MAT2122 Multivariable Calculus (Fall 2016)
Assignment 1 solutions (the total is 62 points)
1.1.18 (5 points). Find the equation of the line passing through the points P = (5, 0, 4) and Q = (6, 3, 2).
Solution: By the general formula this equation is
v(t)
MAT2122 Multivariable Calculus (Fall 2016)
Assignment 2 solutions (the total is 85 points)
2.5.8 (6 points).
Let f (u, v, w) = (euw , cos(u + v) + sin(u + v + w) and g(x, y) = (ex , cos(y x), ey ).
Calculate f g and D(f g)(0, 0).
Solution: The composition
MAT 2371
Fall 2016
Assignment 3
Due on Tuesday, November 1, 2016 in class.
[17]1. Let X be a random variable with the probability mass function
x
2
, x = 1, 2, 3, . . .
f (x) = c
3
and zero, otherwise. Find
(i) Calculate c.
(i) We know
X
f (x) = 1 = c
x
MAT 2371
Fall 2016
Assignment 5
Due on Tuesday, December 6, 2016 in class.
[10]1. 10 independent observations are chosen at random from an exponential
distribution with mean 1.
Calculate the probability that at least 5 of them are in the interval (1, 3).
MAT 2371
Fall 2016
Assignment 4
Due on Thursday, November 24, 2016 in class.
[21]1. Let X be a random variable with exponential distribution with mean 1.
(i) Write the p.d.f. for the random variable X.
exp(x), if 0 < x
f (x) =
0
if Otherwise
(ii)
R
exp(x)
Foundations of Statistical Machine Learning: MAT4996/CSI4103A/MAT5319
Assignment 0: A warm-up
Tanya Schmah
Here are some review questions on concepts that will be used during the course. By the start
of the Thursday lecture in the first week (January 14th
MAT 2371
Fall 2016
Assignment 3
Due on Tuesday, November 1, 2016 in class.
[17]1. Let X be a random variable with the probability mass function
x
2
, x = 1, 2, 3, . . .
f (x) = c
3
and zero, otherwise. Find
(i) Calculate c.
(i) We know
X
f (x) = 1 = c
x
MAT 2371 (Fall 2016)
Introduction to Probabaility
Professor :
M. Zarepour
Email : [email protected]
Bureau : 585 King Edward (KED), Room 207D
Telephone : 562-5800 ext. 3503
Web : http:/aix1.uottawa.ca/~zarepour/
Course Schedule:
Tuesday 10:00 - 11:30 TB
MAT 2371, Final Exam Formula Sheet
1
MAT 2377 (Fall 2016)
Final Exam Formula Sheet
Addition Rule: P (A B) = P (A) + P (B) P (A B)
Conditional probability of A given B:
P (A|B) =
P (A B)
P (B)
Total probability rule:
P (A) = P (A B) + P (A B 0 ) = P (A|
MAT 2371
Fall 2016
Assignment 1
Due on Tuesday, October 18, 2016 in class
20 points for each question.
1. Let A and B be two events such that P (A) = 2/5 and P (B) = 7/10.
(i) Calculate P (A B).
(ii) If A and B are mutually exclusive find P (B|A) and P (A
MAT 2371 Fall 2016
Solution to Assignment 2
Professor: M. Zarepour
Question 1.
(i) We have
P (A B) = P (A) + P (B) P (A B) = P (A) + P (B) P (A)P (B) = 0.4 + 0.7 0.4 0.7 = 0.82
(ii) From the definition we get
P (B|A) = P (B A)/P (A) = 0.
Similarly
P (A|B)
MAT 2371
Fall 2016
Assignment 3
Due on Tuesday, November 1, 2016 in class.
[17]1. Let X be a random variable with the probability mass function
x
2
, x = 1, 2, 3, . . .
f (x) = c
3
and zero, otherwise. Find
(i) Calculate c.
(ii) P (X > 4|X > 2).
(iii) P
MAT 2371
Fall 2016
Assignment 1
Due on Tuesday, September 27, 2016 in class
1. (21 Points) A 4-sided is rolled n times
(i) How many possible outcomes do we have?
(ii) For 0 i n, in how many possible outcomes the side 1 appear i times?
(iii) Prove
n
4 =
n
MAT 2371
Fall 2016
Solution to assignment 1
1.i: Using the multiplication principle the sample space
S = cfw_(x1 , . . . , xn ) : xi = 1, 2, 3, 4
has
4 4 . . . 4 = 4n
possible outcomes.
