EXAM REVIEW FOR MHF 4U 1
ADVANCED FUNCTIONS
Study Tips:
• The best way to study for a math exam is to redo all Unit Review Sheets, Quizzes and Tests.
• Write out all of your formulas in one place to help you remember all of the since no formula will be
given on the exam.
ADDITIONAL REVIEW EXERCISES
Expressions
l.
Factor fully.
a) r'+3r-25x-75
d)
)
e)
e)
r+¥-14x-24
64r'+27
2. Simplify:
3.
f)
30.++17£ -8x-4
°±-8-
-124.¥--125
16¥ +128
() (")y'
«
Eh
Find the exact value of the following.
a)
o)
7ls+V5
c)
log. 88
d)
log,6- log,3+ log,4
toe,(3'·381)
2
alog,(x-7)-log,w+2
4.
Write as a single logarithm:
5.
Le f(r)=((3.2).(5,1),(7.4).(9.3),(11,5)) and g(r)={(1,3).(2.5).(3,7).(4.9).(5.11)}. Deerie:
o»(/+g)()
(-g)()
B)(g·f)(9)
)f(g(3))
Convert to degrees.
117
a)
7.
9.
radians
b)
56 radians
b)
-24°
Convert to radians.
a)
8.
15
420°
Find the exact value of the following.
3n
a)
COS-
d)
SI-
b)
4
c)
.4,()
. 7r
12
Given: sinA=
·(-3.7r)
csl5
-6 3n
2
3n
,<A<2n,and tanB= -- n<BK
7
2
3'
2
Find the exact value of the following.
a)
b)
sec A
cos 2B
10. Given:
c)
d)
1
f(x)= and g(x)=
x-5
11n
tan6
sin(A + B)
tan(A-- B)
x +8
Find:
a)
(f- g))
e)
(s·(g))
i)
(g-DX3)
b)
[5
n
f'(o
j)
(gM-D
c)
(f·g)0)
g)
g'()
k)
d)
(g·f))
h)
(/·f )Go
D
(f·g)(5)
(g·f)(5)
Equations, Inequalities & Identities
• Solve. Exact answers are required, where possible. Otherwise, express answers correct to one decimal place.
Where necessary, state restrictions.
a)
=4x-12
D)
log,(x+D+log,2-log,(x+3)=log,Gr-D)
r-3r
b)
r'--5x=5£ -1
k)
5.8.57°
c)
r+4 +9x+10=0
I)
we"-er[l,]
m)
3"--2(3")-15=0
n)
sin
I
x--44
d)
4-=(
f)
2x
I
l
x-I x-3
0
4(7)=8
o)
cos2x = C0sx (0<x <27)
g)
log,(x+3)=2
p)
/2tan xcosrx= tanx(0<x<27)
h) log,(x+2)=1-log,(x-4)
q)
2cos2x=1 (0<x<27)
i)
r)
-L
2
¥4x+3
log,Gr-5)+l0g,(+3)=1
x-2sinx-3=0(0<x<2)
12. Solve.
a)
x(x+ 1(x-2)(x-4)>0
d)
b)
(x+7(--3)<0
e)
e)
2r+3--1Hx2>6
r+2a
r-9
5
r+3
+
3
()
x--1
13. Prove.
ask)-[5-la[]--«a
a)
.
1+tan6
cos(+sin(=sec6
b)
+-=-}eot ()
I - sec 0 1 + sec6
e) cos 20-cos 0
= sin 0- sin 20
r
I
sin (z -x)cos(n +x)tan(2n -x)
..4
() -.'7S'[[] )
-3[-h.-J
D cos(x + y)cos(x-y)= cos x+cos y -1
1
2
14. If log,a= -and log,,Vb =3x,show that x=-.
6
x
4]
s.».
15. If ht
+k' =23hk,where h>0, k>0, show that log] .l=,(logh+logk
s
Graphs
Determine whether each of the following functions are even, odd or neither.
16.
I
a) J(x)=-
r?+1
a0-[',]
b) g()=2±' +3
x +x
17. Graph the following functions. Determine and label all key features.
a)
y=-x(x-3)x+4)
b)
y=(x-2(+3
c)
y= -
(d)
2
x-1
5.x-3
=2x+1
e)
y=3_1
y=log,(8x)
g)
h)
y=2sin(x-7l'),(-27r�X�2.7r)
3
cos(.!..x n)-1,
+
(-27f � x::; 2n)
y=
2
4
·
..2
sl[]
)
It
1
4
3£
x-1
5-4x-x
x)= 3+.
I8. When is the function, f (
z,,below the horizontal asymptote?
19. State the range, period, amplitude, phase shift and equations of the asymptotes for each of the following functions for
-a[-{
Applications
20. When
r 4r' +ax + bx +l
is divided by (x-1), the remainder is 7. When it is divided by (x+I), the remainder is 3.
Determine the values of a and b.
