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AP CALCULUS SUMMER PROJECT
Part 1: Algebra Review
Write the following equations in slope-intercept form.
1. 2x – 3y = 10
2. 14 – 6y – 3x = 15
3. 6y = x – 10y + 5
4. 72x – 4(x – 6y) – y = 14(x – 6y)
Given the following information, find the equation of the line.
5. Find the slope of the line passing through (-1, 3) and (9, -10)
6. Find the equation of the line passing through (-5, -2) with a slope of
1
3
7. Find the equation of the line passing through (12, -7) and (-4, 11)
8. Find the equation of the line through (8, 7) and perpendicular to 2x – 3y = 10.
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Graph the following linear functions.
9. y = 2x – 1
10. 3x – 2y = 8
11. 4 – 26y = 9(x – 12) – 3y
12. 9(x – 3) – 2x = 10(x + 4)
Solve the following systems of equations.
13.
2
y  2x  3
2x  3y  4
14.
2x  4 y  0
3 x  5
15.
3x  7  6 y
2( x  3 y )  3  1  4( x  2)
1
y  x 2  2x  1
16.
2
y  2 x  3x  4
2
Graph the following systems and identify solutions.
17.
19.
3
y  2x  3
y  x 4  x3  2x 2  x  1
y  3  x2
y  2x  3
18.
20.
y  x 2  2 x  35
y  5  x2
y  x 2  5x  6
y7
Factor the following expressions.
21. 5x 4 y 6  20 x 6 y 4  125xy12  200 x 2 y 7
22. x 2  7 x  12
23. 3x 2  8 x  5
24. x 3  7 x 2  3x  21
Solve the following quadratic equations.
25. y  x 2  6 x  9
26. y  2 x 2  7 x  3
27. y  x 2  15 x  44
28. y   x 2  13x  22
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Graph the following quadratic equations.
Identify: Minimums, Maximums, Intervals of Increasing/Decreasing, Positive/Negative
Intervals.
29. y  x 2  5x  6
30. y  2 x 2  11x  5
31. 3x 2  7 x  y  9
32. y  8 x  x 2  15
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Solve the following Exponential Functions.
33. 4 x  2
34. 27 4t  9 t 1
35. 2 63 x  8 x 1
36. e x  18
37. e x  100
38. e 3 x 7  e 2 x  4e
39. 3(2) x2  1  100
40. 3e 2 x 4  7  12
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41. At the World Championship races held at Rome’s Olympic Stadium in 1987,
American sprinter Carl Lewis ran the 100-m race in 9.86 sec. His speed in meters
per second after t seconds is closely modeled by the function defined by
f (t )  11.65(1  e t / 1.27 ) .
a. How fast was he running as he crossed the finish line?
b. After how many seconds was he running at the rate of 10 m per sec?
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Graph the functions along with it’s inverse.
42. y  e x
43. y  2 x  1
x
1
44. y     3
2
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45. y  e 2 x  4
Part 2: Geometry Review
Draw a Picture.
46. A sphere with radius 4 inches is cut into two equal semi spheres. What is the area
of each of the two semi spheres?
47. A square has a side length of 5 cms. Corners of length 1 cm are cut out in order
to make a box with an open top. What is the volume of that box? What is the
surface area of that box?
48. A rectangular playing field is to be made with a length that is 30 feet longer than
the width. Scouts say that a ratio of length to width should ideally be 1.6.
a. In order to obtain that ratio, what must be the measures of the length and
width?
b. If the designers ignore the ratio because they want an area of 2000 square
feet, what be must the length and width ?
c. If the designers must keep the perimeter of the playing field to 175 feet,
what will be the area of the playing field?
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49. A 55 gallon cylindrical drum is leaking oil at the rate of 0.3 gallons per minute.
The gallon begins leaking when there is only 29 gallons remaining. A worker
notices the leak and brings a hose over to fill up the tank. His hose is pumping oil
at the rate of 0.42 gallons per minute.
a. How much time does the worker have before the drum is completely
empty?
b. If the worker arrives when there is 17 gallons remaining, how long should
it take his hose to bring the amount of oil back up to 55 (do not consider
the leaking drum)?
c. How long will it take him to fill it back up with the drum still leaking?
50. What is the maximum area that can be obtained from a rectangle if the perimeter
is to be 45 inches?
51. A right triangle contains two legs that add up to 20 cms.
a. What is the value of the hypotenuse when one of the legs is 14 cms?
b. What is the maximum area that can be obtained?
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Part 3: Trigonometry Review
Find the values without a calculator.
Angle Measure
θ in degrees
30º
45º
60º
90º
120º
135º
150º
240º
300º
315º
330º
540º
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Angle Measure
θ in radians
cos θ
sin θ
tan θ
Graph the following trig functions below. Be sure to label period, amplitude and phase
shift.
52. y = cos x
Amp =
Amp =
Period =
Period =
P.S. =
P.S. =
V.D. =
V.D. =
54. y = tan x
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53. y = sin x
55. y  3 cos 2 x  1
Amp =
Amp =
Period =
Period =
P.S. =
P.S. =
V.D. =
V.D. =
Verify or solve the following trigonometric identities
56. If cot θ = 0.8, find csc θ.
57. If tan θ = ½, find sin θ.
58. Verify that the following is an identity:
1
1

