Calculus
Summer Packet 2012
(Review of Precalculus concepts)
Name ____________________________
1
Calculus
Summer Packet 2012
(Review of Trigonometry/Precalculus Concepts)
The purpose of this summer packet is to review the Precalculus concepts
that are essential for Calculus.
1. The packet is due August 31, 2012 - NO EXCEPTIONS unless by Teacher
2. The packet is worth 50 points. Points are awarded for accuracy. WORK MUST BE
SHOWN! NO WORK, NO POINTS! Show all work on the question packet in
the given space for each question. Transfer answers to the answer sheet.
3. A test covering the material in the packet may be given on Friday, August 31, 2012.
You are responsible for preparing yourself for the exam.
a. Weblinks have been included in the packet.
b. I will review packet and answer questions, the first week of school.
c. Math help will be available for that 1st week of school.
4. Extra Copies of the packet and answer sheet can be found in the office, guidance
office or you may contact Mrs. Romberger.
2
Calculus Summer Packet
Review of Precalculus
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  3x  2
3
14. y  1  x 2
15. y  x 3  2
16. y  x x  2
17. y 
1
x
4
Calculus Summer Packet
Review of Precalculus
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
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3 x  1 3  2 x  1 13 3 x  1
23. Simplify the expression:
2
3x  1 3
1
24. Solve:
x  1  x  13x  1
1
2
x 1
1
2
2
3
3
 0 for x .
25. Solve: x 2  x  2
6
Calculus Summer Packet
Review of Precalculus
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. I have
attached some web sites to help at the end of this test.
26. If  is measured in radians and cos θ 
(a)

3
5
(b)
3
5
3
, then we know that cos θ  2  
5
(c)

4
5
(d)
4
5
27. For all angles  in degree or radian measure, we know that cos 2 θ  
(a)
sin 2 θ   1
(b)
2sinθcosθ
1 - sinθ
(c)
(d)
1 - sin 2 θ 
28. A line that is perpendicular to y  2x  5 will have slope
(a)
m  2
(b)
m
1
2
(c)
m
1
2
(d)
m2
29. Let a  0 . If we know that loga 3  1.8 , then
(a)
we also know that a  3
(b)
we also know that 3  a
(c)
we also know that
a 3
(d)
we also know that
1.8
30. Which of the following equations is the same as
1.8
a  e, where e is euler’s constant.
3
8

?
x -1 x  2
(a)
3x  2  8x  1
(b)
3x  28x  1  0
(c)
3x  6  8x - 8
(d)
3
8

0
x 2 x 1
31. If the radian measure of  is 3, then we know that the terminal side of  lies in
(a)
Quadrant I
(b)
Quadrant II
(c)
Quadrant III
(d)
Quadrant IV
7
32. Convert
(a)
π
radians to degrees
9
20 
(b)
40 


(c)
60 
80 
(d)
33. If we write ln x x 2  1 as a sum of natural logs, we obtain
(a)
lnx  lnx  ln1
(c)
lnx   
1
1
lnx  1   lnx - 1
2
2
1
lnx - 1
2
(b)
lnx   
(d)
the same expression, because it cannot be simplified.
34. For any positive real number a, we know the following about y  loga x :
(a)
(b)
(c)
(d)
its graph is always increasing
its graph is always decreasing
its graph never crosses the x-axis
its graph never crosses the y-axis
35. 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.

36. One form of the equation from the Law of Sines relating the angles A and C to the
sides a and c
(a)
(c)
sin A sin C 

c
a
sin A c

sin C  a
(b)
sin A sin C 

a
c
(d)
a 2  c2  sin 2 A  sin 2 C 
8
Calculus Summer Packet
Review of Precalculus
37. When the equation of a line is written in the form y  mx  b , the constant b
represents
(a)
(b)
(c)
(d)
the slope of the line
the y-coordinate of the point where the line crosses the y-axis.
the x-coordinate of the point where the line crosses the x-axis.
the change in y divided by the change in x.
38. The best first step in solving the equation 32x1  5 would be
2x  1 root of both sides.
2x 1
2x
rewriting 3
as 3  3
(a)
taking the
(b)
(c)
(d)
taking the cube root of both sides.
taking the natural log of both sides.
39. 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
40. 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
41. The method for solving log2 x  log2 x  1  1 yields two possible solutions, namely
x = 1 and x = -2. From this, we know
(a)
(b)
(c)
(d)
both x = 1 and x = -2 are solutions.
only x = 1 is a solution.
only x = -2 is a solution.
neither x = 1 nor x=-2 is a solution.
9
42. Which one of the following statements is true?
  a
3
(a)
a
(c)
a 3  3 a2
2
3
2
(b)
 1 
a 3 2
a 
(d)
a 3  2a 3
3
2
2
43. For all angles A, B both in degree or radian measure, we know that cos A - B
(a)
(c)
cosAsinB  sinAcosB
2sinAcosB
(b)
(d)
44. Which of the following equals 1 
(a)
cos
(b)
cosAcosB  sinAsinB
1 - sinAsinB
sin 2
?
1  cos 
- cos
(c)
1 - sin
(d)
1  sin
45. For all angles  in degree measure, we know that sin =
(a)
(c)

sec 90

 
cos 90   



sin  90 
(b)
sin 90   
(d)