21.
An open box, no more than 5 cm in height, is to be formed by cutting four identical squares from the corners of a sheet of metal
25 cm by 32 cm, and folding up the metal to form sides. The capacity of the box must be 1575 cm'. What is the side length of the
squares removed?
22. Consider all rectangles with an area of 200 m'. Let x be the length of one side of the rectangle.
a)
b)
Ex press the perimeter as a function of x.
Find the dimensions of a rectangle whose perimeter is 70 m.
23. Determine the intercepts, holes and the equations of all asymptotes with behaviour of y
24.
x)=r+¥ at.x=2
5.
r-2x¥-x+2
= ------- then sketch.
r-x-6
Estimate instantaneous rate of change of each function at the given x value using a centered interval of
a) f(
±0.001 .
b) f(r)=-r'+1atr=3
The population of a town is modelled by P(
t)= 6t +HO1 +3000, where P is the population and t is the number of years
since 1990. Find the average rate of change in population between 1995 and 2005.
26.
Energy is needed to transport a substance from outside a living cell to inside the cell. This energy is measured in kilocaloris per
gram molecule, and is given by: E = 14(logC, -logC,),where C, represents the concentration of the substance outside the
cell and C represents the concentration of the substance inside the cell.
a) Rewrite the formula as a single logarithm.
b) Find the energy needed to transport the exterior substance into the cell if the concentration
of the substance outside the cell is double the concentration inside the cell.
c) What is the sign of Eif C, <C,? Explain what this means in terms of the cell.
27. A ferris wheel with a radius IO m makes 2 rotations in 4 minutes. What is the speed of the ferris wheel in meters per second.
28. A circular arc has length 3 cm, and the radius of the circle is 2 cm. What is the measure of the angle subtended by the arc, in both
radians and in degrees?
f
I
i
0<x<27.(State only the properties that each function has.)
a--[-;]-
I
I I
Answers to MHF4U Exam Review
Check
.a. (x-5)(x+53(x +3)
b. (x-4)(x+3)Xx+2)
11a. 3,2,-2 b.-1, 3±2/2
r
c.-2,-1±2i d. 2±V3
c. (4x+3y)(16x-12xy+9y)
d. (5x+2)2x-1)(3x+2)
e. (x+1)(x-5)6x-x-+1)(+5x+25)
f. (x+2)6x-2)(+4)6x+2+4)
t E2.,2:236
log7
-1
g. 13
h. 5 i. 6 j.no solution
!
2lg8+log5.o.3
71og5-log8
4
3r
n.2
log5 -.
.,s
log3
mh.
1
39
3a. � 5 b
v
'75° 1 d. 10
e.,
2n 4n
n
7n
0.0,-,2n p.0;72r
n 5n 77 1Hn
tad%
b.7
c.((3,-5),(5,-10))
d. ((3,9),(5,12))
7a.717
n
9
13.- 15. (proofs vary)
V
is5
7✓31
a.
ere(---3[-r)
-1
c. Undefined d.
13
16a. neither b. even c. odd
6+2 e. V-6
4
13
-8x+
10a
b.
no amplitude, phase shift = :
r=.
n
Asymptotes:
77
20. a=3, b=6
21. 3.5 cm
400
22a. P(x)=2x+
x
.
-1
y-int=,
VA: x=3, x=-2
24a. 16
b.-108
25. 230 people/year
103-66
39
d.
C.
h
]-
5x-
I
+3
4_8r-80x+201
(c.)
26a. E-1Ao[
z)
b. 0.42 c. neg - cell gains energy
27.%,
12
e.
f.
(x-5)
270
28.0= l.5rad or 6=-degrees
7
e. x"+16+72
r'+5
r
%
g. ±Nr-8
.35 ,-3
X
1
k.
28
b. R:{ye R},Period= n,
Ob] A: y=x-I
91
b. x'-5x+8-40
·
17a.
3
x--5
f
12
Asymptotes
23. x-int=1, -1,2
\
b. 2_ c. 18✓3-26
4 -29+/3)
304+/3)
Phase shirt= " right, No
b. 27.87.2 m
2
-1
3
d. xe[-3,-2]u[3,+)
•2r
15
3
, = 2n ,amplitude
Ji id = 3 ·,
Period
b. re(-·,-7)0(-7,3)
»c[lo.»
100803208.6°
6a. 132°
19a. R:{y/-5<y<1,ye R},
right,
12a. xe(-,-1)u(0,2)0(4,+)
5a. 4
.ts9(7-)
• xi
I. undefined
5
g.
h.
MHF4U - Final Exam Review
Polynomial Functions
1.
Determine the zeros of the function f(x) = 3(x2 -25)( 4x2 + 4x + 1).
2.
The remainder when
3.
Determine the equation of:
+
Cr is divided by x +2 is 28.
