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 #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 6
Due noon Monday, November 4 in the Math 147 dropbox.
1. (a) Fix a > 0. Find the tangent line to the curve x + y = a, where x, y 0,
at a point (x0 , y0 ) on the curve.
(b) Show that the sum of the x and y intercepts of this tangent li
Math 147 Assignment 1
Due by noon Monday, Sept. 16 in the Math 147 dropbox
No calculus is needed to do this assignment.
1. Three young men accused of stealing DVDs make the following statements:
(1) Ed: Fred did it, and Ted is innocent.
(2) Fred: If Ed is
Math 147 Assignment 7
Due noon Wednesday, November 13 in the Math 147 dropbox.
2
x 3
1. Graph f (x) = exp x2 x . You can use a computer program to nd the approximate roots
of the polynomial of the degree 5 polynomial that occurs in the second derivative.
Math 147 Assignment 5
Due noon Monday, Oct. 28 in the Math 147 dropbox.
1. Suppose that f (x) and g (x) are continuous functions on [a, b]. Prove that
h(x) = maxcfw_f (x), g (x) is continuous using the denition of continuity.
2. Fix a number d > 0. A func
Math 147 Assignment 4
Due noon Wednesday, Oct. 16 in the Math 147 dropbox.
5
2
1. Graph the function f (x) = xe x x2 for x = 0. Pay attention to the following:
asymptotic behaviour at and behaviour at 0.
compute the derivative and nd the critical points
Math 147 Assignment 3
Due noon Friday, Oct. 4 in the Math 147 dropbox.
1. Give a careful argument to prove that lim 2x2/3 = 18.
x27
2. Compute the following limits:
5
2
(a) lim
2
x1 1 x
1 x5
sin x
(c) lim
x0 sin 3x
1
sin x
(e) lim cos x2 + 2 .
x 0
x
x
3.
Math 147 Assignment 2
Due noon Wednesday, Sept. 25 in the Math 147 dropbox.
1. Let an =
n2 + n n for n 1.
(a) Show that this sequence converges and nd the limit.
1
(b) Given = 2 1040 , nd an N that works for the N denition of limit.
2. For the following s
Math 147 Calculus (Advanced section)
Professor: Kenneth Davidson
Oce: MC 5080
Email: krdavids@uwaterloo.ca
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
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 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 #3
1) A function is defined by
8
< x + 5a
ax + b
f (x) =
:
2x + 10b
if x 3
if 3 < x < 3
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 resulting funct
1
MATH 147 : Calculus 1
Electronic Assignment #4
Instructions:
Ensure you have completed all of the lectures in Module 1, 2, 3 and 4, as well as the first 4 lectures
in module 5. You will require this information to complete the following questions.
Rea
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
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 Notes. You will require this information to complete the
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
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
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 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 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