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

13. Prove each of the following.

a) sin tan cos

b)

sec

sin cos cottan

c) 22csc2 tan secx x x

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.