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MHF4U Final Exam Review
Exam Time Limit: 2.0 hours Name _________________
Exam Instructions:
1. Answer all questions in the space provided.
2. If you need additional paper or graph paper ask for it, otherwise all answers should be
written on the exam paper.
Part I: Place your answers for Part I in the space provided.
1. The instantaneous rate of change of a function is _____________. (____)
A. the domain divided by the range
B. the change in the dependent variable divided by the change in the independent
variable from one point to another
C. the range divided by the domain
D. how quickly the dependent variable is changing with respect to a unit change in the
independent variable at an instant in time
2. The domain of a function is _______________. (____)
A. the entire set of function values (of the dependent variable)
B. x
C. y
D. the entire set of values (of the independent variable) for which the function is
defined
3. The range of a function is _______________. (____)
A. the entire set of function values (of the dependent variable)
B. x
C. y
D. the entire set of values (of the independent variable) for which the function is
defined
4. Which one of the following is a polynomial function? (____)
A.
23cos6 4
3n x x
B. 3 236 11
4f x x x x
C. 23log 4q x x
D.
3
5 4f x
x
5. The ____ row of differences for quartic functions are all the same. (____)
A. 1st
B. 2nd
C. 3rd
D. 4th
6. Even functions have which of the following characteristics? (____)
A. f x f x B. f x f x C.
1 1f x f xa a
7. The Remainder Theorem states ___________. (____)
A. When a polynomial P x is divided by x b the remainder is P b
B. If 0P b , then x b is a factor of P x
C. The remainder in the division P x b is P b
D. If x b divides P x with a remainder of zero, then x b is a factor of P b
8. The Factor Theorem states ___________. (____)
A. When a polynomial P x is divided by x b the remainder is P b
B. If 0P b , then x b is a factor of P x
C. The remainder in the division P x b is P b
D. If x b divides P x with a remainder of zero, then x b is a factor of P b
9. What is the domain of
9 3
7 14
xh x
x? (____)
A. 2x x
B. 9
7y y
C. 2x x
D. x
10. What is the horizontal asymptote for
10
2 3
xf x
x? (____)
A. 5y
B. 3
2x
C. 10y
D. 2y
11. What is the vertical asymptote for
10
2 3
xf x
x? (____)
A. 2x
B. 3
2x
C. 2
3x
D. 2y
12. In a right angled triangle sin (____)
A. opposite
adjacent B.
opposite
hypotenuse C.
hypotenuse
opposite D.
adjacent
hypotenuse E.
adjacent
opposite
13. In a right angled triangle cos (____)
A. opposite
adjacent B.
opposite
hypotenuse C.
hypotenuse
opposite D.
adjacent
hypotenuse E.
adjacent
opposite
14. In a right angled triangle csc (____)
A. opposite
adjacent B.
opposite
hypotenuse C.
hypotenuse
opposite D.
adjacent
hypotenuse E.
adjacent
opposite
15. In a right angled triangle tan (____)
A. opposite
adjacent B.
opposite
hypotenuse C.
hypotenuse
opposite D.
hypotenuse
adjacent E.
adjacent
opposite
16. In a right angled triangle sec (____)
A. opposite
adjacent B.
opposite
hypotenuse C.
hypotenuse
opposite D.
hypotenuse
adjacent E.
adjacent
opposite
17. In a right angled triangle cot (____)
A. opposite
adjacent B.
opposite
hypotenuse C.
hypotenuse
opposite D.
hypotenuse
adjacent E.
adjacent
opposite
18. 7
tan6
(____)
A. 1
3 B. 1 C. 3 D.
1
3 E. 3
19.
4csc
3 (____)
A. 2
3 B. 1 C. 3 D. 2 E.
2
3
20. 3
cot2
(____)
A. -1 B. 1 C. 3 D. 0 E. undefined
21. To convert degrees to radians, multiply by _________ (____)
A.
360 B.
2
180 C. D.
180 E.
180
22. csc (____)
A.
1
tan B.
1
cos C.
1
cot D.
1
sin E.
1
csc
23. sec (____)
A.
1
tan B.
1
cos C.
1
cot D.
1
sin E.
1
csc
24. cot (____)
A.
1
tan B.
1
cos C.
1
cot D.
1
sin E.
1
csc
25. Which of the following is equivalent to
5sin
6? (____)
A.
3sin
4 B.
7tan
2 C.
5sin
6 D.
15cos
4 E.
4cos
3
26. Using the diagram at the right, sec (____)
A. 146
5 B.
146
11 C.
11
5 D.
5
11 E.
11
146
β 5
11
1
27. sin2
x
(____)
A.
sin
2x B. cosx C. sinx D. cosx E.
csc
2x
28. tan2
x
(____)
A.
cotx B. cosx C. cotx D. cosx E.
csc
2x
29. Determine the equation of the function pictured at the right? (____)
A. 1
4cos2
y x B. 4sin2y x
C. 4cos2y x D. 1
4sin2
y x
E. 1
3cos2
y x
30. Determine the equation of the function pictured at the right? (____)
A. 5cos2 16
y x
B. 5sin2 16
y x
C. 1
5cos 12 6
y x
D. 1
4sin2
y x
E. 5cos4 1y x
31. sinx
(____)
A. sinx B.
cos
2x C. cosx D.