-
?
a. The cubic function with zeros 1, -2, -5 and y-intercept 10.
b. The quartic function with zeros 2,-1, 3 (order 2) and passing through (4,-10)
4.
By calculating the Finite Differences, determine
a. Which type of polynomial function best models this relationship.
b. Determine the minimum and maximum number of zeros and turning points for
this type of function.
i
!
[%.
5.
a) Without using division (long or synthetic), find the remainder when 2x--4x + 7 is
divided by 2x-1
b) x - 5 is a factor of 4-- 5x + k -2 . Find k.
c) Divide using long division (3 -2y +12y -9):( +2)
y
6.
I
Factor fully
a) 64x +125
:
b) 3x'+8 +3x-2
d) 8r-4x-2x+1
c)'-2¥'+2x-1
e) 2+13¥ +23x +12
:!
7.
8.
9.
Solve for x,x eC
c)9z'-4z=0
a) --7+6=0
d)x'+2 --45=0
b) 2r'+x--8x-4=0
e)--4r +2x+3=0
f)3'-7 =2-6x g) (2x-1)=2x-1
Solve for t or x
a) t--2£ --30>0
b)t+21 31<0
c) x(x-2)(x +1)(x +5)2> 0
Find the equation of a polynomial function in standard form, with the given the roots:
a)
3,-2,'2
b)1,1,6
0)3, 1±1
0)0, 2±5, ?
3
10. For the function g(x) = -2r(-2)
a. State the degree of the function and comment on its end behaviour.
b. State the zeros and with the aid of a table of values, sketch the function.
c. State from the graph, the interval(s) where g(x) is decreasing.
d. Find, algebraically where g(x) < O. Illustrate, in colour, on the graph.
e. State the coordinates of any local/absolute max/min points.
4
'II
'
MHF4U -- Final Exam Review
Answers:
1.
zeros are 5, -5,
2.
3.
C=3
4.
5.
6.
7.
-l
2
a) y=-r-6x -3x+10 b) y=-x'+7-13¥'-3x +18
a) The 4" differences are constant (value is 24). Therefore it is quartic.
b) Minimum number of zeros is 0, Maximum number of zeros is 4, minimum number
of turning points is 1, maximum number of turning points is 3
a) 5.25
b) -94.6
c) Quotient is 3y-2 and Remainder is 6y-5
a) (4x +5y)(16x - 20xy +25y) b) (x +1)(3x -1)(x+2) c) (x-1)(x +1)
d) (2x-1(2x+1) e) (x+1)(2x +3)(x +4)
a)
b) x=#,
2 �1
) x=,±-:
12 3
e)
x=3,,_I±s[5
'
f) X = 1
2
8.
a) - 1 < t < 0, t >3
9.
a) y=
=0 ± 2
3
_
d) x :.. . - 5 ± ✓5i
.x 3
X
= 2 ± ✓i2
g) X = ± 1, ..!_
3
=
b)0<1<1,1<-3
2x-3x-1lx+6
d) y=3'-10-5
10.
)
C)z='
=
2
2
c) x<-5,-1<x<0,x2>2
r-8x +13x-6 c) y=x-5 +8x-6
b) y=
+2x
a) Degree is 5, For end behaviours, as x -» +oo,y » -oo and as x -» -0,y -» +0o
b) Zeros are O and 2
X
y
-1
18
0
0
1
-2
2
3
0
-54
eee
!
c) The function is decreasing when x < 1.2 (approximately from graph) and when x > 2
d) g(x)<0 when 0<r <2 and x>2
e) Local minimum at approximately (1.2, -2.2) and local maximum at (2,0)
2
Review Questions Rational Functions
Multiple Choice
2
1.
-51+4
What is the x-intercept of the function /(x) = ----?
- l
a.
b.
x=landx=4
x=l
=4
C.
d.
x= 4
+3
2. What is the domain of the function f() = , ?
-1
6]
=
,
C.
a. i A=,1,1
sf rR)
d.
b. (x]
I,r e R)
=
3.
What is the oblique asymptote for f() =
a.
- -3r+2
x+2
y=12
?
1.-,¢
C,
)'=7-4 .
.
r+2
b.
y=x-5
1 d·�·
d.
z
-
+2
4. Which of the following functions does not have a vertical asymptote?
c.
a.
3+3
'!
-4
)!=-2
hp
x -5
b.
d.
x
z
5.
r+2
2
)'± -
x
-5
Which of the following functions does not have a horizontal asymptote?
a.
c.
+1
?_,,,a-1
ad
•
z y=
2-l
-5
+
b.
d.
•
3r-5+2
'-1
= _.
2
2r -5
+8
6.
.
(-3)(+ 2)
What are the zeros of r() = ?
.·
x(x+5)(- 1)
a.
=3-2-5,1,0
b. =-5,l
c.
d.
=3,-2
x=0,-5,1
g(r)
7.
