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 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 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 Week 2 Integrals Invloving Trig
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 functions. T
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 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 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
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 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 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 Final Exam Information
Date/Time/Location: Monday, April 20, 7:00pm9:30pm, Theatre Auditorium
ONLY the Casio FX-300MS Plus calculator will be allowed during the test.
(This stipulation is stated in the course outline.)
All answers require full jus
MA104 Midterm Information
Date and Time: Monday, Mar. 2, 7:008:20 pm
Location: BA201 (Bricker Academic)
Topics: Sections 7.2, 7.3, 7.5, 7.7, 7.8, 6.2, 6.3, 8.1, 8.2, 9.3, 10.110.4.
Refer to the course outline for corresponding assigned Practice Problem
MA104 Mock Final Exam
Name:
* Please remember that mock tests are meant as a means of providing an extra set of practice questions
and basis for a review class. Do not study for the exam based solely on the topics covered by the mock
test! Go back through
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 Midterm 0 from the origin
I r =
_ . olar equation
5. The oriented curve 01 in the ﬁgure is the Splral WIth p
(0, 0) to the point (7T, 7T) in polar coordinates (7‘,
P (7T , 7T )
l l - ' 1 e nation
1 [6 marks] (a) Find the area of the region that
-’ 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 Lab 8 Notes
1. Absolute Convergence
P
P
The series
an is called absolutely
convergent if the series
|an | is convergent and is called conditionally
P
convergent if it converges but
|an | does not.
P
P
We then have the following theorem: If
an is abs
MA104 Integration Review
1. Brief Table of Antiderivatives
(Text: 4.9)
n+1
R
xn dx =
x
+C
n+1
R
sec x tan x dx = sec x + C
R 1
dx = ln |x| + C
x
R
ex dx = ex + C
R
csc2 x dx = cot x + C
R
csc x cot xdx = csc x + C
R
ax
ax dx =
+C
ln a
R
1
dx = sin1 x + C
Strategies for testing series
X
Arn
n=0
Geometric series
(
A
converges to 1r
diverges
X
if |r| < 1
if |r| 1
1
p
n
n=1
p-series
(
converges if p > 1
diverges
if p 1
Divergence test
If it is obvious that limn an 6= 0, use this test. Even better, use this t
MA104 Week 5 Surface Area; Separable Dierential Equations
1. Surface Area (Text: 8.2)
The method used to find the volume of a solid of revolution had us integrate with respect to x if the solid was
generated by revolving a region about an axis parallel to
MA104 Lab 9 Notes
1. Power Series Representations of Functions
Note: Review last weeks Lab Notes for the definition of a power series.
1
The function f (x) =
can be represented as a power series:
1x
P
1
= 1 + x + x2 + x3 + =
xn , |x| < 1.
1x
n=0
1
as a po
MA104 Midterm Test
Page 1 of 7
Student Number:
The questions on this page are True/False or Multiple Choice.
Circle the correct answer. No justification is required.
Z
Z
f (x) dx is convergent, then
(1 mark) 1. If f (x) is continuous on [0, ) and
1
gent.
MA104 Midterm Test
Page 1 of 6
Student Number:
1. Evaluate the following integral.
Z
[6 marks]
tan3 x sec3 x dx
Z
[6 marks] 2. By using a suitable trigonometric substitution, evaluate the integral
x2
1
dx.
4 x2
Over
MA104 Midterm Test
Page 2 of 6
Z
[7 mar
Wilfrid Laurier University
LAU RI E R MA104 midterm test
Inspiring Lives: Fall term, 2016
M MC SOLUTION FACE
Name: "
Student number:
m-nnnnn-u-
m-nn-
_-II
_-
Instructor: Dr. Giuseppe Sellaroli
Date of exam: Tuesday, October 25, 2016
Start time: