MATH 147 Assignment #4
Solutions 1) a) Find the derivative of each of the following functions. i) y = cos(3x2 ) + arcsin(x) ii) f (x) = iii) f (x) = x Solutions i) We have y = 6x sin(3x2 ) +
1 . 1x2 t
MATH 147 Assignment #3 Solutions
1) A function is dened by if x 3 x + 5a ax + b if 3 < x < 3 f (x) = 2x + 10b if x 3 where a and b are constants. Find values of a and b that will ensure that f is cont
MATH 147 Assignment #4
Due: Friday, November 20 1) a) Find the derivative of each of the following functions. i) y = cos(3x2 ) + arcsin(x) ii) f (x) = iii) f (x) = x
tan(x) ex sin(x)
2) a) Show that f
MATH 147 Assignment #1 Solutions
1) Prove that: a) 12 + 22 + + n2 = for each n N. Solution: (5 marks). Let P (n) be the statement that 12 + 22 + + n2 = n(n + 1)(2n + 1) . 6 n(n + 1)(2n + 1) 6
We rst s
MATH 147 Assignment #3
Due: Friday, October 30 1) A function is dened by if x 3 x + 5a ax + b if 3 < x < 3 f (x) = 2x + 10b if x 3 where a and b are constants. Find values of a and b that will ensure
MATH 147 Assignment #2
1) For each of the following sequences determine if it converges or diverges. If it converges nd the limit:
n a) cfw_ n+1 c) cfw_ n + 1 n e) cfw_ n cos(n) 2n+1
b) cfw_ sin(n)
Chapter 4
Limits of Functions and Continuity
4.0 Some Denitions
Below is our working denition of function. Informal definition. A function is a rule that assigns each element in a given set X a unique
Math 147 Assignment 5 Solutions
1. Let h(x) = maxcfw_f (x), g (x) and let c [a, b]. Let > 0. By the continuity of f , there is
a 1 > 0 so that
|f (x) f (c)| < if |x c| < 1 .
Similarly by the continuit
MATH 147 Assignment #4
Due: Monday, November 28
Hand in 2,3,5,7,8,9,11,12
1) a) Find the derivative of each of the following functions.
i) y = cos(3x2 ) + arcsin(x)
ii) f (x) =
tan(x)
ex
sin(x)
iii) f
1
MATH 147 :Calculus 1 Advanced
Electronic Assignment #1
Instructions:
Ensure you have read and completed Chapter 1: A Short Introduction to Mathematical Logic and
Proof in the Course Lecture Notes.
MATH 147 Assignment 5
Note: These questions need not be handed in but some may appear on the
final examination.
1) a) Show that | sin(u)
u |
|u3 |
6
b) Let
h(x) =
for every u 2 R.
sin(x)
x
1
for x 6=
Math 147 Assignment 5 Solutions
y
1
y
. So the tangent line
1. (a) By implicit dierentiation, + = 0. Thus y =
x
2x 2y
y0
(x x0 ).
at (x0 , y0 ) is T (x) = y0
x0
(b) Setting x = 0, the y -intercept i
MATH 147 Supplementary Material
Kenneth R. Davidson
1. The Language of Mathematics
The language of mathematics has to be precise, because mathematical statements must be
interpreted with as little amb
Math 147 Assignment 4 Solutions
1.
5
2
lim e x x2 = e0 = 1. Therefore we look at
x
2
2
e5u2u 1
(e5u2u 1)(5 + 2u)
x =set u=1/x lim
lim xe
= lim
u0
x
u0
u
5u + 2u2
2
d
(e5u2u 1)
ev 1
= 5 ev
= 5 lim
=s
Limits of Real Numbers
1. Limits
The notion of a limit is the basic notion of analysis. Limits are the culmination of an innite
process. It is the concern with limits in particular that separates anal
Math 147 Assignment 2 Solutions
( n2 + n n)( n2 + n + n)
an =
+nn=
n2 + n + n
n
1
=
=
.
1
n2 + n + n
1+ n +1
1. (a) Calculate:
n2
Therefore using properties of limits,
1
1
1
=.
=
lim an = lim
n
n
2
1
MATH 147 Assignment #2
Due: Friday, October 15
1) For each of the following sequences determine if it converges or diverges.
If it converges state the limit. If it diverges briefly explain why this is
1
MATH 147 : Calculus 1 Advanced
Electronic Assignment # 3
Instructions:
Ensure you have read and completed Chapter 4: Limits of Functions and Continuity up to
Sections 4.2.4 in the Course Lecture No
MATH 147 Assignment #1
Due: Monday, September 26
1)
a) The statement p _ p can be translated as either p is true, or not p is true.
Prove the statement
p _ p
by completing the following truth table. (
Math 147
Instructor: L.W. Marcoux
Assignment Three.
Due: Friday, September 29, 2017
Question 1. Equivalent formulations of countability
Let S be a non-empty set. Prove that the following statements ar
Math 147
Instructor: L.W. Marcoux
Assignment Two.
Due: Friday, September 22, 2017
Question 1. Equivalent formulations of Induction
Prove that the following statements are equivalent:
(a) The Well-Orde
Math 147
Instructor: L.W. Marcoux
Assignment One.
Due: Friday, September 15, 2017
Question 1. Logic
Three young men accused of stealing DVDs make the following statements:
Ed: Fred did it, and Ted is
Mathematical Induction:
Principle of Mathematical Induction
Axiom: (Principle of Mathematical Induction)
Suppose that S N is such that
I
I1) 1 S
I
I2) if n S, then n + 1 S.
then S = N.
University of
Sets and Boolean Algebra
Products of sets:
Definition:
Given two sets X , Y , define the product of X and Y by
X Y = cfw_(x, y ) | x X and y Y .
x is called the x-coordinate of (x, y ).
y is called th
MATH 147 Final Examination Information
Friday, December 9, 2016
Time: 7:30-10:00 P.M.
Room: M3 1006
Note: No Aids or Calculators are permited.
1) The exam covers material up to and including Taylors T
Lecture 15, Oct. 7
15.1 Theorem. Bolzano-Weierstrass Theorem Every bounded sequences has a convergent sub-sequence.
15.2 Definition. We say that R is a limit point of cfw_an if there exists a sub-seq