Winter Term, 2015
Name: 52 l-U~l\'3 W _
Section: _ _.
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 104 Calculus II
Midterm March 2, 2015
Instructors:
A. Allison (Section C - 8:30 am MWF)
S. Bauman (Section B - 2:30 pm MWF)
Time Allowed: 80 m
MA104 Lab 3 Numerical Integration; Improper Integrals
Concepts
1. Numerical Integration (Text: 7.7)
Numerical [approximate] integration is useful when there is no way to find an exact value of a definite integral
b x2
e
a
because the integrand has no know
Winter 2011
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 104 -
Calculus II
Midterm Test -February 11, 2011
Instructor: A . Allison
Time Allowed: 80 minutes
Total Value: 60 marks
Number _of Pages: 7 plus cover page
Instructions:
Non-programmabl
MA104 Lab Report 3 - Approximate Integration; Improper Integrals
Name:
Student Number:
Lab
2
1. [8 marks] Consider the function f (x) = ex , 0 x 4. Dene f in Maple:
Winter 2015
f:=(x)->?;
(a) Approximate the area between f (x) and the x-axis using Simpson
MA104 - Final Examination Page 3 of 9
-___________________
[5 marks] 3. For each of the following, indicate in the space provided whether the statement is true (T)
or false(F). No justication for your answer need be shown.
Every bounded
Fall 2009
Name:
WILFRID LAURIER UNIVERSITY
Waterloo, Ontario
Mathematics 104 Calculus II
Midterm Test October 29, 2009
Instructor: Mr. Bauman
Time Allowed: 80 minutes
Total Value: 75 marks
Number of Pages: 7 plus cover page
Instructions:
Non-programmable,
MA104 Lab 2 Additional Integration Techniques
1. Trigonometric Identities
The focus of the lab assignment this week will be to use known techniques of integration and apply them in a
particular manner to integrate expressions containing trigonometric func
MA 104 Week 6 Report Differential Equations; Parametric Equations
Name:
Lab:
Winter 2006
x + xy 2
ex2 y
and it has an x-intercept at x = 2. [Note: You do NOT need to express y explicitly as function of x.]
1. [6 marks] Find a formula for the curve y = f (
MA104 Lab Report 4 - Volumes of Revolution; Arc Length
Name:
Student Number:
Lab
Winter 2015
2
1. [5 marks] Consider the region R in Quadrant I thats bounded by y = 4 x , y = 0, and x = 0.
(a) Sketch R on the axes provided.
(b) Using the disc method, stat
MA104 Lab Report 10 - Power Series; Taylor and MacLaurin Series
Name:
Student Number:
Lab
Winter 2015
n2
by completing the following:
n
n=1 2
1. [8 marks] Determine the sum of the series
(a) Use term-by-term dierentiation to express
1
as the sum of an inn
MA104 Lab Report 1 - Review of Integration Techniques
Name:
Student Number:
Lab
Winter 2015
x
, and y = 1.
4
(a) Sketch and shade the region bounded by the given equations.
1. [4 marks] Consider the equations y =
x, y = 8
(b) Integrating with respect to
MA104 Lab Report 11 Taylor and MacLaurin Series; Multivariable
Functions
1. Taylor and Maclaurin Series (Text: 11.10)
If f (x) [ an infinitely dierentiable function on an interval about a ] can be represented by a power series,
f (n) (a)
n
f (x) =
cn (x a
MA104 Lab Report 9 Strategies for Testing Series; Power Series
1. Absolute Convergence
The series
|an | is convergent and is called conditionally
an is called absolutely convergent if the series
convergent if it converges but
|an | does not.
We then have
MA104 Lab Report 5 - Surface Area; Exponential Models; Separable Dierential
Equations
Name:
Student Number:
Lab
Winter 2015
1. [10 marks] Consider the solid of revolution formed by rotating the region between the curves f (x) = sin2 (2x)+1
and g(x) = e3x
MA104 Lab Report 7 - Sequences and Series
Name:
Student Number:
1. [6 marks] Consider the sequence dened by an =
Lab
Winter 2015
2 5n2
, n 1.
