MA104 Week 6 Report Separable Differential Equations
Name:
Student Number:
Spring 2012
1. [5 marks] The Malthusian Law of Population Growth is given by P 0 (t) = kP (t), where k is constant and P (t)
is the size of the population (in millions) at time t.
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MA104 Lab Report 9 - Taylor and MacLaurin Series
Name:
Student Number:
Spring 2016
1. [8 marks] Consider the function f (x) = ln(1 + 2x).
(a) Manually determine T2 (x), the 2nd degree Taylor polynomial o
MA104 Lab 3 Notes - Volumes; Arc Length; Surface Area
1. Volumes of Revolution (Text: 6.2, 6.3)
In general, the volume of a solid is the sum of its cross-sectional areas so that:
V = lim
n
n
X
A(xi )x =
Rb
a
A (x) dx.
i=1
Then, for example, if a solid reg
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MA104 Lab Report 4 - Differential Equations, Parametric Equations
Name:
Student Number:
Spring 2016
1. [7 marks] When a skydiver jumps out of an airplane, gravity will accelerate the skydiver downwards.
MA104 Lab 9 Notes - Taylor and MacLaurin Series
1. Taylor and Maclaurin Series (Text: 11.10)
If f (x) [ an infinitely differentiable function on an interval about a ] can be represented by a power series,
P
f (n) (a)
n
f (x) =
cn (x a) , then it can be sh
MA104 Lab 8 Notes - Power Series
1. Power Series
A series of the form
P
cn xn = c0 + c1 x + c2 x2 + c3 x3 + is called a power series, where the cn s are
n=0
constants called the coeff icients of the series. Thus, a power series is similar to a polynomial
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MA104 Lab Report 1 - Additional Integration Techniques
Name:
Student Number:
Lab
Spring 2016
1. [6 marks] Evaluate the following integrals. Show all of your work.
/4
Z
sec4 tan4 d
(a)
0
Let u = tan . The
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MA104 Lab Report 5 - Polar Coordinates
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Student Number:
Spring 2016
1. [12 marks] Consider the curve (x2 + y 2 )3/2 = 18x2 + 6y 2 in Cartesian coordinates.
(a) Show that the curve can be expressed i
MA104 Lab 6 Notes - 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 sequ
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MA104 Lab Report 10 - Multivariable Calculus
Name:
Student Number:
Spring 2016
1. [6 marks] Consider the function f (x, y, z) = x3 ey/z .
(a) A function f (x, y, z) is said to be positively homogeneous o
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MA104 Lab Report 8 - Power Series
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Student Number:
Spring 2016
(1)n (2x + 3)n
P
.
n ln n
n=2
Remember to check convergence at endpoints and state intervals of convergence using interval notation.
1
MA104 Lab 4 Notes - Differential Equations; Parametric Equations
1. Proportionality
When we say that a given quantity, say y, is proportional to another quantity, say t, the relationship is
denoted by y t. This means that the value of y will be some const
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MA104 Lab Report 7 - Absolute Convergence; Strategies for Testing Series
Name:
Student Number:
Spring 2016
2
3 3 X (n!)
1. [7 marks] Consider the series s =
.
2 n=0 (2n + 1)!
(a) Use the Ratio Test to
MA104 Lab 5 Notes - Polar Coordinates
1. Polar Coordinates (Text: 10.3)
We have represented points in the plane by ordered pairs (x, y) called Cartesian coordinates which give directed
distances [ left/right, above/below ] from two perpendicular axes [the
MA104 Lab 2 Notes - Approximate Integration; Improper Integrals
1. Numerical Integration (Text: 7.7)
Numerical [approximate] integration is useful when there
hR is 2no way to find ani exact value of a definite integral
b x
because the integrand has no kno
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MA104 Lab Report 6 - Series
Name:
Student Number:
1. [5 marks] Consider the series
P
an =
n=1
(a) Express
P
Spring 2016
27 2n2
P
.
3n+1
n=1
an as a difference between two geometric series and explain wh
MA104 Week 2 Report Additional Integration Techniques
Name:
Student Number:
1. [8 marks] Consider the integral
Z
Spring 2012
f (x)dx where f (x) = x3 1 + x2
3=2
:
(a) Fill in the following with the trig. substitution that could be made to assist in evalua
MA 104 Week 4 Report Improper Integrals; Numerical Integration
Name:
Student Number:
Spring 2012
1. [11 marks] Evaluate each of the following integrals, or show that the integral diverges.
(a)
1
Z
xe
x
dx
= lim
N !1
0
N
Z
x
xe
x
; then du = dx; v =
0
0
xN
MA104 Week 5 Report Applications of Integration
Name:
Student Number:
Spring 2012
1. [12 marks] Suppose R is the region that's above the x-axis and within the circle (x
5)2 + (y + 3)2 = 25.
(a) Plot R using Maple (a sketch is not required).
with(plots):
e
MA 104 Week 7 Report Parametric Equations; Polar Coordinates
Name:
Student Number:
Spring 2012
1. [10 marks] In general, a cubic Bezier curve is determined by four control points P0 (x0 ; y0 ); P1 (x1 ; y1 );
P2 (x2 ; y2 ); and P3 (x3 ; y3 ) and is de ned
MA104 Week 10 Report Sequences and Series
Name:
Student Number:
30 ln n
;n
3 + 2 ln n
(a) Plot the rst 20 terms of the sequence fan g using Maple:
1. [9 marks] Consider the sequence de ned by an =
a:=(n)->?;
pts:=seq([n,a(n)],n=1.20);
From the plot, does
MA104 Week 11 Report Convergence Tests
Name:
Student Number:
1
P
1. [5 marks] Consider the series
1
P
3n 2 1
: Use the Comparison Test and a relevant p-series
4
n=1 15n + 14
1
P
an =
n=1
bn to determine whether the series
n=1
Spring 2012
1
P
an converges
MA104 Week 12 Report Ratio/Root Tests and Absolute/Conditional
Convergence; Power Series
Name:
Student Number:
Spring 2012
p1
2
3 3 X (n!)
1. [7 marks] Consider the series s =
.
2 n=0 (2n + 1)!
(a) Use the Ratio Test to show that the series
1
X
2
(n!)
con
MA104 Week 13 Report Power Series Continued; Taylor and MacLaurin
Series
Name:
Student Number:
1. [7 marks] Recall Question #4(a) from Week 12 where the series
for all x 2
57
;
22
:
1
P5
n=1 6
Spring 2012
1
n1
2x
was shown to converge
6
(a) Find the funct
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MA104 Lab Report 3 - Volumes, Arc Length, Surface Area
Name:
Student Number:
Spring 2016
1. [10 marks] Consider the curve y = 4 x2 for x [0, 2].
State the definite integral(s) required to determine each
MA104 Lab 1 Notes - 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 trigonomet
MA104 Lab 7 Notes - Absolute Convergence; Strategies for Testing
Series
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
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MA104 Lab Report 2 - Approximate Integration, Improper Integrals
Name:
Student Number:
Lab
Spring 2016
1. [7 marks] Evaluate each of the following integrals, or show that the integral diverges.
Z
1
dx
MA104 Lab 10 Notes - Multivariable Calculus
1. Partial Derivatives (Text: 14.3)
Given a differentiable function of 2 variables, say f (x, y), the partial derivatives of f (x, y) using limits are given
by:
fx (x, y) = lim
h0
f (x + h, y) f (x, y)
f (x, y +
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