Which of the following statements is false for the function f()
-- where both g and h are polynomial
h(r)
functions?
a. It is possible to have both a vertical asymptote and a horizontal asymptote.
b. It is possible to have both an oblique asymptote and a horizontal asymptote.
c. It is possible to have both a vertical asymptote and an oblique asymptote.
d. It is impossible to have an oblique asymptote, a horizontal asymptote, and a vertical
asymptote.
=
+l
r-?
8.
What is the horizontal asymptote of f() •
a.
)y=?
b.
y=-l
c.
d.
y=l
y=l
9.
r-3
What is the vertical asymptote of f(x) = - ?
-8x+15
a.
=3and x= 5
c.
=5
b. r=3
d.
=0
•
-9
What are the vertical asymptotes of f(x) = ?
-7+ 10
a.
=2and=5
c.
=3and=-3
b. 3=-Zand
-5
d. x= l
1. Which of the following functions has an oblique asymptote?
�
a.
C.
3+3
r-3x-4
)'= ----;
-5- 14
r-4
·,
b.
d.
-4
10.
=
'=
=
z
+3
s¢
(- 3)(x+4)
12. Which of the following functions has a vertical asymptote at x = 2 and a horizontal asymptote of y = 1?
2
a.
C.
+3
-6x+
?z -4
2
-
+i3+Z
b.
d.
3
2
)l=-- .
·
C
4
-2
y=
•
4+4
13.
Which of the following functions has {x] e } as its domain?
2
a.
C.
+5-(
y=,
7-1
b.
2
y=
14.
+
-4+3
- +6x+ 8
d.
y=,
++3
What is the domain of the following function?
x+5+4
lit-°
_____ !
/2
./'
\
·-----
C.
d.
Short Answer
l 5.
Under what condition does a rational function have an oblique asymptote?
16.
What are the zeros for the function f() = "
°',
(-9)(4 2)
-
I
x(-1)
(- 2)(+ 5)
17. What are the vertical asymptotes of the function () =
·
(
5)
IA•
'
(+1)
18. What is the domain of the function f() = ?
-1y' ,4
,''
' } · ' -'
),-(,. - �•-.)1' ,. + •:')
19. What are the x-intercepts of the function f() =
-4
.,
20. What is the y-intercept of the function f() =
r-2r+l
+3
?
21. Draw the graph of a rational function that has a vertical asymptote at x = 3 and a horizontal asymptote at
y=2
x+2
22. What are all the asymptotes of the function f() = _?
-'
2
23.
What is the domain of the function f(r) =
--]
"
? How many vertical asymptotes does this function
(- 1)(+ 2)
have?
24. Create a rational function with vertical asymptotes at = 5 and x = -3, a horizontal asymptote at y = D, and an
x-intercept of 2. Describe your process in creating the function.
2
-
+3-Z
25. Find the equations of all asymptotes for the rational function f(:) = +2
k
26.
Find the equation for the following graph in the form rtr)= ,
+b+c
6-4.-2
2
4
6
•. � . ! -· ·-
.s
27.
For the rational function r()=
--4
,determine the equations of all asymptotes.
,
r-3x+2
28. Create a rational function that has a domain of {x]x + -2,7,r e } and a vertical asymptote at r = 7. Describe
the graph of this function.
29.
Using domain, intercepts, and asymptotes, draw the graph of ft) =
+2-15
Review Questions Rational Functions
Answer Section
MULTIPLE CHOICE
1. ANS:
2. ANS:
3. ANS:
4. ANS:
5. ANS:
6. ANS:
7. ANS:
8. ANS:
9. ANS:
10. ANS:
11. ANS:
12. ANS:
13. ANS:
14. ANS:
D
B
B
C
A
C
B
D
C
A
B
D
B
D
SHORT ANSWER
l 5. ANS:
The degree of the numerator must be exactly one more than the degree of the denominator.
16. ANS:
The zeros are x = 9 and
= -2.
17. ANS:
The vertical asymptotes are
=
18. ANS:
The domain is {]x
I,x e F}
19. ANS:
The x-intercepts are x = , 3,- S.
20.
ANS:
1
The y-intercept is -.
3
21.
ANS:
6
-4 -2
6
-2
-4
-6
8
and
= 5.
MHF4U -- Final Exam Review
Rational Functions
1.
2
Find the vertical and horizontal asymptotes for J (x) _x+
3x-2
2.
Given: f(x)=
2
.Find the domain, intercepts and vertical and
x-2x -5x+6
horizontal asymptotes. Sketch the graph.
.
Given: g(x)=
a.
b.
c.
d.
4.
5.
x-2
x +5x+6
Determine x and y intercepts.
State the domain.
State any asymptotes.
Are there any holes?
-
·
x-3
Given:
x-9
a. Find the x and y intercepts.
b. Find the domain.
c. Find any asymptotes.
d. Are there any holes?
