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PREREQUISITES FOR
CALCULUS (MATH1113) AND
PRECALCULUS (MATH1102)
This pamphlet describes the prerequisites for MATH1102 (Precalculus Mathematics) and
MATH1113 (Introductory Calculus I) and provides practice questions on the
prerequisite topics (a separate pamphlet is available for Statistics and Finite
Mathematics). The formal prerequisite of MATH1102 is Grade XI and XII Math AND the
Mathematics Placement Test, or MATH0020 passed in the last three years with a grade
of C or higher). The formal prerequisite of MATH1113 is either MATH1103, or high
school precalculus mathematics AND the Mathematics Placement Test.
A separate
brochure with information on the Mathematics Placement Test is available from the
Mathematics Department.
IF YOU HAVE NOT YET TAKEN THE PLACEMENT TEST the questions here will give you
practice in the required topics. Note that the actual placement test consists of
multiple choice questions so that it can be marked quickly. CALCULATORS ARE NOT
NEEDED AND NOT PERMITTED for the placement test (otherwise students with more
sophisticated calculators would have an advantage).
IF YOU HAVE ALREADY TAKEN THE PLACEMENT TEST or a prerequisite course the questions
here cover the topics that you will be expected to know at the start of the course.
It is strongly recommended that you review these questions just before the course
begins so you will find it easier to learn the new course material.
One major
reason why students have trouble with these courses is that they are weak on the
prerequisites. Since mathematics is cumulative, difficulty with algebra will cause
a lot more difficulty with calculus. Note that calculators are required for these
questions and for both courses.
Prerequisites for MATH1102 (Precalculus Mathematics)
Covered in MATH0020 (College Algebra), or in high school mathematics.
A. Polynomial Expressions
B. Linear Equations in One Variable
C. Equations of Lines
D. Systems of Equations in Two Variables
E. Factorization
F. Quadratic Equations
G. Rational Expressions
H. Square Roots
Prerequisites for MATH1113 (Introductory Calculus I)
Covered in MATH1102/1103 (Precalculus), or Grade XI, XII, & precalculus mathematics
0. ALL PREREQUISITE TOPICS FOR MATH1102 (above)
A. Functions
B. Quadratic Functions
C. Polynomial and Rational Functions
D. Exponents and Radicals
E. Conic Sections
F. Exponential and Logarithmic Functions
G. Trigonometry
References: For MATH1102 try textbooks with titles like "College Algebra", and, for
MATH1102 or MATH1113, titles like "Algebra and Trigonometry" or "Precalculus". Some
are in the university library (mostly near QA154).
MATH1102 PREREQUISITE QUESTIONS
A. Polynomial Expressions
1) Evaluate 4x2y - 3xy2 for x = -2, y = 3.
2) Simplify (5ab-4b)-(3ab+2a-6b).
3) Multiply (6x-2)(x2-3x+5).
B. Linear Equations in One Variable
4) Solve and check: 4(3x+5) = -5(x+13).
5) Jo has 23 coins. She has twice as many nickels as pennies and three
more dimes than pennies. How many of each type of coin does she have?
6) Solve and check: 9x - 7(x+200) = 6600.
7) The length of a rectangle is two centimetres more than twice its
width. The perimeter is 34 centimetres. Find the dimensions.
8) Solve: 3-2x < x+9.
C. Equations of Lines
9) Graph the linear equation 4x - 5y = 10.
10) Find the slope of the line through (-5,2) and (6,-1).
11) Give the equation in standard form ax+by=c of the line through the
points (2,-1) and (4,6).
12) Give the equation in standard form ax+by=c of the line with slope
-5/6 and y-intercept 4.
D. Systems of Equations in Two Variables
13) Solve by graphing: 14) Solve by elimination: 15) Solve by substitution:
x - y = 8
5x + 3y = 10
6x - y = 9
3x + 2y = 9
6x + 4y = 15
y = 3x - 7
16) Alice spent 6 minutes on each factoring problem and 3 minutes
on each evaluation problem. She spent a total of 42 minutes
on 9 problems. How much time did she spend on factoring?
E. Factorization: for questions 17-24, factor the expression completely.
17) Factor 15a2b3c - 45ab3c2 + 9b4c3
18) Factor 6x2 - 12x + xy - 2y
2 6
4
19) Factor a b - 49c
20) Factor 27s3 - t6
2
21) Factor t - 10t + 24.
22) Factor 2x2 + x - 10.
4
2
23) Factor x + 3x - 18
24) Factor 36x3 + 6x2 - 12x
F. Quadratic Equations
25) Solve by factoring: 2x2 - 31x = 16. 26) Solve by any method: x(x-3) = 10.
27) One side of a right-angled triangle is 4 cm less than the hypotenuse and
the other is 2 cm less than the hypotenuse. Find the lengths of all sides.