46. If 0 < a < 1 , then we know that the graph of y = a x
(a)
(b)
(c)
(d)
always passes through the point 1,0  .
always passes through the point - 1,0  .
has a horizontal asymptote along the x-axis.
has a vertical asymptote along the y-axis.
47. A point (a, b) is known to lie on the graph of a line. If we reach another point on
the line by moving three units to the right and two units down from (a, b), then the
slope of this line is
(a)
(c)
3
2
3
m
2
m
(b)
(d)
2
3
2
m
3
m
10
Calculus Summer Packet
Review of Precalculus
48. 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   
 2 
49. The exact value of cos 
 is:
 3 
(a)
3
2
(b)
2
2
(c)

(c)
1
2
3
2
d)

d)
1
1
2
 3 
50. The exact value of sin  is:
 2 
(a)
-1
(b)
0
51. Find the exact value of tan -1 - 1 and cos -1 - 1 .
(a)
3
,
4
52. If cos  
(a)
-3
10
53. If sin  
(a)
5
6
(b)

4
,0
(c)
3 3
,
4 2
d)
-
,
4
-3
3
 
and    
, then find cos   .
5
2
2
(b)
5
5
(c)
-2 5
5
d)
 5
5
1


and  lies in Quadrant II, find the exact value of sin    .
3
6

(b)
3 2 2
6
(c)
3 2 2
6
d)
3 1
2
11
54. What are the first four positive solutions of the equation sin 2  
(a)
(c)
 5 13 17
,
,
(b)
,
6 6 6
6
 2 7 8
,
,
,
3 3 3 3
d)
1
?
2
 5 13 17
,
,
,
12 12 12 12
 5 7 11
,
,
,
6 6 6 6
55. Find all solutions in the interval 0 2π for the equation 2cos 2  1  0 .
(a)
 7
4
,
4
(b)
3 5
,
4 4
(c)
 3 5 7
,
4 4
,
4
,
4
d)
 5
,
3 3
56. 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.
(b)
57. The terminal side of  
(a)
(c)
1500 ft.
(c)
500 ft.
d)
750 ft.
23
lies in
3
Quadrant I
Quadrant III
(b)
(d)
Quadrant II
Quadrant IV
58. If we know that the solutions of the equation u2  5u  6 are u = 2 and u = 3, then
2
what are the solutions to the equation 3x  2  53x  2  6 = 0 ?
(a)
(c)
4
5
and x =
3
3
x  2 and x = 3
x
(b)
(d)
4
5
and x = 
3
3
only x  2
x
59. Let a > 0. As long as m and n are both positive, we know
(a)
(c)
loga mn   loga m  n 
loga mn   nloga m
(b)
(d)
loga mn   loga m  loga n 
loga mn   mlog a n 
12
Calculus Summer Packet
Review of Precalculus
60. Let a > 0. If we know that 2, 5 lies on the graph of y = a x , then we know that
(a)
(c)
5  a2
2  a5
(b)
(d)
a  52
loga 2  5
61. 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.
62. 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
(a)
(b)
the graph of g is the reflection of the graph of f about the y-axis
the graph of g never crosses the x-axis
63. A function g is the inverse of a function f provided
(c)
fgx   x and gfx   x whenever these expressions are defined
(d)
both f and g have the same domain.
64. If f(x) = x2  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
65. x  1 and y  1 is a solution to the system of equations:
(a)
TRUE
(b)
x 2  3x  y 2  5
2x 2  4x  y  7
FALSE
13
66. If we were to graph the function f(x) = 3x2  1 on the interval 1  x  2 , then we
would
(a)
(b)
(c)
(d)
- 1, 2 and an open circle at the point 2, 11
place an open circle at the point - 1, 2 and an closed circle at the point 2, 11
place an closed circle at the point - 1, 2 and an open circle at the point 2, 11
place an closed circle at the point - 1, 2 and an closed circle at the point 2, 11
place an open circle at the point
3
67. If Arctan     , then we know
5
(a)
(c)
3
5
3
sin   
5
cot   
(b)
(d)
3
5
3
tan   
5
tan   
68. The range of y  2 sin 2x - 3  5 is:
(a)
 2, 2
(b)
(c)
0, 5
d)
 
 2 2 
 3 , 3 
3, 7
 
69. Find the exact value of 2 sin 15  cos 15 
(a)
0
(b)
1
2
(c)
2
2
d)
3
2
70. 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.