Create a function that has a graph with the given features.
a. Vertical Asymptote at x =-2 and horizontal asymptote at y = 1
b. Two vertical asymptotesat x=3 and x = 4, with a hole at x=l
c. Vertical Asymptote at x =-3, horizontal asymptote at y = O, no x-intercept and
a y intercept at -2
d. No vertical asymptote and x-intercepts at 2 and -1
6.
Graph f(_' I
rind the domain, the intercepts, the location of any
X
asymptotes and describe the function's behaviour near these asymptotes.
7.
Simplify and state restrictions.
x+1
x
a.---x
8.
x+l
3
b
x-x
· x 2 - 2.x-3
2
x-x
2x-6
b.'='
Solve and state any restrictions.
a.
'\_+3
x--3 x+4
5
d.O= , 2
hr+4
x-1
x+1
x-l
c.0=8%_200
r
I
e. 0=2x( +1?-1
to=2-!
X
MHF4U -- Final Exam Review
9. The estimate revenue and cost functions for the manufacture of a new product
are (x)=-28+15x and C(x)=5x+8.
a) Express the average profit function AP(x) = P(x) , in two different forms.
.
X
b) Explain what can be determined from each form.
c) What is the domain of the function in this context?
d) What are the break-even quantities?
10. The value of a car, t years after it is bought, is modeled by
a)
b)
c)
d)
v,
_2100+84,j50.
1+05t
State the domain of V(t), determine any intercepts as asymptotes, and sketch the function.
What will the car be worth in the long run?
After how many years will the car be worth $1500?
Find the·average rate of change in the value of the car between 2 and 5 years and the
instantaneous rate of change at 2 years.
-r+9 and its reciprocal on the same grid.
11. Sketch f (x) =
Answers:
1.
2
1
VA: x=- HA: y=-
2.
3
3
Domain: D={xlx e R,x + -2,x + 1,x + 3}, x-intercept is O and y-intercept is 0,
VA:r =l,x=3,x=-2 HA: y=0
3.
a) x-intercept is 2 and y-intercept is '
3
c) VA: x = -3,x =-2
4.
d) There are no holes.
a) There is no x-intercept and y-intercept is
c) VA: x =-3
5.
HA: y = 0
b) Domain D= {xhxe R,x + -3,x + -2}
a) y=
X
x+2
HA: y = 0
b) y=
±3
b) Domain D = {lxe R,x + ±3}
d) There is a hole at x = 3 .
x-l
(x - l)(x - 3)(x + 4)
--6
c) Y = (x+3)
d) y=(x-2)0x+1)
2
·.! -·-
MHF4U - Final Exam Review
6.
Domain: D= {xlx E R, x+0},
x-intercepts are ± 4 , there is no y-intercept,
VA: x = 0,
HA: none, Oblique Asymptote: y = x
I
/
Behaviour of function near asymptotes:
£
/
X - +0,y-) X
°
X -) -0o,y-)X
As
--00
x -0,y-+0
2x +1
i
,x±0,-l
7.
a)
8.
a) x=x±3,-4
b) x=-3or x=l, x±±l
d) x =±', no restrictions
2
ex
) =.-,no restrictions
9.
x(x +1)
-5
3 '
a) AP()_2
,
(x-9(x-D
x
oblique asymptotes
h
b) -2, x+0,±1,3
+✓ 3
..
3
c) r =4.3, r±0
t) x=±/5,x0
or AP()= -2+10
·x
b) From the factored form, you can find zeros and from the form
-
10.
/
x-)0,y
c) The Domain is: D = {xlx > 0,x eR}
g(x)+'you
can find
h(x)
d) x =4,x = 1
a) D= {rlt -2,t e R}, t-intercept is --27.11 years and V-intercept is $2250, VA:t = -2,
HA: V(t) = 166
b) Worth $166 in long run
c) 1.12 years after purchase d) Average Rate of Change
between t=2 and t=5 is: -$147.86. Instantaneous Rate of Change at t=2 years is -$260.50
11.
3
MHF4U -- Final Exam Review
Trigonometry Review
1. Sketch -2n <0 <2n of:
a)y-2sle-[
s)
v-;a[@{)
y-4cs[lo-rh+3
d) y=-3cos(40 +r)-2
z sea«o-a[o-{Jnsemvaar]'.,]
a) Find the average rate of change (to 5 decimals) of the function from ""
3
3
What does this
tell us about the graph?
b) Find the instantaneous rate of change (to 5 decimals) of the function at ., What does this tell us
3
about the graph?
