FOR ALL STUDENTS
TAKING Calculus
2013-2014
SUMMER REVIEW PACKET
1
Dear Student and Parent/Guardian,
The math department at Tarpon Springs High School wants you
to be successful in Calculus. This summer packet is designed to
help you reach these goals by reviewing necessary skills.
Be sure to follow the key information below when completing
this packet:
The packet is due when you return to school in August.
Every problem must be completed. None left blank.
The packet is worth 10 times a regular homework grade.
Work must be shown to receive credit – no work, no points.
Final answers must be circled.
Use any resources available to you: Internet, Text Books,
etc.
When you return in August, you will have the opportunity
to ask questions. Math Help will also be available during
the first week.
We hope that you have an enjoyable summer and return to
school ready to be successful in Calculus!
Helpful Websites
www.glencoe.com
www.wolframalpha.com
www.purplemath.com/modules
www.khanacademy.org
apcentral.collegeboard.org
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Calculus Summer Packet
In exercises 1 – 4, match the equation with its graph.
1. y   12 x  2
______
2. y  9  x 2
______
3. y  4  x 2
______
4. y  x 3  x
______
In exercises, 5 - 12 find any intercepts.
5. y  x 2  x  2
__________________
6. y 2  x 3  4 x
7. y  x 2 25  x 2
__________________
8. y   x  1 x 2  1 __________________
__________________
10. y 
9. y 

32 x
x

11. x 2 y  x 2  4 y  0 __________________
x 2  3x
3x  12
__________________
__________________
12. y  2 x  x 2  1 __________________
In exercises, 13 - 17 sketch the graph of the equation. Identify any intercepts, test for
symmetry and find domain and range.
13. y  3 x  2
3
Calculus Summer Packet
14. y  1  x 2
15. y  x 3  2
16. y  x x  2
17. y 
1
x
4
Calculus Summer Packet
18.
Find equations in General Form of the lines passing through (-2, 4) and having the following
characteristics.
a) Slope of
7
16
b) Parallel to the line 5 x  3 y  3
c) Passing through the origin
d) Parallel to the y-axis.
19. Find equations of the lines passing through (1, 3) and having the following characteristics.
a) Slope of
2
3
b) Perpendicular to the line x + y = 0
c) Passing through the point (2, 4)
d) Parallel to the x-axis
20. Find the domain and range of each function.
a)
y  36  x 2
b) y 
c)
7
2 x  10
x2 ,
y
2  x ,
x0
x0
5
Calculus Summer Packet
21. Simplify the expression:
2 x  1
22. Simplify the expression:
x  13 4 x  9  16 x  9x  12
x  6x  13
1
2
  x  2 2 x  1
1
2
23x  1 3  2 x  1 13 3x  1
23. Simplify the expression:
3x  12 3
1
24. Solve:
x  1
1
2
x  1x  1

x3  1
1
2
2
3
3
 0 for x .
25. Solve: x 2  x  2
6
Calculus Summer Packet
Numbers 26 to the end is part of a diagnostic tool to help you judge your familiarity with precalculus
topics. If you do not remember some of the topics, go on the web and look up a lesson.
26. For all angles  in degree or radian measure, we know that cos 2 θ  
(a)
sin 2 θ   1
2sinθ cosθ 
(b)
(c)
1 - sinθ 
(d)
1 - sin 2 θ 
27. Let a  0 . If we know that loga 3  1.8 , then
a  31.8
(a)
we also know that
(c)
we also know that a  3
28. Convert
(a)
3  a 1.8
(b)
we also know that
(d)
we also know that a  e, where e is euler’s constant.
π
radians to degrees
9
20 
(b)
40 