3n2
(a) Plot the rst 20 terms of the sequence cfw_an using Maple:
a:=(n)->?;
pts:=seq([n,a(n)],n=1.20);
From the
MA170 Week 9 Lab Notes
General Annuities (4.1):
An annuity where payments are made more or less frequently than interest is compounded. There are two methods
used to deal with such annuities:
1. Change of Rate:
Replace the given interest rate by an equiva
MA104 Lab Report 11 - Taylor and Maclaurin Series; Multivariable Functions
Name:
Student Number:
Lab
Winter 2015
1. [10 marks] Consider f (x) = 4 cos(2x).
(a) Find the Taylor series for f (x) centered at a =
n
0
f n (x)
2
fn
22 cos(2x)
f (x)
22
1
23 sin(2
MA104 Lab Report 6 - Parametric Equations; Polar Coordinates
Name:
Student Number:
Lab
Winter 2015
1. [6 marks] Suppose the position of a particle at time t is given by x1 = 4 cos t, y1 = 2 sin t, 0 t 2 and
the position of a second particle at time t is g
MA104 Lab Report 8 - Series
Name:
Student Number:
1. [8 marks] Consider the series
an where an =
n=1
Lab
Winter 2015
n2
.
e n3
(a) Verify that the Integral Test can be applied to this series.
[Note: You may use Maple to nd any derivatives: f:=(x)->?; f (x
MA104 Lab Report 9 - Absolute Convergence; Power Series
Name:
Student Number:
1. [11 marks] Consider the series s =
1
3
an where an =
n=1
n=1
n
1
3
s=
n=1
n+1
3
4
n1+1
1
3
=
n=1
1
3
=
3
16
n=1
=
n=1
n
1
3
(a) Determine the sum of the series s =
2
3
4
1
4
MA104 Lectures 32, 33 and 34
November 28 December 2, 2016
Partial derivatives [14.3]
Given a function of two variables f (x, y), we define the partial derivative with respect to x at a, b,
denoted by
f
(a, b) = x f (a, b) = D x f (a, b) = fx (a, b)
x
as t
MA104 Lab Report 10 Power Series; Taylor and MacLaurin Series
1. Power Series Representations of Functions
1
can be represented as a power series:
1x
The function f (x) =
1
= 1 + x + x2 + x3 + =
xn , |x| < 1.
1x
n=0
1
as a polynomial [ of infinite degree
MA104 Lab Report 7 Sequences and Series
1. Sequences (11.1)
A sequence can be defined by a function whose domain is the set of positive integers [ or the set of natural
numbers, N = cfw_1, 2, 3, . . . ] and where f (n) = an gives the nth term of the seque
MA104 Lab Report 8 Series
1. Integral Test
The Integral Test states: If a function f is continuous, positive and decreasing on [1, ) and an = f (n),
then the series
an is convergent if and only if the improper integral
n=1
1
if
1
f (x) dx is convergent; t
-’ 1 P 5_of6
A104 Midterm ‘ age
6. Determine whether each of the following sequences {an} is convergent or divergent. t_F1nd
the limit if the sequence converges. Fully justify your answers and/or computa ions.
[3 marks] (b) a,” —
G6 7, ‘
at 1 ‘9‘ _
MA104 Lecture 26
November 14, 2016
Power series [11.8]
We call power series a series of the form1
c n x n = c 0 + c 1x + c 2x 2 +
n=0
where x R is a variable and c n are some coefficients only depending on n. In general the series may
converge for some v
MA104 Lecture 24
November 9, 2016
Ratio and root test [11.6]
P
Almost all the tests we considered require the series n=N
an to have non-negative terms an 0 (at
least in the long run). For a generic series with both positive and negative terms, we can cons
MA104 Lecture 27
November 16, 2016
Representation of functions as power series [11.9]
When studying power series, our goal is to use them to represent functions. For example, we know
n
1
1
x when |x | < 1, that is we can represent the function 1x
that 1x