2
x
71. The period of y  tan   is:
3
(a)

3
(b)
2
3
(c)
3
(d)
6
14
Calculus Summer Packet
Review of Precalculus
  7
72. Find the exact value of sin -1  sin
  9
(a)
2
9
(b)

 

7
9
9
2
(c)
(d)
9
5
73. 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)
74. If fx 
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.
x  1 , then f  f  f8 
{  means multiply}
(a)
(c)
6
27
(b)
(d)
2
3
75. Suppose an ant is sitting on the perimeter of the unit circle at the point (0, -1).
2
If the ant travels a distance of
in the clockwise direction, then the coordinates
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 


76. What can be said about the function y  x 2  1/ x 2  x  2?
(a)
(b)
(c)
(d)
The
The
The
The
function has two vertical asymptotes, one at x = -1, the other at x = 2.
function has exactly one vertical asymptote at x = 2 and a horizontal asymptote at y = 1.
function has exactly one vertical asymptote at x = -1 and no horizontal asymptotes.
function has no vertical asymptotes and a horizontal asymptote at y = 1.
15
77. The horizontal translation of the function f(x)  2  4cos4x  5 is
(a)
(b)
(c)
Exactly two units to the right compared to the basic cosine function.
Exactly five units to the left compared to the basic cosine function.
Exactly 1.25 units to the right compared to the basic cosine function.
(d)
Exactly

18
(a)
18
78. cos 

2
units to the left compared to the basic cosine function.
 isin 

18
18
= using DeMoivre’s Theorem
(b)
3  4i
(c)
9i
(d)
-1
79. 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
80. 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
16
Calculus Summer Packet
Review of Precalculus
Helpful Web Sites
Rational/Irrational Numbers
http://www.themathpage.com/aPreCalc/rational-irrational-numbers.htm
http://www.regentsprep.org/Regents/math/rational/Lrat.htm
Absolute Value
http://www.purplemath.com/modules/absolute.htm
http://www.regentsprep.org/Regents/math/absvalue/Labsolute.htm
Midpoint
http://www.regentsprep.org/Regents/math/midpoint/Lmidpoint.htm
Pythagorean Theorem
http://www.purplemath.com/modules/perimetr3.htm
Scientific Notation
http://www.regentsprep.org/Regents/math/scinot/page1.htm
Adding Polynomials:
http://www.regentsprep.org/Regents/math/polyadd/sp_add.htm
http://www.purplemath.com/modules/polyadd.htm
Subtraction Polynomials:
http://www.regentsprep.org/Regents/math/polyadd/sp_subt.htm
http://www.purplemath.com/modules/polyadd2.htm
Multiplying Polynomials:
http://www.regentsprep.org/Regents/math/polymult/Smul_bin.htm
http://www.purplemath.com/modules/polymult.htm
http://www.regentsprep.org/Regents/math/polymult/Strinom.htm
Radicals:
Simplifying:
http://www.regentsprep.org/Regents/math/radicals/Lsimplify.htm
http://www.purplemath.com/modules/radicals.htm
Addition/Subtraction of Radicals:
http://www.regentsprep.org/Regents/math/radicals/Laddsubt.htm
17
Multiplication/Division of Radicals:
http://www.regentsprep.org/Regents/math/radicals/Lmultdiv.htm
http://www.purplemath.com/modules/radicals3.htm
Exponents:
Multiplication
http://www.regentsprep.org/Regents/math/polymult/rule_pmu.htm
http://www.purplemath.com/modules/exponent.htm
Raising to a power:
http://www.regentsprep.org/Regents/math/polymult/rule_pow.htm
Negative Exponents:
http://www.purplemath.com/modules/exponent2.htm
Rational Exponents:
http://www.purplemath.com/modules/exponent5.htm
Slope
http://www.regentsprep.org/Regents/math/glines/Llines.htm
Equations of a line:
http://www.regentsprep.org/Regents/math/line-eq/EqLines.htm
http://www.purplemath.com/modules/slopyint.htm
Graphing Inequalities
http://www.regentsprep.org/Regents/math/ginequal/GrIneqa.htm
http://www.purplemath.com/modules/ineqgrph.htm
Solving equations:
http://www.regentsprep.org/Regents/math/solveq/LSolvEq.htm
Solving inequalities:
http://www.regentsprep.org/Regents/math/solvin/LSolvIn.htm
http://www.purplemath.com/modules/ineqlin.htm
Graphing Inequalities:
http://www.regentsprep.org/Regents/math/ginequal/GrIneqa.htm
http://www.purplemath.com/modules/ineqgrph.htm
Quadratic Formula
http://www.purplemath.com/modules/solvquad4.htm
18
Calculus Summer Packet
Review of Precalculus
Factoring:
Common Factoring:
http://www.regentsprep.org/Regents/math/factor/Lfaccom.htm
http://www.purplemath.com/modules/simpfact.htm
Binomial Factoring:
http://www.regentsprep.org/Regents/math/factor/Lfactps.htm
http://www.purplemath.com/modules/specfact2.htm
Trinomial Factoring:
http://www.regentsprep.org/Regents/math/factor/Ltri1.htm
http://www.regentsprep.org/Regents/math/factor/Ltri3.htm
Logarithms:
http://www.regentsprep.org/Regents/math/algtrig/ATE9/logs.htm
Trigonometry:
http://www.regentsprep.org/Regents/math/algtrig/math-algtrig.htm
19
Calculus “2012” Summer Packet
Answer Sheet
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