28) Solve using the quadratic formula: 2(x2-3x) = 5
G. Rational Expressions: for questions 29-33, simplify the expression.
29 (a)
(b)
30
(b)
31 (a)
32
H. Square Roots
34) Evaluate
35) Simplify (a)
(a)
(b)
(b)
(c)
(c)
(d)
(where a,b $ 0)
Answers to MATH1102 Prerequisite Questions
1) 102 2) 2ab+2b-2a 3) 6x3-20x2+36x-10 4) x = -5
5) 5 pennies, 10 nickels, 8 dimes
6) x = 4000 7) 12cm x 5 cm
8) x > -2 or (-2,4) 9) Intercepts at (0,-2) (2 1/2, 0) 10) -3/11
11) 7x-2y=16 12) 5x+6y=24 13) (5,-3) 14) (-5/2,15/2) 15) (2/3,-5)
16) 30 minutes (5 problems) 17) 3b3c(5a2-15ac+3bc2) 18) (x-2)(6x+y)
19) (ab3-7c2)(ab3+7c2)
20) (3s-t2)(9s2+3st2+t4) 21) (t-6)(t-4)
2
22) (2x+5)(x-2) 23) (x -3)(x2+6)
24) 6x(3x+2)(2x-1)
25) x =
29) (a)
31) (a)
34(a) 9
-1/2, 16 26) x = 5, -2 27) 6 cm, 8 cm, 10 cm
28)
\
2
7x / 9y (b) -2x / x+1
30) 4(x+8) / x(x-6)(x+2)
t / t+6
(b) x+4 / 5x-1 32) 4 / x-2 33) 2(9-x) / 8+9x
(b) -5 (c) not a real number (d) 4 35(a) 3%5 (b) %6/3 (c) 5a3b5%2
MATH1113 PREREQUISITE QUESTIONS
A. Functions
1) f(x) = 3x2+8
Find (a) f(-2) (b)
2) f(x) =
Find the domain.
3) For children between ages 6 and 10, height y (in cm) is frequently a
linear function of age t (in years). The height of a certain child is
122 cm at age 6 and 128 cm at age 7. Express y as a function of t.
4) For f(x) = 3x and g(x) = 2x2-5x find (a) (fBg)(x) (b) (gBf)(x)
5) f(x) = 3x + 5 Find the inverse function f-1.
6) f(x) =
x+4
if x # -1 Find f(-2), f(5) and f(-1).
9 x2
if x > -1
7) For f(x) = *2x+7* - 4 solve f(x) = 11
B. Quadratic Functions
8) f(x) = 3x2-6x+2 Find the vertex and intercepts and sketch the graph.
9) A library is suffering from overcrowding. Surveys have shown that when
there are 200 seats in the library, students can study at an average
rate of 12 pages per hour, but that for every 25 seats added, the
resulting distractions slow the average study rate by 1 page per hour.
How many seats should the library contain to maximize the total number
of pages studied there when all seats are occupied?
10) Solve x2-x $ 12
11) Find the domain of
C. Polynomial and Rational Functions
12) Solve (x-1)2(x+3) > 0
13) f(x) = x3+x2-5x+3 Factor f(x) completely.
14) Graph f(x) = (x-1)2(x+3)
15) Solve
16) For
find all int
and asymptotes, then sketch the graph.
D. Exponents and Radicals
17) Simplify (a) (5x2y-3)(4x-5y4)
(b) (27a6)-2/3
18) Rationalize the denominator of
19) Factor and simplify
20) Solve
21)
(3x+1)6(1/2)(2x-5)-1/2(2) + (2x-5)1/2(6)(3x+1)5(3)
Find the inverse function.
22) Sketch the graph of
E. Conic Sections
23) Find the distance between points P1 = (-2,5) and P2 = (3,4)
24) Graph x2+y2+2y = 4x+4
25) Graph 4x2+9y2-32x-36y+64 = 0
F. Exponential and Logarithmic Functions
26) Solve the equation 4x-3 = 84-x
27) Sketch the graph of f(x) = 3x-1
28) The number of bacteria in a certain culture grows exponentially, so
that the number f(t) of bacteria t hours after noon is given by
f(t)=500(20.5t). Estimate the number of bacteria at 2 p.m.
29) Evaluate where possible a) log525 b) log2%8 c) log100 d) ln(e-3)
30) Simplify a) 2loga5 + loga20 - loga100 b) (loga9)2 + (loga2)2
31) Solve a) 10x = 25 b) log9x = 0.5 c) logx36 = 2
32) Rewrite using log (i.e., common logarithms log10) a) log21000 b) ln(10)
33) When a certain drug is taken orally, the amount A (in milligrams)
present in the bloodstream after t hours is predicted to be
A = 100(1 - ½t), for 0#t#10. How much of the drug is predicted to be
present in the bloodstream after 2 hours?
34) Solve log2x + log2(x+2) = 3
35) Find the inverse of f(x) = 2ln(x)-7
36) Find the domain of f(x) = log(x2-2x)
37) Find the inverse of
G. Trigonometry
38) Find the radian measure equivalent of (a) 120E (b) -45E (c) 450E
39) Find sint, cost, and tant for (a) 3ð/2 (b) -ð (c) 9ð/4
40) Find the values of all six trigonometric functions given that
cost = -1/4 and tant > 0.
41) Verify the identity cos2t (sec2t - 1) = sin2t
42) Find sinè, cosè and tanè for the angle è in standard position with the
point P(-2,1) on its terminal side.
43) A right triangle has an angle è, a hypotenuse of 7, and the side
opposite to è of 4. Find sinè, cosè and tanè.
44) Find the exact value a) sin(2ð/3)
b) cos(-5ð/4)
45) Find the exact value of all angles è in the interval [0,2ð) that satisfy
the equation sin è = -1/2.
46) Find the amplitude, the period and the phase shift and sketch the graph
of f(x) = 1.5 sin(3x + ð).
47) Find all solutions of cot è = %3
48) Find all solutions in the interval [0,2ð) of tan2x sinx = sinx
49) Find the exact value of sin(3ð/8) using the half-angle formula.
50) Given á and â are second quadrant angles with siná = 3/5 and tanâ = -2,
find sin(á+â).
51) Verify the identity cos(á-â) - cos(á+â) = 2sinásinâ
Answers to MATH1113 Prerequisite Questions
1(a) 20 (b) 6a+3h
2)[1/3,4)
3) y = 6t + 86.
4) f(g(x)=6x2-15x g(f(x))=18x2-15x
5) f-1(x)=(x-5)/3
6) f(-2)=2 f(5)=25 f(-1)=3
7) x= 4 or -11
8) Vertex at (1,-1) Zeros (6±%12)/6.1.58,0.42 Graph is a parabola, opening
upwards, y-intercept 2.
9) 250 seats (which maximizes studying at 2500 pages).
10) (-4,-3]c[4,4)
11) (-4,1]c[2,4)
12) (-3,1)c(1,4)
13) (x-1)2(x+3)
14) Cubic polynomial, zeros at -3,1; y-intercept 3.
15) (-4,-1/2]c(3,4)
16) f(x)=(2x+1)/(x+3) with a hole at x=2 (2,1). x-intercept x=-1/2
vert.asymptote x=-3 f(0)=1/3 horizontal asymptote y=2
17(a)20y/x3 (b)1/9a4
18)(t+10%t+25)/(t-25)
19) (3x+1)5(39x-89)/(2x-5)1/2
20) 4
21) f-1(x)=(x-1)2+2(x$1)
22)graph of f(x)=%x (half-parabola) shifted right 2 and up 1
23)%26
24) Circle with centre (2,-1), radius 3
25) Ellipse centre(4,2) a=3 b=2
26) x=18/5.
27) Graph of 3x shifted 1 unit to the right, through (1/3,0), (1,1) and (2,3)
28) 1000 bacteria
29a)2 b)3/2 c)undefined d)-3
30a) loga5 b) cannot be simplified
31a) log(25) b) 3 c) 6
32a)log(1000)/log(2) b) 1/log(e)
33) 25 milligrams
34) 2
35) f-1(x)=e(x+7)/2
36) (-4,0)c(2,4)
37) f-1(x)=log2(1+3/x)
38a) 2ð/3 b) -ð/4 c) 5ð/2
39 a)sint=-1 cost=0 tant is undefined b)sint=0 cost=-1 tant=0 c)sint=%2/2
cost=%2/2 tant=1
40) sint=-%15/4 tant=%15 csct=-4/%15 sect=-4 cott=1/%15
41)cos2t (sec2t - 1) = cos2t(tan2t)=cos2t(sint/cost)2=cos2tsin2t/cos2t=sin2t
42) sinè=1/%5 cosè=-2/%5 tanè=-1/2
43) sinè=4/7 cosè=%33/7 tanè=4/%33
44 a) %3/2 b) -%2/2
45) 7ð/6, 11ð/6
46)Amplitude 1.5 Period 2ð/3 Phase shift -ð/3 One cycle between -ð/3 and ð/3
47) ð/6+2ðn, for all integers n.
48) 0, ð, ð/4, 3ð/4, 5ð/4, 7ð/4
49) %(2+%2)/2 50) -11 / 5%5
51) cos(á-â)-cos(á+â)=cosácosâ+sinásinâ-(cosácosâ-sinásinâ)
=cosácosâ+ sinásinâ-cosácosâ+sinásinâ=2sinásinâ