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 tan(x) ex sin(x)
ii) The Quotient Rule shows that f (x)
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 show that P (1) holds. But this is true since 1= 1(1 + 1
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 that f is continuous for all x. Sketch the graph of the
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 (x) =
1 x2 sin( x2 ) if x = 0 is dierentiable every0 i
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) n n d) e ! n
Solutions: [25 marks (5 marks each)]
n a)
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 value in a set Y . The set X is called the domain of t
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 continuity of g , there is a 2 > 0 so that
|g (x) g (c)| < if |x
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 is y0 + x0 y0 ; and setting T (x) = 0, the x-intercept i
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= 0
.
for x = 0
Show that h(x) 1 =O(x2 ) as x ! 1 and the
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 ambiguity as possible. Indeed, the rigour in mathematics i
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
=set v=5u+2u2 5 lim
2
u0
v 0
5u + 2u
v
dv
So y = x 5 is t
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 analysis from algebra. In
this section, we will deal with l
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
1+0+1
1+ n +1
n
1
|n n2 + n|
1
|an 2 | =
=
(b) Estimat
Math 147 Assignment 1 Solutions
1. (a) Ed is lying as Fred is innocent. Fred is telling the truth as Ed is not guilty. Ted is
lying as no one is guilty.
(b) Ed said Fred did it, so Fred is guilty. Ed said Ted is innocent, so Ted is innocent. Ed
must be in
Math 147 Calculus (Advanced section)
Professor: Kenneth Davidson
Oce: MC 5080
Email: [email protected]
Oce hour: Thursdays 10:0011:00 in MC 5080, or by appointment.
TA: Jordan Hamilton
Oce: MC 5071
Oce hours: Mondays 2:003:00pm, Tuesdays 10:0011:00 am
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. You will require this information to complete the follo
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 (x) = x
x2 sin( x12 )
0
continuous at x = 0.
2) a) Sho
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
so:
n
b) cfw_ sin(n)
a) cfw_
n+1
n
en
c) cfw_ n + 1
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 Waterloo and others
Brian Forrest
Mathematical Inductio
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 the y -coordinate of (x, y ).
University of Waterloo and
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 Theorem and the
Big O notation. The exam will be weighte
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-sequence cfw_ank with
limn ank =
LET LIM(cfw_an ) = cfw_
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. (The statement is proved if the column with the
heading p p
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 (x) = x
x2 sin( x12 ) if x 6= 0
is differentiable at x
Mathematical Induction II:
Principle of Strong Induction.
Recall:
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 Waterloo and others
Brian Forrest
Mathematical In
Preliminary Notation:
Sets and Boolean Algebra
Definition: Given A X define
(
1 if x A,
A (x) :=
0 if x
6 A.
A (x) is called the characteristic function of A.
University of Waterloo and others
Brian Forrest
Sets and Boolean Algebra
Preliminary Notation: