revolved about the x-axis.
( c ) The sectionof a certain solidcutby any plane perpendiculartothe x-axisisa circle withthe
endpointsof its diameterlying on the parabolas y2
= x and x2
= y. Findthe volume of the
solid
31. 6. An object moves with velocity v ( t ) = t2
– 8t + 7
( a ) Write a polynomial expression for the position of the particle at any time t ≥ 0
( b ) At what time (s) is the particle changing direction?
( c ) Find the total distance traveled by the particle from time t = 0 to t = 4
CALCULUS AB
SECTION I, Part A
Time – 60 Minutes
Numberof questions – 30
A CALCULATOR MAY NOT USED ON THIS PART OF THE EXAMINATION
Directions: Solve each of the following problems, using the available space for scratchwork. After
examiningthe formof the choices,decidewhichisthe bestof the choicesgivenandfill inthe
correspondingoval onthe answersheet. No creditwill be givenfor anythingwritteninthe
test book. Do not spend too much time on any one problem.
In this test: Unless otherwise specified, the domain of a function f is assumed to be the set of all real
numbers x for which f ( x ) is a real number.
1. ∫ cos(2𝑡) 𝑑𝑡
𝑥
𝜋
4
=
( A ) cos (2x)
( B )
sin(2𝑥)−1
2
( C ) cos ( 2x ) – 1
( D )
𝑠𝑖𝑛2(𝑥)
2
32. 2. What are the coordinates of the point of inflection on the graph of y = x3
– 15x2
+ 33x + 100 ?
( A ) ( 9 , 0 )
( B ) ( 5 , - 48 )
( C ) ( 9 , - 89 )
( D ) ( 5 , 15 )
3. If 3x2
– 2xy + 3y = 1 , then when x = 2 ,
𝑑𝑦
𝑑𝑥
=
( A ) - 12
( B ) - 10
( C ) -
10
7
( D ) 12
4. ∫
8
𝑥3
𝑑𝑥
3
1 =
( A )
32
9
( B )
40
9
( C ) 0
( D ) -
32
9
5. Whichof the followingintegralscorrectlycorrespondstothe areaof the shadedregioninthe figure
above? f ( x ) = 5 ; g ( x ) = 1 + x2
( A ) 1
( B ) 4
( C ) 8
( D ) 10
33. 6. lim
𝑥→0
𝑥−𝑠𝑖𝑛𝑥
𝑥2
=
( A ) 0
( B ) 1
( C ) 2
( D ) The limit does not exist.
7. If f(x) = x2
√3𝑥 + 1 , then f’(x) =
( A )
9𝑥2+2𝑥
√3𝑥+1
( B )
−9𝑥2+4𝑥
2√3𝑥+1
( C )
15𝑥2+4𝑥
2√3𝑥+1
( D )
−9𝑥2−4𝑥
2√3𝑥+1
8. What is the instantaneous rate of change at t = - 1 of the function f, if f(t) =
𝑡3+𝑡
4𝑡+1
?
( A )
12
9
( B )
4
9
( C ) −
4
9
( D ) −
12
9
9. ∫ (
4
𝑥−1
)
𝑒+1
2 𝑑𝑥=
( A ) 4
( B ) 4e
( C ) 0
( D ) - 4
34. 10. A car’s velocityisshownonthe graphabove. Which of the followinggivesthe total distancetraveled
from t = 0 to t = 16 ( in kilometers ) ?
( A ) 360
( B ) 390
( C ) 780
( D ) 1,000
11.
𝑑
𝑑𝑥
𝑡𝑎𝑛2
(4x) =
( A ) 8 tan( 4x )
( B ) 4 sec4
(4x)
( C ) 8 tan (4x)sec2
(4x)
( D ) 4 tan (4x)sec2
(4x)
12. What is the equation of the line tangent to the graph of y = sin2
x at x =
𝜋
4
?
( A ) y -
1
2
= (𝑥 −
𝜋
4
)
( B ) y -
1
√2
= (𝑥 −
𝜋
4
)
( C ) y -
1
√2
=
1
2
(𝑥 −
𝜋
4
)
( D ) y -
1
2
=
1
2
(𝑥 −
𝜋
4
)
13. If the function f (x) = {
3𝑎𝑥2 + 2𝑏𝑥 + 1; 𝑥 ≤ 1
𝑎𝑥4 − 4𝑏𝑥2 − 3𝑥;𝑥 > 1
is differentiable for all real values of x, the b =
( A ) -
11
4
( B )
11
4
( C ) 0
35. ( D ) -
1
4
14. The graph of y = x4
+ 8x3
– 72x2
+ 4 is concave down for
( A ) - 6 < 𝑥 < 2
( B ) x > 2
( C ) x <−6
( D ) - 3 – 3 √5 < 𝑥 <−3 + 3√5
15. lim
𝑥→∞
ln(𝑥+1)
log2 𝑥
=
( A )
1
ln 2
( B ) 0
( C ) 1
( D ) ln 2
16.
The graph of f(x) isshowninthe figure above. Whichof the following could be the graph of f´(x) ?
36. 17. If f(x) = ln ( cos (3x) ) , then f´(x) =
( A ) 3 sec ( 3x )
( B ) 3 tan ( 3x )
( C ) - 3 tan ( 3x )
( D ) - 3 cot ( 3x )
18. If f(x) = ∫ √𝑡2 − 1
3𝑥+1
0 dt , then f´( - 4 ) =
( A ) - 2
( B ) 2
( C ) √153
( D ) 0
19. A particle movesalongthe x-axissothat itspositionat time t, inseconds, is given by x(t) = t2
– 7t +
6. For what value ( s ) of t is the velocity of the particle zero ?
( A ) 1
( B) 6
( C ) 1 or 6
( D ) 3.5
20. ∫ sin(2𝑥)𝑒 𝑠𝑖𝑛2 𝑥
𝜋
2
0
( A ) e – 1
( B ) 1 – e
( C ) e + 1
( D ) 1
21.
