MATH 128 Winter 2013
Assignment 5
Topics: Linear dierential equations, separable dierential equations, applications of
dierential equations
Due: Wednesday, February 13th, 12:30 pm
Instructions:
Print your name and I.D. number at the top of the rst page o
UNIVERSITY OF WATERLOO
FINAL EXAM
WINTER TERM 2013
Student Name (Print Legibly)
(family name)
(given name)
Signature
Student ID Number
COURSE NUMBER
MATH 128
COURSE TITLE
Calculus 2 for the Sciences
COURSE SECTION(s)
001 002 003 004 005 006
DATE OF EXAM
M
UNIVERSITY OF WATERLOO
FINAL EXAM
WINTER TERM 2014
Student Name (Print Legibly)
(family name)
(given name)
Signature
Student ID Number
COURSE NUMBER
MATH 128
COURSE TITLE
Calculus 2 for the Sciences
COURSE SECTION(s)
001 002 003 004 005 006
DATE OF EXAM
W
MATH 128
Calculus 2 for the Sciences
Fall 2016
MWF 1:30-2:20 MC 4061
Tutorial: Mondays 5:30-6:50 in MC 4059
Instructor: Herbert Tang, Ph.D
Email: [email protected]
Phone: 519-888-4567, ext. 30177
Office: MC 6513
Office hours: MF 10-12 in MC 6471 (or by
UNIVERSITY OF WATERLOO
Final Exam
Fall Term 2013
Student Name (Print Legibly)
(family name)
(given name)
Signature
Student ID Number
COURSE NUMBER
MATH 128
COURSE TITLE
Calculus 2 for the Sciences
COURSE SECTION
001
DATE OF EXAM
Wednesday, December 11, 20
MATH 128 Winter 2013
Assignment 1 Solutions
1. Evaluate the following.
ex + 1
dx
2ex
(a)
1 ex
+
dx
2
2
x ex
+C
=
2
2
ex + 1
dx =
2ex
x5 x3 + 7dx
(b)
Let u = x3 + 7 so then du = 3x2 dx. Noting that x5 = x3 x2 and x3 = u 7 we get
1
x5 x3 + 7dx =
3
1
=
3
1
MATH 128 Winter 2015
Assignment 6 Solutions
1. Sketch the parametric curve given by the following parametric equations. Show your
reasoning/work.
(a) x = sin (t), y = 2 cos (t).
y
We have that = cos (t) so then
2
y
2
1 = sin2 (t) + cos2 (t) = x2 +
2
From
Math 128
ASSIGNMENT 1
Winter 2010
Submit all problems by 8:20 am on Wednesday, January 13th in the drop boxes across from MC4066, or in class depending on your instructor's preference. Late assignments, or ones put into the wrong drop slot will not be mar
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Lecture 2: Integration by Substitution 5.5, Integration by Parts 7.1
1st Integration Technique: Integration by Substitution
If u = g(x) is differentiable on some interval and f is continuous on the range of g, then
Z
Z
0
f (g(x)g (x)dx = f (u)du = F (u) +
Lecture 3: Friday, January 8, 2016
Last Lecture
integration by substitution examples
integration by parts formula and examples
udv = uv
Todays Lecture
integration by parts (7.1)
examples
denite integrals
integration by reproduction
trig integrals (7.2)
v
Lecture 4: Monday, January 11, 2016
Last Lecture
integration by parts
examples
denite integrals
integration by reproduction
trig integrals involving sine and cosine
sin5 x cos xdx
sin5 xdx
sin2 x cos5 xdx
What if the powers of sine and cosine are both eve
Lecture 2: Wednesday, January 6, 2016
Last Lecture
welcome, introductions, oce hours, course information
Review
FTC I & II (5.3, 5.4)
If f is continuous on [a, b], then
x
d
f (t)dt = f (x)
Part 1:
dx a
b
f (x)dx = F (b) F (a) = F (x)|b
a
Part 2:
a
Where F
Example:
Determine if the following series converge or diverge.
