Summer Math Program 2015
Calculus 2015 – 2016
NAME_____________________________________________________
This project is to be completed during the summer months and turned in to your Calculus teacher on September 11, 2015.
Completed packets turned in earlier will receive extra credit. All students are responsible for completing the
assignment, which will be graded and counted as the first test for the new year. Please read the directions for each problem
and be sure to answer fully explaining all work. Reference sheets and graph paper are included in the packet. If extra space
is needed to complete any problem, use the reverse side of the paper labeling the problem number clearly. If you should
misplace the packet, you may download a copy from the district website www.linden.k12.nj.us .
SOLVE THE FOLLOWING
1. 3w – 4 = 5
2. x3 – 7x2 + 12x = 0
SHOW YOUR WORK.
3. Find the equation of the line going through
(-3, 4) and (3, 10).
4. Find the equation of a line going through (2, 1)
slope
5. Find the equation of the line perpendicular to
y=
3
.
2
6. Divide (2x3 – 4x2 + 3x) by (x – 3).
2
x – 5 and passes through (2, 1).
3
7. Find [f(g(x))] for f(x) = 4x – 1 and
g(x) = 2x + 5
8. Solve the system of equations
3x + 5y = 7
3x + 2y = 1
9. Solve x –
x2=6
10. Solve
3  2x  5
11. State the domain and range of {(2, 1), (3, 4), (4, 8)}. Explain why or why it is not a function.
12. Solve 2 = (y + 3)(2y – 2)
13. Find the domain of f(x) =
5x 1
x  8 x  15
2
14. Use the quadratic equation formula to solve x2 + 3x – 5 = 0
with a
15. Find the critical point of y = 3(x – 2)2 + 3. Is it a max or min?
16. Find the critical point of y = -2(x + 1)2 – 4. Is it a max or min?
17. Explain in your own words when you would use the Law of Sines to solve for the missing parts of a triangle. When
would you need to use the Law of Cosines?
18. Solve 5 = e0.4t for t
19. Find the inverse of f(x) = 2x + 7
20. Graph the following functions on one set of axes. Describe the changes with regard to vertices and position on axes
y = x2
y = x2 + 2
y = (x – 3)2
y = (x – 3)2 + 2
Use the information from the previous problem to discuss the changes below
Compare
# units shift left # units shift right # units shift up
21. y = x3 to y = -2x3
22. y = x3 to y = (x – 5)3
23. y = x3 to y = x3 + 12
24. y =
# units shift down
x to y = x  15 - 9
Fill in the table for the given trig equations.
Amplitude
25.
26.
27.
28.
Period
Phase shift
y = -3 sinx
y = cos4x
y = cos x + 2
y = - 4 sin2x
29. y = 10sin(
x
– 4 ) – 5
4
30. Graph y = 2 cos
1
x
2
31. How can you tell whether f(x) defines a function
by looking at its graph?
32. A parabola y = ax2 + bx + c has vertex (4, 2). If
(2, 0) is on the parabola, then find the value of
abc.
33. A rectangular field is twice as long as it is wide.
Express the area of the rectangle as function of its
perimeter P.
34. A vendor sells large sodas for $.70 and small sodas for $.50. One afternoon, he sold 1000 sodas for a total of $580.
How many large sodas did he sell?
35. Find the future value to the nearest dollar of $5000 invested at 8% for 5 years in an account that compounds interest
continuously.
36. The distance from city A to city B is 150 miles. From city A to city C is 90 miles. Which of the following is
necessarily true?
a. The distance from B to C is 60 miles
b. Six times the distance from A to B equals 10 times the distance from A to C
c. The distance from B to C is 240 miles
d. The distance from A to B exceeds by 30 miles twice the distance from A to C
e. Three times the distance from A to C exceeds by 30 miles twice the distance from A to B
Be sure to explain your thinking.
37. Solve the triangle
Y
16
X
24
30°
y
38. Solve 2sin2x + sin x – 1 = 0
39. If sin A =
2
and A is in Quadrant II, find cos A.
