Tutorial 6 - Review of integration
MTHE 224, Fall term, 2014
Integration by part and by substitutions
Integration by part
udv = uv
vdu
Problem 1.
1. Evaluate
12x
dx
1 + 4x2
2. Evaluate
12x
dx
1 + 4x
Integration by partial fractions
A rational function f

Tutorial 7
MTHE 224, Fall term, 2015
Problem 1. Solve the following Linear rst order dierential equations
1. (x + 1)
dy
+ (x + 2)y = 2xex
dx
2. y + (tan x)y = cos2 x
3. (x2 1)
dy
+ 2y = (x + 1)2
dx
Problem 2. Solve the following separable dierential equat

Week 3: Probability Distributions
Goals:
Introduce several probability distributions of discrete random variables: geometric, binomial, negative binomial.
Study basic denitions of continuous random variables
Suggested Textbook Readings: Chapter 3: 3.4-3

Week 2: Probability
Goals:
Review counting techniques
Study conditional probability
Study probability distributions for discrete random variables
Dene expected value and variance of discrete random variables
Suggested Textbook Readings: Chapter 2: 2.3

Week 1: Descriptive Statistics & Introduction of Probabilities
Goals:
Review numerical summaries of data sets
Review measures of variabilities of data
Study probability theories
Suggested Textbook Readings: Chapter 1: 1.3, 1.4, Chapter 2: 2.1, 2.2
Prac

Week 4: Normal Distributions and Condence Intervals
Goals:
Study normal and standard normal distributions
Introduce condence intervals
Study condence intervals based on normal distributions
Suggested Textbook Readings: Chapter 4: 4.3, 4.6; Chapter 7: 7

Week 7: Analytical Method for Solving First-order DE
Goals:
Review Eulers method, direction eld
Study Autonomous dierential equations.
Solve simple rst-order DE: separable and linear equations.
Suggested Textbook Readings: Chapter 2: 2.1 - 2.3, 2.6
Pra

Tutorial 4
MTHE 224, Fall term, 2015
Problem 1. The cdf for a continuous random variable X is
0 x < 2
3
3
F (x) = 1 + 32 4x x , 2 x < 2
3
2
1 2 x
1. Find P (X > 0.5)
2. Find P (1 X 1)
3. Find the median
4. Find pdf function f (x), sketch both pdf and cdf

Homework 5
MTHE 224, Fall term, 2015
Due Monday (Nov. 23th) 4:30pm
Homework Questions:
Problem 1. Classify the following DE as separable, exact (or non-exact but can be
turned into exact by a suitable integrating factor), linear, homogeneous, or Bernoulli

Homework 1 Solutions
MTHE 224, Fall term, 2015
Problem 1. There are many dierent ways to program these questions.
(a)
StrengthData=xlsread(strength.xls);
sample=randsample(StrengthData,60);
samplemean=mean(sample);
samplemedian=median(sample);
q1=quantile

Tutorial 8
MTHE 224, Fall term, 2015
Problem 1. For the following equations, nd M (x, y) or N (x, y) so that the equation is exact. In each
case, it is enough to nd one expression.
1. M (x, y) dx + (sin x cos y xy ey ) dy = 0
2. (yexy 4x3 y + 2) dx + N (x

Homework 2
MTHE 224, Fall term, 2015
Due Friday (Oct. 16th) 2:30pm
Homework Questions:
Problem 1. A consumer organization that evaluates new automobiles customarily
reports the number of major defects in each car examined. Let X denote the the number
of m

Homework 2 Solutions
MTHE 224, Fall term, 2015
Problem 1.
1. The probability mass function is given below
x
0
1
2
3
4
5
6
7
8
9
10
p(x)
p(0) = 0
p(1) = F (1) F (0) = 0.3
p(2) = F (2) F (1) = 0.4 0.3 = 0.1
p(3) = F (3) F (2) = 0.4 0.4 = 0
p(4) = F (4) F (3

Homework 4
MTHE 224, Fall term, 2015
Due Friday (Nov. 13th) 2:30pm
Homework Questions:
Problem 1. Dams are used for hydroelectric power across Canada and around the
world. Part of the design of the dams must take into account both the amount of water
in a

Tutorial 3
MTHE 224, Fall term, 2015
Keep at least 4 decimal places for your answers, and check your answers in Problem 1 and 2 by MATLAB.
Problem 1. Suppose that only 0.1% of all computers of a certain type experience CPU failure during the
warranty peri

Homework 3
MTHE 224, Fall term, 2015
Due Friday (Oct. 23rd) 2:30pm
Homework Questions:
Problem 1. The probability density function of X, the lifetime of a certain type of
vehicle (in years), is given by
f (x) =
kxex/5 x > 0
0
otherwise
(a) Determine k.
(b