(c)
60 
80 
(d)
29. If we write ln x x 2  1 as a sum of natural logs, we obtain
1
ln x - 1
2
(a)
ln x   ln x   ln 1 
(b)
ln x   
(c)
1
1
ln x    lnx  1    ln x - 1 
2
2
(d)
the same expression, because it cannot be simplified.
30. Suppose we know that a = 5 cm, b = 3 cm, and A = 53  in a certain triangle.
According to the Law of Sines,
(a)
(b)
(c)
(d)

angle B must have approximate measure .48 .
angle B must be obtuse.
there are two triangles which meet the criteria.
there is exactly one triangle which meets the criteria.
31. The best first step in solving the equation 32x 1  5 would be
(a)
taking the 2x  1 root of both sides.
2x 1
2x
(b)
rewriting 3
as 3  3
(c)
(d)
taking the cube root of both sides.
taking the natural log of both sides.
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Calculus Summer Packet
32. Which of the following equations is the same as 2x 2  3x  1  0 ?
(a)
3
9

2 x 2  x   - 1 = 0
2
16 

(c)
3

2 x   - 1 = 0
2

(b)
3
9 9

2 x 2  x   - - 1 = 0
2
16  8

(d)
3

2 x   - 1 = 0
2

2
2
33. A good first step in solving the equation 2x  1  2x  1 would be to rewrite the
equation as
(a)
2x  2x
(b)
(c)
2x  1 = 2x + 1
(d)
2x  1  2x  1  0
2x  12  2x  1
34. Which one of the following statements is true?
a 
(a)
a
(c)
a 3  3 a2
2
3

3
2
(b)
 1 
a 3 2
a 
(d)
a 3  2a 3
35. Which of the following equals 1 
(a)
cos
(b)
3
2
2
sin 2
?
1  cos
- cos
(c)
1 - sin
(d)
1  sin
36. For all angles  in degree measure, we know that sin =
(a)
(c)
cos90    
sec90    
(b)
(d)
sin90    
sin  90  
37. If we know that  is such that sin  
(a)
(c)
4
5
4
csc  
3
cos  
(b)
(d)
3
5
and tan  
3
, then we know
4
5
4
4
cos  
5
sec  
8
Calculus Summer Packet
 2 
38. The exact value of cos
 is:
 3 
(a)
3
2
2
2
(b)
(c)

(c)
1
2
3
2
d)

d)
1
1
2
 3 
39. The exact value of sin  is:
 2 
(a)
-1
0
(b)
40. Find the exact value of tan -1 - 1 and cos -1 - 1 .
(a)
3
,
4
41. If cos  
(a)
-3
10
42. If sin  
(a)
5
6
(b)

4
,0
(c)
3 3
,
4 2
d)
-
,
4
-3
3
 
and    
, then find cos  .
5
2
2
5
5
(b)
(c)
-2 5
5
d)
 5
5
1


and  lies in Quadrant II, find the exact value of sin   .
3
6

3 2 2
6
(b)
(c)
3 2 2
6
d)
3 1
2
43. What are the first four positive solutions of the equation sin2  
(a)
(c)
 5 13 17
,
,
,
6 6 6
6
 2 7 8
,
,
,
3 3 3 3
(b)
d)
1
?
2
 5 13 17
,
,
,
12 12 12 12
 5 7 11
,
,
,
6 6 6 6
9
Calculus Summer Packet
44. Find all solutions in the interval 0 2π  for the equation 2cos 2  1  0 .
(a)
 7
3 5
,
4 4
(b)
,
4 4
(c)
 3 5 7
,
4 4
,
4
,
4
d)
 5
,
3 3
45. A ship, off-shore from a vertical cliff known to be 200 feet high, takes a sighting
of the top of a cliff. If the angle of elevation is found to be 15 degrees,
approximately how far off-shore is the ship?
(a)
3000 ft.
1500 ft.
(b)
46. The terminal side of  
(a)
(c)
(c)
500 ft.
d)
750 ft.
23
lies in
3
Quadrant I
Quadrant III
(b)
(d)
Quadrant II
Quadrant IV
47. If f(x) = 5x + 4, then the inverse of f will
(a)
(b)
(c)
(d)
subtract 4 from its input, then divide by 5.
divide its input by 5, then subtract 4.
divide its input by 4, then subtract 5.
subtract 5 from its input, then divide by 4.
48. If a population of lemmings is growing at a relative annual rate of 2.2%, how many
lemmings will there be in five years, assuming the initial population is 500? Round
to the nearest lemming.
(a)
(c)
556
557
(b)
(d)
555
558
49. If f(x) = x 2  1 , then f  fx  is given by the formula