5
40. If tan B =
12
and B is in Quadrant I, find sin 2B.
5
Z
41. Change
7
radians to degree measure.
6
42. Change
200 to radian measure.
43. Find the reference angle for -
44. Solve
120 .
x5 =3 x3
Use reflections, rotations, transformations of y = x2 to sketch the graphs in #’s 45 – 48.
45. y = 1 + (x – 2)2
46. . y = 2 – (x + 1)2
47. y = -2(x + 1)2 – 3
48. y = x2 + 6x
Given f(x) = 2x and g(x) = x2 + 1, find formulas and domains for #’s 49 – 53.
49. f + g
50. f – g
51. fg
52.
f
g
53. f(g(x))
54. f(x) =
x  5 and g(x) = 2x + 8, find f(g(-3)).
55. A locomotive travels on a straight track at a constant speed of 40mph, then reverses direction and returns to its starting
point, traveling at a constant speed of 60mph.
a.
b.
c.
What is the average velocity for the round trip?
What is the average speed for the round trip?
What is the total distance traveled by the train if the trip took 5 hours?
56. Given that f(x) = 2x + 1, find a function g(x) such that f(g(x)) = x.
Use the equation y = 1 +
x to answer the #’s 57 – 60.
57. For what values of x is y = 4?
58. For what values of x is y = 0?
59. For what values of x is y
 6?
60. Does y have a minimum value? A maximum value? If so, define them.
61. If you had a device that could record the temperature of a room continuously over a 24 - hour period, would you
expect the graph of temperature versus time to be a continuous (unbroken) curve? Explain.
62. If you had a computer that could track the number of boxes of cereal on the shelf of a market continuously over a one
week period, would you expect the graph of the number of boxes on the shelf versus time to be continuous (unbroken)
curve? Explain your reasoning.
63. The diagram below shows a field ABCD with a fence BD crossing it. AB = 15m, AD = 20m, and angle BAD =
110  BC = 22m and angle BDC = 30 .
A
D
C
(a) Calculate the length of BD.
(b) Calculate the size of angle BCD.
One student gave the answer to (a) “correct to 1 significant figure” and used this answer to calculate the size of angle
BCD.
(c) Write down the length of BD correct to 1 significant figure.
(d) Find the size of angle BCD that the student calculated, giving your answer correct to the 1 decimal place
64. Determine the asymptotes and/or holes for the following graph
f(x) =
2 x 2  5x  3
x2  2x 1
65. In economics, the demand function relates the price per unit of an item to the In economics the demand function
relates the price per unit of an item to the number of units that consumers will buy at that price. The demand, q, is
considered to be the independent variable, while the price, p, is considered to be the dependent variable.
Suppose that in a certain market, the demand function for widgets is a linear function
p = -0.75q + 54,
where p is the price in dollars and q is the number of units (hundreds of widgets in this case).
a.
What is the slope of this function? Explain the meaning of the sign of the slope in practical terms
b.
Find the p- and q- intercepts for this function. What is the significance of
these intercepts in the context of the problem?
66. Celia has $ 20000 to invest. There are two different options that she can choose.
Option 1: The investment grows at a rate of 3.5 % compound interest each year.
Option 2: The total value of the investment increases by $ 800 each year.
The money is to be invested for 15 years.
(a) Copy and complete the table below giving the values of the investments to the nearest dollar for the first
4 years.
Year
0
1
Option 1
20000
20700
Option 2
20000
20800
2
3
4
(b) Calculate the values of each investment at the end of 15 years.
(c) If Option 1 is chosen find the total number of complete years before the values of the investment is first
greater than $ 25000.
(d) If Option 2 is chosen calculate the percentage increase in the
investment for the final year.
67. The amount of potassium-42 present is given by P(t) = 12 e
a. How much potassium-42 is there initially?
b.
How much is present after 4 hours?
c.
How long before only 30% is present?
d.
When will it be all gone? Explain
.055t
where (t ≥ 0 in hours).