cos
2x E.
sec
2x
32. The graph at the right is ______. (____)
A. coty x
B. cscy x
C. tany x
D. siny x
E. secy x
33. What is the amplitude of2
3cos2 53
y x
? (____)
A. 2
right3
B. 3 C. 5 down D. 2 E. -2
34. What is the amplitude of sin 64
y x
? (____)
A. left4
B. 2 C. 6 up D. 1 E. -1
35. What is the period of 4sin5 24
y x
? (____)
A. 4 B. 2
5
C. right
4
D. 2 down E. 5
36. The graph shown is ________. (____)
A. 2xy
B. 3xy
C. 4xy
D. 1
4
x
y
E. 1
3
x
y
37. The graph shown is ________. (____)
A. 2xy
B. 3xy
C. 1
2
x
y
D. 1
3
x
y
E. 1
4
x
y
38. Write h p 3 in its equivalent logarithmic form. (____)
A. p h log3 B. h p log3 C. 3 logp h D. log3 p h
39. Write loga b c in its equivalent exponential form. (____)
A. cb a B. ca b C. ac b D. bc a
40. Simplify 1
32 82
22 2log log logx y z into one logarithm. (____)
A.
3 2 8 22log x z y
B. 3 2 8
2 2log
x z
y
C. log2
2 2
8
x y
z
FHGG
IKJJ
D. log2
32 8
2
x z
y
FHG
IKJ
Part 2: Show all work you do in the space provided for each question in part 2.
1. Use the graph at the right for
each of the following questions.
a) Determine Albert’s
average velocity from 2 to
6 seconds.
b) Determine Albert’s
instantaneous velocity at 4
seconds.
2. Use differences to determine the degree of the polynomial functions that the following
tables represent.
x y
-2 27
-1 -1
0 -7
1 -3
2 -1
3 -13
x y
-2 43
-1 0
0 -7
1 -2
2 15
3 68
a) b)
3. Solve completely by factoring.
a) 3 22 5 6 0x x x b) 4 3 23 8 4 16x x x x
4. Solve without using technology.
a) 2 4 32 0x x b) 22 11 6 0x x
5 -5
-5
5
10
10
y
x
-10
-10
5. Sketch the function
5
2 8f x
x
. Be sure to
indicate all asymptotes and
intercepts (if any). Show how to
get each behavior near any
asymptotes.
6. Find the asymptotes for each of the following functions.
a) 6 3
4 12
xf x
x
b)
2
7 1
2
xf x
x x
7. Solve each of the following rational equations.
a)
1 7 4
3 5m m
b)
3 18
4 2 2 4
x
x x x x
8. Use a numerical method to approximate the instantaneous rate of change for the
function
2 7
3
xf x
x where x = 2.8.
9. The radius of the earth is 6380 km. A space shuttle 300 km above the earth travels
around the planet through an angle of 11
6
radians. How far does the shuttle move?
10. Use a compound angle identity to rewrite 3 3
sin sin cos cos3 4 3 4
as one single
trigonometric ratio and then find its exact value.
11. Determine the average rate of
change for the function at the right
from where x = 0.2 to x = 1.1.
12. Solve on the interval 0 2 .
a) 22sin 7 sin 4 0
b) 22cos 3cos 2 0
x
y
14. The sunset time varies sinusoidally with time. The sunset time (S) in the town of Parry
Sound can be modelled by the equation 2
1.75cos 172 18.4365
S d
, where d is the
day of the year. How long a period of time (in days) is the sun setting earlier than 7:00
p.m.? You may use technology to solve this problem. If you do use technology, include a
neat labelled sketch with your solution.
15. Evaluate 6log 93 . Explain how you can check this answer.
16. Solve each of the following.
a) 4 50x
b) 59 221x
17. Evaluate without the use of the log function on a calculator. (Hint: use the log laws)
a) 9 9log 3 log 27 b) 53log 25
18. Simplify each of the following using the three laws of logarithms.
a) 3 3log 144 log 6
b) 37 74log 2 log 8x xy
c) 31log 5log 2log
3z x u
19. Solve each of the following.
a) 9 9log ( 5) 1 log ( 3) x x
b) 24 4 4log ( 2) log 10 log ( 5 14) x x x
c) 3 19 27x x [Write the answer in exact form]
d) 25 4 5 5 0x x
20. The magnitude, M, of an earthquake is measured using the Richter scale, which is defined
as M log .o
I
I I is the intensity of the earthquake, and Io is the intensity of a standard
low level earthquake.
a) What is the Richter Scale reading for an earthquake that is 45000 times more
powerful than the standard low level earthquake?
b) How much more powerful is a 6.8 Richter Scale earthquake than a 4.0 scale quake?
21. A 60 mg sample of a radioactive isotope decays to 50 mg after 4.7 days.
a) Determine the half-life.
b) How long will it take for the amount
of the isotope to decrease to 20 mg?
5 -5
-5
5
10
10
y
x
-10
-10
22. Given 24 2 1f x x x and 3xg x .
a) Graph h x f x g x
b) Solve the inequality 24 2 1 3xx x
c) Find and simplify g f x and f g x
d) Explain why you would expect g f x and f g x to be different.