1.ii: To fill the n coordinate (x1 , . . . , xn ), we need to choose
MT1320D Calculus 1
Final Exam
Professor: Jose Malag
on-L
opez
16 April 2011
NAME:
STUDENT NUMBER:
No calculators or other electronic aids allowed.
No notes, books or other papers allowed.
Answer all questions in the space provided. You must justify you
2013 Spring/Summer: MAT1320 3x
Calculus
Instructor : Yuqing Zhang
07/11/2013
07/11/2013
Midterm2 Review
Midterm2
problems
Name:
Student ID:
Instructions.
1. You must show all your work. You get zero point if you just state the answer without showing
neces
2013 Spring/Summer: MAT1320 3x
Calculus
Instructor : Yuqing Zhang
07/11/2013
Midterm2 Review problems
Name:
Student ID:
Instructions.
1. You must show all your work. You get zero point if you just state the answer without showing
necessary intermediate st
Review Answer keys these are not complete solutions
1 Domain of ex is R, range of ex is y > 0, x intercept is none, y intercept is
(0, 1)
Domain of ln x is x > 0, range of ln x is R, x intercept is (1, 0), y intercept
is none.
Their graphs can be found on
2013 Spring/Summer: MAT1320 3x
Calculus
06/04/2013
Instructor : Yuqing Zhang
Midterm Exam One Review probles
Name:
Student ID:
Instructions.
1. Have your photo ID in front of you while you work. Communication with others is prohibited.
2. Calculator is no
2013 Spring/Summer: MAT1320 3x
Calculus
06/04/2013
Instructor : Yuqing Zhang
Midterm Exam One Review probles
Name:
Student ID:
Instructions.
1. Have your photo ID in front of you while you work. Communication with others is prohibited.
2. Calculator is no
Problem 6 Solutions ( my solutions are sketchy, you are supposed to provide
more details)
f (a + h) f (a)
h0
h
a)f (a) = lim
Geometrically, f (a) is the slope of the tangent line to the graph of y = f (x)
at the point (a, f (a).
Physically, f (a) is the i
Problem 1 correction
Here is what problem 1 will look like. You can ignore the origianl Problem
1 on the review sheet.
(1) Find the domain of the function and sketch the graph.
(a) f (x) = 4 3x
Domain is R and you nend to sketch its graph.
(b) f = sin x
d
Homework 3 Due June 20
1. Find the exact value of each expression.
a) eln 3
b) e2 ln 3
c) log10 25 + log10 4
f ) cos
d) ln e
e) sin
3
3
3
3
3
g ) arcsin
h) arcsin
i) arccos
j ) arctan 3
2
2
2
2. Solve the following equation for x.
1
3
2). ln(x + 1) + ln(x
HW1 Solutions ( my solutions are sketchy, you are supposed to provide more
details)
1 a) x2 1 > 0, x2 1 = 1
Therefore x > 1, x < 1, x = 2
b) |x 1| 1
Therefore square both sides we have (x 1)2 1
x2 2x 0
x(x 2) 0
Therefore 0 x 2
c) 1 x2 = 0
Therefore x = 1
HW2 Solutions
1 limx0 f (x) = limx0 (x + 2) = 2
limx0+ f (x) = limx0+ (2x2 ) = 0 The two one sided limits are dierent,
therefore the function is discontinuous at 0. (2 points)
limx1 f (x) = limx0 (2x2 ) = 2
limx1+ f (x) = limx0+ (2 x) = 1 The two one side
HOMEWORK 4 (DUE JULY 9)
x2
1 (1) Estimate the area under the graph of f (x) = e 2 from x = 0 to x = 2
using four approximating rectangles and rignt endpoints.
x2
(2) Estimate the area under the graph of f (x) = e 2 from x = 0 to x = 2
using four approxima
MAT1320 first midterm TOPICS
Real numbers and absolute values
Solving equations and inequalities
The exponential function, logarithmic function and reciprocal function
Trigonometric functions (Appendix C)
Limits (2.2, 2.3)
Continuity (2.4)
Infinite limits
MAT1320C
Solution to Midterm 1 (A)
Fall 2012
Solution to Test 1 (version A)
MAT1320C, Fall 2012
Total = 20 marks
1. (3 marks) The following is a table of some values of two functions y = f (x) and y = g(x).
x
f (x)
g(x)
1
4
2
2
3
4
3
2
3
4
1
1
Find (i) (