(a)
y = x 2  1 x 2  1 x  (b)
y = 2x 2  2
(c)
y = x 4  2x 2
(d)
y = x 4  2x 2
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Calculus Summer Packet
x 2  3x  y 2  5
50. x  1 and y  1 is a solution to the system of equations:
(a)
TRUE
(b)
2x 2  4x  y  7
FALSE
3
51. If Arctan    , then we know
5
(a)
(c)
3
5
3
sin  
5
cot  
(b)
(d)
3
5
3
tan  
5
tan  
52. The range of y  2 sin 2x - 3  5 is:
(a)
 2, 2
(b)
(c)
0, 5
d)
 
 2 2 
 3 , 3 
3, 7 
 
53. Find the exact value of 2 sin 15  cos 15 
(a)
0
(b)
1
2
(c)
2
2
d)
3
2
54. If y  2  3sin4x  1 , then we know
(a)
(b)
the midline of the sinusoid is y=3.
the amplitude of the sinusoid is 2.
(c)
the period of the sinusoid is
(d)
the horizontal translation of the sinusoid is one unit left.
x
55. The period of y  tan   is:
3
(a)

3
(b)
2
3

2
(c)
3
(d)
6
11
Calculus Summer Packet
  7
56. Find the exact value of sin -1  sin
  9
(a)
2
9
(b)

 

7
9
(c)
9
2
(d)
9
5
57. If the average rate of change for a function f on the interval [2, 5] is -3, then we
know that
(a)
(b)
(c)
(d)
the function is increasing on the interval [2, 5].
the function is decreasing on the interval [2, 5].
the function f has a turning point in the interval [2, 5].
the slope of the line connecting the points 2, f 2  and 5, f 5 is -3.
58. Suppose an ant is sitting on the perimeter of the unit circle at the point (0, -1).
2
in the clockwise direction, then the coordinates
If the ant travels a distance of
3
of the point where the ant stops will be
(a)
(c)

3 1


 2 ,2


1 3
 ,

2 2 


(b)
(d)
 1 3
 ,

 2 2 



3 1


 2 , 2 


59. Suppose you deposit $1,000 into an account which pays 4% annual interest,
compounded quarterly. Approximately, how long will it take for the amount of
money in the account to double?
(a)
(c)
About 25 years
About 17.3 years
(b)
(d)
About 17.4 years
About 25.2 years
60. In a triangle, suppose we know that side b = 3 feet, side c = 2 feet, and that angle
A  140  . According to the Law of Cosines, the length of side a is approximately
(a)
(c)
17.6 feet
4.7 feet
(b)
(d)
22 feet
3.6 feet
12
Calculus “2012” Summer Packet
Answer Sheet
21. ___________________
42. ___________________
1. ___________________
22. ___________________
43. ___________________
2. ___________________
23. ___________________
44. ___________________
3. ___________________
24. ___________________
45. ___________________
4. ___________________
25. ___________________
46. ___________________
5. ___________________
26. ___________________
47. ___________________
6. ___________________
27. ___________________
48. ___________________
7. ___________________
28. ___________________
49. ___________________
8. ___________________
29. ___________________
50. ___________________
9. ___________________
30. ___________________
51. ___________________
10. ___________________
31. ___________________
52. ___________________
11. ___________________
32. ___________________
53. ___________________
12. ___________________
33. ___________________
54. ___________________
13. ___________________
34. ___________________
55. ___________________
14. ___________________
35. ___________________
56. ___________________
15. ___________________
36. ___________________
57. ___________________
16. ___________________
37. ___________________
58. ___________________
17. ___________________
38. ___________________
59. ___________________
18. ___________________
39. ___________________
60. ___________________
19. ___________________
40. ___________________
20. ___________________
41. ___________________
13