ln( n )
a) 2
n 1 n
c)
2
n 0
3 n 1 2 n
3
3n 2 5n
b) ln
4n 2 1
n 1
1
1
d)
n
n 1
n 1
1
11.4 Comparison Test
1
1 1 1
Consider the series n
3 5 9
n 1 2 1
1
1 1 1
Comparing this term-by
11.2 Series
A series is just a sum of the terms in a sequence:
Sequence
cfw_a1 , a 2 , a n ,
Series
a1 a 2 a n
We will often use summation
(sigma) notation so that we can
write the sum more concisely
an
n 1
Given that a sequence is an infinite number of
Lecture 1: Review: FTC I & II, Integration by Substitution 5.3 - 5.5
Recall:
The area problem and the denite integral:
n
b
f (x )x
i
f (x)dx = lim
n
a
i=1
The Fundamental Theorem of Calculus (FTC)
If f is continuous on [a, b], then
Part 1:
d
dx
x
f (t)dt
11.3 Integral Test
Another way of showing the harmonic series diverges
is to represent it as the area of blocks with unit width
1
and left endpoint on the curve y .
x
The total area of the blocks is
1
1 1 1
n 1 2 3 4
n 1
How does this
compare to the
are
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Lecture 3: Integration by Parts Examples 7.1, Trig Integrals 7.2
Example 2.5:
R
x2 cos 3x
Let u = x2 , dv = cos 3xdx = du = 2xdx, v = sin33x
R
R 2
2
3x
23 x sin 3xdx
x cos 3x = x sin
3
Let u = x, dv = sin 3xdx =
du = dx, v = cos33x
Z
Z
x cos 3x 1
x2 sin
Lecture 4: Trig Integrals 7.2, Trig Substitutions 7.3
What if the powers of sine and cosine are both even?
3
Example 4.1
cos2 xdx
0
1
1
Use half angle identity: cos2 x = (1 + cos(2x) (and/or sin2 x = (1 cos(2x)
2
2
3
2
cos xdx =
0
=
=
=
=
Example 4.2
1 3
Lecture 12: Arc Length 8.1, Area of Surface of Revolution 8.2
Last lecture:
The length of the curve y = f (x) on a x b is
dy
dx
1+
L=
a
2
dx
dy
b
2
dx
The length of the curve x = f (y) on c y d is
d
L=
1+
c
dy
2
Example 12.1: Determine the length of x = (
Lecture 13: Monday, February 1, 2016
Last Lecture
nished discussion of calculating arc length of a curve
dy
dx
2
1+
dx
dy
2
1+
dx
dy
2
1+
b
If y = f (x) on a x b, then L =
a
d
If x = f (y ) on c y d, then L =
c
surface area
if rotating about the x axis: S
Lecture 12: Friday, January 29, 2016
Last Lecture
nished discussion of volumes of solids of revolution
one more cylindrical shell example - rotating about a line other
than x or y axis
tips for calculating volumes of solids of revolution
arc length
The le
Lecture 11: Volumes (Cylindrical Shells) 6.3, Arc Length 8.1
Example 10.3: Find the volume of the solid obtained by rotating the region bounded by x = y 2 ,
x + y = 2 about y = 3.
x = y 2 , x = 2 y = y 2 = 2 y = y = 1, 2(x = 1, 4)
If we use vertical recta
Lecture 10: Review of Volumes (Washers/Disks) 6.2, Volumes
(Cylindrical Shells) 6.3
Volumes of Solids of revolution 6.2:
Example 9.5: Find the volume of the solid obtained by rotating the region bounded by y = x2 + 4
and y = 2x2 about y = 0.
2x2 = x2 + 4
Lecture 10: Monday, January 25, 2016
Last Lecture
improper integrals - comparison theorem:
If f and g are continuous functions with 0 g (x) f (x) for x a.
Then
a) If
f (x)dx is convergent, then
a
b) If
g (x)dx is convergent.
a
g (x)dx is divergent, then
a