3. State two co-terminal angles for each of the following. Your answers must include n.
2TC
a) -
b)
3
11n
6
c)
7
d)
5
3n
4
4. Find each function value.
a) csc0 if sine=
}s
b) cot
if cseo='
3
5. Find each of the following. Use exact values only.
7
$ SIn-
a
b) sec
4
11n
6
2n
37n
c) cot -
d)
c) cot 6 =-I
d) tano =[
3
COS
4
6. Find all values for 0<0 <2n.
6
a) cos 6 =
2
b) sec0= -[2
7. Solve for0<0 < 2n.
✓3
a) cos20 =
2
a[{ '
='
s a[·-5l-'
b) sin20
2
o[;]'✓
o=[·{)-}
2
3
MHF4U -- Final Exam Review
8. Solve for0<x<2n.
a) cos±-'
4
b) 2sin2x + sin x-1 = O
c) 10cos(2x) + 700s(2x)= 6
d) 4cos(2)-1= 0
e) 3tan'x = 1
f) 2tan'x + tan x-3 = 0
g) 6cos2x - sin X -4 = 0
h) 2cos x = 1-
sin x
i) 2sin x =-COS X + 1
9. Write as a trig function of the related angle and evaluate:
o[-)
so[)
10. Write as a trig function of the co-related (co-function) angle and evaluate.
a) sec
».[;-5
(117!)
,
11. Write as a single trig function.
a) sin AcosB -cos Asin B
d) cos 3M
g)
j)
-sin 3M
. X
X
4 Sin-cos--
2
2
tan 3x - tan 4x
I -tan 3xtan 4x
2tanx
I-tan x
b) cos2AcosA-sin2AsinA
c)
e) 1 Osin xcosx
oz[$)
») 2cos
'5!'
;4
i) 2 tan acos a
12. Evaluate the following.
a) sin 45 cos15- cos 45 sin 150
7
c)
b)
7[
7[
•
7[
•
7[
COS-COS--Sin-SIn-
4
12
4
12
T
tan + tan
12
6
T
T
12
6
d) cos15 + sin15°
I--tan tane) cos15 - sin 15°
•
f) (cos 15 + sin 15°
TT
g ) sIn
12
n) ta !
12
2
MHF4U -- Final Exam Review
13. Simplify.
a)
c)
(sin x + cos x)(sin x -cosx) + I
b)
.5
sIn x
sin(x-30°) + cos(60° - x)
sin x
d)
tan xsin' x
sec x
sin2a
cosa
f) (sin x +cos.x)' -sin 2x
e) 2sin a +cos2a
g) 2sinxcos x-2sin xcos.x
14. Develop a formula for sin 30 in terms of sin 0.
15. If sins" and cosy '',where 0<±<" and " <y<n, fnd
5
13
2
2
b) tan(x + y)
a) sin(x-y)
the following.
c) sec(x - y)
16. If cot=-' nor " ax<n, find the following.
3
2
b) cos 2x
a) tan 2x
c) csc 2x
d) sin 4x
X
e) tan -2
17. Prove the following identities.
l+tan'x
tan
a) =-CSC
x
c)
cos x -sin'x
4
b) ----- = cos X
I-tan x
2
cos'x-sin'x
cos.x-sinx
.
----- = 1 + sm xcosx
cos(x -y)
cosxcosy
e) "-=l+tanxtany
)
d) cos(x + y)cos(x-y) =cos x-sin y
4;-») «al-)5~[3E)
2
"a/j
1-cos2x
.
=SIn 2 •
tanx
h) 2cosx-2cos'x = sin xsin2x
«) [f-»,-«as
j) cot0 +tan6 = 2csc20
sin 2t
sin t
cos2t
cost
) SeCl=-
3
i:
MHF4U -- Final Exam Review
18. Sketch y = cot x for -2n <x<2n.
19. a) State the equations of the vertical asymptotes for
i) y = secx for xeR
ii) y = tan x for xcR
b) State the local minimums for y = csc x for xsR.
c) State the period of y = tan 2x.
Answers
1. (a)
(b)
"
V
(note: vertical scale is 1, horizontal scale is
7
6
)
c)
(note: vertical scale is 1, horizontal scale is 1C)
(d)
T
(note: vertical scale is 1, horizontal scale is )
4
TT
(note: vertical scale is 1, horizontal scale is )
8
2.
7
(note: vertical scale is 1, horizontal scale is -)
6
a) ARC = -0.95493 -This tells us the slope of the secant between the
points(;
,2 )and(2; ,1}
4
MHF4U - Final Exam Review
b) IRC = -1.73726 -- This tells us the slope of the tangent at "
23n
n
8 ·%
3
1
5.a)
77
b) -2
6. a)
IHn
6'6
% 3 Sr
c)
a
nr
'«
~ n 5n 3n
n 5n 7n 11n
W
n
9, a)·-cos=-
7a±
5n
I
b) Cos-=---
V5
4
11. a) sin (A - B)
b) cos(3A)
c) tan(2x)
d) cos(3M)
g) -2sinx
h) -sin0
i) sin(2a)
j) -tanx
s)' o1 o1 eN?
2
13. a) 2
c)
5
°
13
(c) -
12
13
e) 1
d) 2sina
1s.e-"65
14. 3sin6 -4sin'0
(s)
2
"
56
e) 5 sin(2x)
6-7
%
%4°
2
b) sin'x
16. (a) "
'
47
f) 0, .2r
3
4
2
6
2
11r
TT
10. a) csc- = 2
1
5n
e 3
b) -tan=--I
T
12. a)
4r
3'3
5n 7n 11n
3es
·;1; ·{as
6
a
1% ",
8.a)z%·¢.«
1Hnr
1
5n 13n 17n
.)5%
5n
o2
« 3+4'47
4'4
n 1Hn 13n 23n
3
9nr
-
s 2/
3
,
1ln
4
f) 1
%.
46
cos3
) 2+5-!
2
1 . 4 ·x
g ) -sin
2
65
¢
(d) 120
169
16
(e)
W13+2
MI-2
y ==cot.r
18.
\
7
19. (a) ( x= (2n+1)z.ne I
\
7t
(ii) x = (2n+1)--,n e I
2
1l
local minimums at (x,1) where x = -+2nn,ne I
\
(b)
\
(c) Period is
2
2
5
Factoring Special Polynomials
:±2y +y =(x±
,.- -
a
-y =(-y)j + y)
±= ( ± y)( ; y +y)
o
'
. t
cN
r
I'
$
r
-«Ir
&le
t
t
Rules for Exponents
-x
Jt
.
I
E'
¢
,.,
I
F
e
... 5l
4¥
<'
log, x is written as ln x
log, e=1
=
log.=1
log, a
a·la11,.:r = X ·
=
n·ls
log
hoe.[5;]-oe. +-toe•
5/o
log,
log. b
It
e
t'
Ii
I
m
t
5le
F
y
m
m
5/
4]es
e·
e
cos(A ± B) = cosA cos.B
Double-Angle Formulas
t
I
es
k; /rN
ttCl
r
k/le
a;
t /s
"'
t
k; /o
tr
tan2A. ?tanA
1-tan A
e
0
Fundamental ldentities
IC·
:?.....
sin 0 =1-cos 0
cos0=1--sin@
sec O=1+tan 0
csc 0 =1+cot 0
·
¢
» ;
f
t
t
t
•
r
r
t
]
¢
0
r
l
I
g
JI:
f
i
%±
-;:
po(.
I
'
I
ss[5
a
m
9
le
t
n
.
l(
)t'.
0
f
t
)(
..
e
I
�
4/ [;
t
4
mt
r
r
r
e]cl.
ll
e
�
t
r
t
ss[;
0
-.
-.t
•r
«
«df
•
¢
t
4
):(
Ji:
5
?-
I
t
,-(
%
)t
/ ·[ r!I
•
r
r
I -.
' .
t
)(
4]s 4/% e
0
ss
f..
N
...
�
I
4[s e]
r
D
.........
l
R.
z
t
r
i
·-
r
m
.. I
IN]s 42[ 4
):(
r::t
r!
r
o
n
i
I
eI yt
7
l:l
I
[€
[N;
ss
-
X
f
rt[r
r
¥
f
•
)(
I
4/
r
4/s -]e
rt
t
sec0= cot0=-±
cos6
tan6
)I;
...
f
I -- -
ti
):(
It
e
csc@= -.%
•
.J
0
�
2]¢ rt]r
I
m
I
/ [ �-
I
m
'g
4/ ·F rtr
.t
t
eI
.4
I
l(
I
t
...It
,.,
1
t
c
4
sin 2A = 2 sinA cos A
cos24 = cos A - sinA
= 2cosA--1
=1-2sin4A
Cl
Cf
I'
«
.tanA±tanB
tan(A±B).
· 1tanAtan B
.... - . .... I ..
rt]rt
«v
sinA sinB
r
rt[eN
•
•
cos6
I
I
o
...
Addition and Subtraction Formulas
sin(A± B) = sinA cosB ± cos A sinB
6+ cos' 6=1 tan6=
/s
I
[C
j:(
di
t
sin@
t
):(
«
2
I
0
rt
It
c
-- ·--
log,"= n log,
• 2
sin'
):(
t:t
I
P..
t
sv[< (
I
li
):(
«]
r
.-E
�
4v
4
ti
e.=x
log, (y) = log, + log. y
....
-
X
e!
I
I
r
e 5le ]e ]
I
I
«
x
I
g
z
j
�
m
/s [ �
¢
4¥
-
li
• •
a,
t
I
m
; $ls"
y=log,x
=x
log, is usually written as log x.
