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 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
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
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 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 5: Wednesday, January 13, 2016
Last Lecture
trig integrals (7.2)
Strategy for evaluating
sinm x cosn xdx
a) n is odd: save one cosine and use cos2 x = 1 sin2 x to express
the remaining cosines in terms of sine. Then let u = sin x.
b) m is odd, sav
Lecture 6: Friday, January 15, 2016
Last Lecture
I trig substitutions
I integral contains (or powers of these)
p
p
p
a2 b 2 x 2 ,
a2 + b 2 x 2 or
b 2 x 2 a2
I In general:
Integral contains
Substitution
Apply Identity
a
a2 b 2 x 2
x = sin ,
1 sin2 = cos2
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 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) +
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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
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
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
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
Lecture 7: Monday, January 18, 2016
Last Lecture
trig substitutions: integral contains ax 2 + bx + c
complete the square rst!
partial fractions:
applies only to rational functions
2
4
6x + 8
dx vs
+
2 + 3x + 2
x
x +1 x +2
partial fraction decomposition ha
Lecture 14: Separable DEs 9.3, Mixing Problems 9.3
A First-Order Separable DE is a DE that can be written in the form
dy
= g(x)f (y)
dx
Examples:
dy
= x sin y is separable
dx
dy
y = yex is separable
dx
dy
= y(ex + 1)
dx
dx
= t + x is not separable
dt
How
Lecture 14: Wednesday, February 3, 2016
Last Lecture
dierential equations
a DE is an equation involving an unknown function and one or
more of its derivatives
examples
the order of a DE is the order of the highest derivative
the general solution of a DE r
Lecture 13: Dierential Equations 9.1, Direction Fields 9.2
A Dierential Equation (DE) is an equation involving an unknown function and one or more of
its derivatives.
Examples:
dP
= kP
dt
This DE describes the law of natural growth (solution: P (t) = Cekt
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