Math 2312.3S7
Precalculus Final Exam Review
These questions came from the 4 exams. Work through them once using your notes if
necessary. Work through them a second time without using your notes. These are the
same type of questions you will see on the final exam. You should only study the concepts
on these types of questions. No other concepts will be included. The exam is less than
20 questions.
1.
The population, P, in thousands of Rivendell is given by P (t ) 
a.
Find the population at t = 0 and t = 8 months.
b.
Sketch the graph of the function.
c.
Find the horizontal and vertical asymptotes.
d.
What is the meaning of the horizontal asymptote?
e.
What is the maximum population?
500t
where t is in months.
2t 2  9
x 2  4x  5
. Identify all asymptotes and intercepts.
x 3
2.
Sketch the graph of f (x ) 
3.
Let f (x )  x 2  1 . Find f-1, algebraically, and graph both functions.
4. Prove that f (x )  1  x 2 and g (x ) 
x 2  1 are inverses of each other.
5. Solve the following log2(x2 + x) – log2(x2 – x) = 1.
1
6.Solve e 5x  3  10
4
7. If you invest $2000 at a rate of 12% compounded continuously, how long will it take your
money to double?
8. Find the exact solutions of the equation 10cos x  5 2  0 for 0, 2  .
9. Big Dog Snowboard Co. charges $15 for equipment rental plus $35 per hour for snowboarding
lessons. Half-Pipe Snowboards, Inc. charges $40 for equipment rental plus $25 per hour for
lessons. Write a system of equations and determine the number of hours that the cost of
equipment and lessons are the same for each company.
10. Graph 2 periods of the equation y = 2 cos 3(x + ) - 1. State the amplitude, period,
vertical and phase shifts, and domain and range. Label graph with all critical information.
y = 2 cos 3(x + ) - 1
A:_________ P:__________
EP:_________ CI:_________
VS:_________ PS:_________
Domain:___________ Range:___________
11. Write two different equations for the graph.
12. Write an equation for an ellipse whose major axis is of length 10 and has foci at (6, 1), (2, 1).
13. A spacecraft is in a circular orbit 150km above the Earth. Once it obtains the velocity
needed to escape Earth’s gravity, the spacecraft will follow a parabolic path with the focus at
the center of the Earth. Suppose it obtains its escape velocity above the North Pole. Assume
the center of the Earth is at the origin and the radius of the Earth is 6400km. Write an
equation for the parabolic path of the spacecraft.
Note: Application questions will not include any hyperbolas. All other conics are fair
game.
14. Put the conic x2 + 4y2 + 4x – 24y + 20 = 0 into standard form, then graph. Identify the
vertices, co-vertices, and foci.
15. Write the partial fraction decomposition of the following rational expressions:
a.
x 1
3x  14x  15
2
b.
3x  1
x2 x
16. A line has parametric equations x = t2 - 4 and y 
equation by eliminating the parameter.


17. Using exact values only, convert  3,
t
. Write the resulting rectangular
2

 from polar to rectangular form.
6
18. Using exact values only, convert (-1, 1) from rectangular to polar form.
19. Find the center & the radius of the sphere 9x2 + 9y2 + 9z2 – 18x – 36y - 72z + 68 = 0.
20. A high school baseball player hits the ball at a height of 3 feet. The initial velocity for the
ball is 75 miles per hour at an angle of 20°. The 6 foot center field fence is 350 feet from
home plate. Given the equations and viewing window, will the player hit a home run? (The models
for projectile motion are in feet per second)
These are the equations of the motion of the ball:
The equations for the fence are:
x1T = (110cos20o))T
Y1T = -16T2 + (110sin(200))T + 3
x1T = 350
y1T = 6 – 3T
a. Sketch a basic graph of the situation. Does the player hit a home run?
b. Find the time it takes the ball hits the ground.
c. What is the maximum height of the baseball, and when does the ball reach this maximum
height?
d. Adjusting only the minimum angle (guess and check), can the player hit a home run? What is
the angle needed?
e. Adjusting only the initial velocity (guess and check) can the player hit a home run? What is
the initial velocity needed?
Note: I will not give you the equations or the window on the test. I will give you the
models for the equations. You will have to write the equation, but you will not have to
convert the units as I did here. The units will already agree.
21. Determine whether u = <-1, 3, -1> and v = <2, -1, 5> are orthogonal, parallel, or neither.
22. Find the angle between u = <0, 2, 2> and v = <3, 0, -4>.
Note: You will have to put the vectors into component form first before finding the angle.