1
2
sec x csc 2 x
1  tan 2 
59. Verify that the following is an identity:
 tan 2 
2
csc 
60. Verify the following:
12
sin A cos A

1
csc A sec A
61. Verify the following: sin( 270  x)   cos x
62. Verify the following: 1  cos 2 A 
2
1  tan 2A


63. Verify the following: sin   x   cos x
2

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Part 4: Pre Calculus Review
Find the inverse of each of the following.
64. f ( x)  3 x  4
65. j ( x)  3 x / 3  2
66. h( x)  1  log 3 ( x  2)
Solve each of the following.
67.
3  27 2 x 1
1
68.  
3
x 5
 81x  2
69. log x 3  1
70. Find the future value and interest eared if $8906.54 is invested for 9 years at 5%
compounded
a. Quarterly
b. Annually
c. Monthly
d. Continuously
71. Find the present value of $5000 if interest is 3.5% compounded quarterly for 10
yrs.
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72. Use the given functions, identify domain and range, and evaluate at the given
point.
Function
Domain
Range
f(0)
f(-1)
y  3x  2
f ( x)  9 x 2  3x  2
h( x ) 
3
x
g ( x)  3 x  4
73. Let V be our function for the Volume of a 3 foot high cylindrical Gatorade cooler.

The volume of a cylinder is given by V (h)  r 2 h .
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a. What do you know about V(h), r, and h? (Check the box for
variable or constant)
Variable
Constant
V(h)
r
h
b. Find V(4), when r = 10.
c. Find the Volume when the cooler is full.
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x
f(x)
g(x)
h(x)
-3
8
0
7
0
3
4
3
2
4
-3
6
4
1
5
-3
7
-3
6
2
Evaluate each of the following using the table above.
74. f(-3)
75. g(7)
76. h(2)
77. g(h(7))
78. h(f(2))
79. f(g(-3))
Solve the following equations.
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1
80. log 4 x 
81. log x 3 5 
2
3
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82. 2 x  1  log 6 6 x
83. A ball is dropped from the top of a house h feet above the ground, where h
is the initial height. It’s position in terms of time is
s(t )  16t 2  32t  48 where s is feet above the ground and t is time in
seconds.
a. Find the time it takes for the ball to reach the ground (set s(t) = 0).
b. What is the average velocity for the interval starting when the ball is
dropped to the when the ball hits the ground (Average Velocity is given by
s s 2  s1
)?

t t 2  t1
84. A ship is sinking to the bottom of the sea floor given by the position
 t 2  5t  150
function s(t ) 
3t  5
i. What is the depth of the water if the ship starts at the top of the
sea?
ii. At what time will the ship be at the bottom of the sea?
iii. What is the ship’s average velocity over the interval starting when
the ship starts sinking (t = 0) and ending when it hits the bottom?
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85. Paris’s Eiffel Tower was constructed in 1889 to commemorate the 100th
Anniversary of the French Revolution. The right side of the Eiffel Tower
has a shape that can be approximated by the graph of the function defined
by
x
f ( x)  301ln
207
i. Explain why the shape of the left side of the Eiffel Tower has the
formula give by f (-x).
ii. The short horizontal line at the top of the figure has length 15.7488
ft. Approximately how tall is the Eiffel Tower?
iii. Approximately how far from the center of the tower is the point on
the right side that is 500 ft above ground?
86. The shadow of a vertical tower is 40.6 m long when the angle of elevation
of the sun is 34.6º. Find the height of the tower.
cos(  x)
and then using
sin(  x)
the appropriate relationships for cos(-x) and sin(-x).
87. Show that cot(-x) = -cot(x) by writing cot(  x) 
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88. The maximum afternoon temperature in a given city might be modeled by
xt
t  60  30 cos ,
6
Where t represents the maximum afternoon temperature in month x, with x = 0
representing January, x = 1 representing February, and so on. Find the maximum
afternoon temperature to the nearest degree for each month.
a. January
b. April
c. May
d. June
e. August
f. October
89. A coil of wire rotating in a magnetic field induces a voltage
t 
E  20 sin(
 ) where t is in seconds.
4
2
i. What will the voltage be after 10 seconds?
ii. When will the voltage be 0?
iii. What is the maximum voltage/ minimum voltage?
90. 3 – 6 = ?
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