0
r
rt Ir
Sl /
I
t%
-
¥
;] 4g]a
I
e
, c
Logarithms
-
:t(
.-(
r
r
I'
Ill'
-
4
l(
t
C
q]
3
t
r
wt
#
4
«
$
s1n 0
Logarithmic Scales
±
«
-es'] -os}]
4=
P±?)"
z-vs@l
A=P (+
)j
i
\
MHF4U -- Exam Review
Exponential and Logarithmic Functions
1. Sketch the following
(a) y =302)
(b) y=-402)'
(d) y= 23
(c) y=-(2)
(fy=2?-3
sr»-{3J
(a) 2logx
(b) y= -logx+2
(c) y= log(3x)
(@d) y=log (+9)
(e) y= log(2r-6)
(·)
2. Sketch the following
y=log['r+2]
g) y=-log(-2)+2
3. Express in logarithmic form
(a) 5 = 25
(b) 6"° = 1
I
1
(c) 49° =7
(d) 125 = 25
4. Express in exponential form
(a) log,I= 0
3
(c) log,27=--
(b) log.(±)=-2
2
5. Evaluate the following
(a) log; 49
(b) lo9,16
(I) log, 576 -log724
(c) log%V27
() log32
(e) log1
( log% }
(m) log¢ 18 +log¢ 12
'
6. Solve
(b) logVx = log1-3log2 (c) log,x- log, 4= log, 12
(a) logx = log5 + 2 log3
7.
Express as a single log:
1
I
1
2
3
4
(a) -log,x +-log,y--log,z
1
(b)--[log, x +3log, y]-2[log, a+ log, b]
2
1
MHF4U- Exam Review·
8.
Solve
(a) 3'5
(b) 2 +302")-10=0
(c) 6
-2(6)-15=0
(e) 6(10)+10 --2= 0
9. A sample of 500 cells in a medical research lab doubles every 20 min.
a) Determine a formula for the number of cells at time t, where t is measured in minutes.
b) How long will it take for the population to reach 18 000? Answer correct to 2 decimal
places.
10. In 1987 there were about 130 000 cell phone users in Canada. In 1999, there were about 10
million cell phone users. What is the percent increase per year? Answer correct to 2 decimal
places.
11. A sample of 700 cells in a medical research lab triples every 30 min.
a) Determine a formula for the number of cells at time t.
b) How long will it take for the population to reach 18 000? Answer correct to 2 decimal
places
12. A sample of radioactive iodine-131 atoms has a half-life of about 8 days. Suppose that 1 000
000 iodine-131 atomsare initially present.
a) Determine a formula for the number of atoms at time t, where t represents number of days
b) How long will it take for the sample to reach 180 000 atoms? Answer correct to 2 decimal
places.
13. A new car costs $23 000. In 5 years it will be worth $9500. What is the rate of depreciation
· per year? Answer in percent, correct to. 2 decimal places.
14. Most of Canada's earthquakes occur along the west coast. In 1949, there was an earthquake
in the Queen Charlotte Islands that had a magnitude of 8.1 on the Richter Scale. In 1997
there was an earthquake in south-western B.C. with a magnitude of 4.6 on the Richter Scale.
How many times as intense as the 1997 earthquake was the 1949 earthquake? Answer
correct to 2 decimal places.
15. The loudness L in decibels (} of a bel) of a sound of intensity / is defined to be
I
L=10log
Io
where I, is the minimum intensity audible to the human ear
a) The loudness level of a heavy snore is 69 dB. How many times is this more intense than
conversational speech at 60 dB? Answer correct to 2 decimal places.
b) Sound is 316 times less intense if earplugs are worn. What would the decibel level of
snoring be if earplugs were worn? Answer correct to the nearest dB.
16. Given the function y = 4, determine:
a. the average rate of change from t=5 seconds to t=6 seconds.
b. the instantaneous rate of change at t=5 seconds.
2
MHF4U -- Exam Review
Solutions
1. (a)
(b)
(c)
(d)
t
(f)
(e)
2. (a)
(g)
(b)
(c)
·····1
. .. , f
• +.2#
.
·1 .
,
�... ,, /
(
[-
i
.- .l
?
¢
·)
+
f
(f)
-2
I
(g)
'1
.
� �
;;�;,�: -1::::r,,. . . -':F"..:. .::;jJ.. :.=:-=r.: :::: =:=-; =:
->-,f
j'
+
·j j
!
+
l
3
MHF4U -- Exam Review
3. (a) log,25=2 (b) log.1=0
(c) log,7-,
2
(d) log.25=3
2
(b)63='- (c)9° =27
36
}
; to
5 (a)2 ()4
4. (a) 3° =1
« () ; () 3s 0?2 «!'3 (e "3
(I) 2
(m)3
='
+a4ls±£]
6. (a) x=45
(b)
8. (a) x=1.23
(b) x=1
64
(c) x=48 (d)x=3
(e)64
.('?9
(c) x=0.9 (d) x=-1 (e) x= -0.3
I
9. (a) N() = 500(2)3°
(b) 103.40 min
ii
I
11. (a) N() = 700(3)3
10. 43.61%
= 100oooo[']' (s)
I
(b) 88.67 min
I
12.
(a) N(
13. 16.21%
16.
14. 3162.28
19.79 days
15. (a) 7.94 (b) 44 dB
(a) 3072 (b) 1419.57
4