Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
Math 105  Section 201  Homework Problems
Set 2: due March 2, 8 am, inclass
R5
8. (a) Evaluate 1 (t2 t 2)dt by taking the limit of a Riemann sum.
R1
(b) Evaluate 0 udu by taking the limit of a Riemann sum with subinterval endpoints xj = (j/n)2 .
9. Eval
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
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Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
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Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
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Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
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Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 26: REVIEW FOR MIDTERM 2
MINGFENG ZHAO
March 13, 2015
Example 1. Evaluate lim
n
X
6(k 1)2
r
1+2
(k 1)3
.
n3
n3
1
First, you should split off a x = in the sum, then
n
r
r
n
n
X
1 X 6(k 1)2
6(k 1)2
(k 1)3
(k 1)3
=
.
1+2
1+2
3
3
2
n
n
n
n
n3
n
k=1
k=
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 31: THE COMPARISON TESTS AND POWER SERIES
MINGFENG ZHAO
March 25, 2015
Theorem 1 (Divergence test). If a series
X
series
ak diverges.
X
ak converges, then lim ak = 0. That means, if lim ak 6= 0, then the
k
k
Theorem 2 (Integral test). If f (x) is
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 29: THE DIVERGENCE AND INTEGRAL TESTS
MINGFENG ZHAO
March 20, 2015
For a sequence cfw_ak , the associated series is
X
ak , and the sequence of the partial sums of this series cfw_Sn is given
k=1
by:
n
X
Sn =
ak = a1 + a2 + + an .
k=1
Example 1. L
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 34: WORKING WITH TAYLOR SERIES
MINGFENG ZHAO
April 01, 2015
Definition 1. Suppose a function f (x) has derivatives of all orders on an interval centered at point a, then the Taylor
series for f (x) centered at a is:
X
f (k) (a)
k=0
k!
(x a)k := f
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 30: THE RATIO, AND COMPARISON TESTS
MINGFENG ZHAO
March 23, 2015
Theorem 1 (Divergence test). If a series
X
series
ak diverges.
X
ak converges, then lim ak = 0. That means, if lim ak 6= 0, then the
k
k
Theorem 2 (Integral test). If f (x) is a cont
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 27: SEQUENCES
MINGFENG ZHAO
March 16, 2015
Sequence
Definition 1. A sequence cfw_an is an ordered list of numbers o the form
cfw_a1 , a2 , a3 , , .
A sequence maybe be defined with an explicit formula of the form an = f (n) for n = 1, 2, 3, . A s
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 24: CONTINUOUS RANDOM VARIABLES
MINGFENG ZHAO
March 09, 2015
The cumulative distribution function
Definition 1. A random variables is a function from the set of some outcomes to (, ).
Definition 2. The cumulative distribution function, or simply C
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 25: EXPECTED VALUE, VARIANCE, AND STANDARD DEVIATION
MINGFENG ZHAO
March 11, 2015
Cumulative distribution function and probability density function
Theorem 1. Let X be a continuous random variable, then
I. For the cumulative distribution function
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 32: POWER SERIES
MINGFENG ZHAO
March 27, 2015
X
Theorem 1 (Divergence test). If a series
series
X
ak converges, then lim ak = 0. That means, if lim ak 6= 0, then the
k
k=1
k
ak diverges.
k=1
Theorem 2 (Integral test). If f (x) is a continuous, pos
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
Math 105  Section 201  Homework Problems
Set 1: due January 26, 8 am, inclass
1. Let r = ha, b, ci be a vector and denote x , y , z to be the smallest angle
between r and the positive x, y, and z axes respectively. Note that
this means all three angles
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2013
LECTURE 33: POWER SERIES AND TAYLOR SERIES
MINGFENG ZHAO
March 30, 2015
X
ck+1 
, then the radius of convergence of the power
ck 
k=0
X
X
1
series
ck (x a)k is R = . In particular, the power series
ck (x a)k absolutely converges for all x a < R,
r
k
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
1. Find the maximum of
2
(b) f (x, y) = ex
Math 105: Problems 2
(a) f (x, y) = x2 6x cos y + 9
+y 2 4
with
xy
g(x, y) = x2 + y 2 4 = 0
For part (b), rst establish the result using a Lagrange multiplier, then redo the calculation
by avoiding such a multip
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
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Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
Math 105, Assignment 4
Due 20150316, 4:00 pm
In the following problems you are expected to justify your answers unless stated otherwise.
Answers without any explanation will be given a mark of zero. The assignment needs to be
in my hand before I leave t
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
MATH 105 101
Assignment 1 Solutions
Due date: September 18, 2014
MATH 105 101 Assignment 1 Solutions
1. Find all vectors in R3 of length 3 that are normal to the plane 5x 12z = 5. (3 marks)
Solution: Let v be a vector in R3 of length 3 that is normal to t
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
MATH 105 101
Assignment 4 Solutions
Due date: October 30, 2014
MATH 105 101 Assignment 4 Solutions
All work must be shown for full marks.
1. (14 marks) Evaluate the following indenite integrals:
a) (7 marks)
et 9 e2t dt
b) (7 marks)
2
dx
x 3 x + 10
Soluti
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
MATH 105 101
Assignment 2 Solutions
Due date: October 2, 2014
MATH 105 101 Assignment 2 Solutions
All work must be shown for full marks.
1. Find both rstorder partial derivatives of the function g(x, y) = y sin (x
point ( 2 , 49). Simplify your answers.
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
Math 105, Assignment 2
Due 20150128, 4:00 pm
In the following problems you are expected to justify your answers unless stated otherwise.
Answers without any explanation will be given a mark of zero. The assignment needs to be
in my hand before I leave t
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
STAT 200, Lang Wu
1
Chapter 6. Introduction to Inference
6.1. Estimating with Confidence
Suppose that we wish to estimate the average age (denoted by ) of the population in Vancouver. We can randomly choose a sample of (say) n = 1000 people
in Vancouver,
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
1. Harris recently installed a spam filter software, but he still saw spam emails in his
inbox. He made a daily record of the number of spam emails that were delivered to
his inbox over the past 20 days. The following is a frequency histogram for his data
Integral Calculus with Applications to Commerce and Social Sciences
MATH 105

Winter 2015
1. Circle the most appropriate answer: [2 marks each]
a) Form a data set that consists of four integer numbers from 1 to 10 (inclusive,
without repeats).
Among all the possible data sets that can be formed (as described in the above),
which of the followi