University of Toronto Scarborough
Department of Computer & Mathematical Sciences
MATA30: Calculus I - Midterm Test
Examiner: Sophie Chrysostomou
Date: Friday, November 23, 2012
Duration: 110 minutes
DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO.
FAMI
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #9
A. Homework problems from the lectures(+) :
1. Find the derivatives of f (x) = sec(1 + x3 )101 ) and g (x) = sin(cos(tan(sec(x2 + 1)
f (
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #7
A. Homework problems from the lectures(+) :
1. Show that f (x) = x3 x 1 has a root. Does f (x) = 1/2 have a solution? What
about f (x) =
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #12
A. Homework problems from the lectures :
1. Let f (x) = 6x4 20x3 6x2 + 72 x + 12. Using the rst derivative test, determine the
characte
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #8
A. Homework problems from the lectures(+) :
1. Find the derivatives of f (x) = 3
x
1/3
ex
x and h(x) = 2 .
x
solution :
f (x) = (3x )
x
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #15
You are expected to work on this assignment prior to your tutorial in the week of February
3 . You may ask questions about this assignmen
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #13
You are expected to work on this assignment prior to your tutorial during the week of
Monday Jan. 20th.
In your tutorial in the week of M
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Term Test II Info
The second term test is scheduled for Saturday March 1 1:00 PM- 3:00 PM in the Gym.
The second test will cover all the material cove
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #16
10-15 Integrals a Day
Keep Low Marks Away
So do not complain about the homework
You are expected to work on this assignment prior to your
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2011-2012
Solution to Assignment #15
A. Homework problems from the lectures :
3
7x(x2 + 1) 1 dx
1. Evaluate
1
3
x(x2 + 1) dx = 24
solution : In the lectures we es
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #14
A. Homework problems from the lectures :
1. Express the following as an integral:
n
lim
(1 +
n
i=1
solution : If x =
3i 2(1+ 3i ) 3
n
)
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #13
A. Homework problems from the lectures :
1. Sketch the graph of each of the functions below showing:
1)
3)
5)
7)
9)
domain
symmetry
hor
METHODS OF INTEGRATION
We have seen that in order to nd the exact area under the graph of a function f (x) it is
necessary to determine an antiderivative for f (x), i.e., a function F (x) with the property
F (x) = f (x). F (x) is also called an indenite i
2.1 THE LIMIT OF A FUNCTION AT A POINT
2.1
THE LIMIT OF A FUNCTION AT A POINT
The Tangent Line Problem
DEFINITION: The tangent line (or simply the tangent) to a curve at a
given point is the straight line that just touches the curve at that point, so
that
THE DERIVATIVE
3.1
DERIVATIVES
Problem: Find the slope of the tangent line to the curve of f (x) = x2 at
the point (1, 1) and at the point (a, a2 ).
68
c 2013 by Sophie Chrysostomou
3.1 DERIVATIVES
The Tangent Line Problem (revisited)
We now generalize ho
FUNCTIONS
1.1
FUNCTIONS
DEFINITION: A function, f , with domain D, is a rule that associates,
to each element x D a single real number, y or f (x).
The domain of f is often written as dom (f ). Also, x dom (f ) is often
expressed as f is dened at x.
The
APPLICATIONS OF DIFFERENTIATION
4.1
LHOPITALS RULE
DEFINITION:
The xa
lim
f (x)
0
has an indeterminate form of the type , if
g (x)
0
lim f (x) = xa g (x) = 0.
lim
xa
The xa
lim
f (x)
has an indeterminate form of the type
, if
g (x)
lim |f (x)| = xa |g (x)
INTEGRALS
5.1
THE AREA PROBLEM AND THE DEFINITE
INTEGRAL
n
Notation to be used in this section:
i=1
ai = a1 + a2 + a3 + + an
4
4
2
i2 = 12 + 22 + 32 + 42 = 30
ai =
EXAMPLE: If ai = i then
i=1
i=1
The following will be useful:
1.
n
1=n
i=1
2.
n
i=
i=1
3.
n
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #11
A. Homework problems from the lectures :
1. Suppose that f (x) is continuous and dierentiable on the open interval (a, b). If f (x) < 0
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #10
A. Homework problems from the lectures :
1 sin x
2.
x 0
x
x
solution : This has the indeterminate form , so LHpitals rule doesnt apply
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Term Test Info
Time:
Saturday November 23rd 1:00-3:00 pm.
Location:
You write in the room: If your last names starts with:
IC 230:
A - CHA
IC 130:
CHEN
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #3
You are expected to work on this assignment prior to your tutorial in the week of September 30th. You may ask questions about this assignm
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #4
You are expected to work on this assignment prior to your tutorial in the week of October
7th. You may ask questions about this assignment
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2012-2013
Assignment #1
You are expected to work on this assignment prior to your tutorial during the week of
September 16th. You may ask questions about this ass
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #2
You are expected to work on this assignment prior to your tutorial in the week of September 23rd. You may ask questions about this assignm
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30F
Fall 2013
Assignment #0
To prepare for the diagnostic test you could
Complete this assignment, the Review and the More High School Material Review
exercises on W
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #6
You are expected to work on this assignment prior to your tutorial in the week of October
28th. You may ask questions about this assignmen
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
2013-2014
Assignment #5
You are expected to work on this assignment prior to your tutorial in the week of Monday
October 21st. You may ask questions about this as
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #3
A. Homework problems from the lectures:
1. Prove the identity: tan x + cot x = sec x csc x
solution :
tan x + cot x =
sin x cos x
sin2 x
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A30Y
Solution to Assignment #2
A. Homework problems from the lectures:
1. Give the graph of the ceiling function.
solution :
o
5
The Ceiling Function
o
4
o
3
2
o
1o
-2