Solutions to Assignment 2 for Math 202 [A01]
Due Day: Tuesday October 27, 2015 at the class, late assignments will
not be accepted
There are five problems for you to solve and hand in. Illegible disorganized or
missing solutions will receive no credit. Pl

Math 202 A01
Tutorial 11
1. Solve
y 00 8y 0 + 20y = 100x2 26xex
by undetermined coefficients.
2. Solve
y 00 + y = sec2 x
by variations of parameters.
3. Solve the following initial value problem:
y 00 + 4y 0 + 4y = (3 + x)e2x ,
y(0) = 2, y 0 (0) = 5.
4. S

Math 202 A01
Tutorial 12
1. Use the Laplace transform to solve the initial value problem:
y 0 y = 2 cos 5t, ,
y(0) = 0.
2. Use the Laplace transform to solve the initial value problem:
y 00 4y 0 = 6e3t 3et ,
y(0) = 1, y 0 (0) = 1.
3. Use the Laplace trans

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial III
Thomas Section 12.5,14.1-14.2. Friday, September 30, 2016.
A1
Let L be the line through Q(0, 0, 0) with direction v = h1, 4, 2i, and let P be the plane with
equation 2x + z = 5. Does L intersect P ? If

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial II
Thomas Section 12.4-12.5. Friday, September 23, 2016.
A1
Let a = 3i 2j + k, b = i k, c = 3j and d = 6i + 4j 2k.
1. Compute a b and b a. How are they related? Does this make sense?
2. Compute a b and b a

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial I
Thomas Section 12.1-12.3. Friday, September 16, 2016.
A1:[4 points]
Describe the set of points in R3 (ie. three dimensions) which satisfy the equations x2 +z 2 = 4
and y = 0. What is the set of points wh

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial IV
Thomas Section 14.3-14.4. Friday, October 7, 2016.
A1
Compute the first order partial derivatives for f (x, y, z) = z 2 logx (y).
A2
Compute the second order partial derivatives for f (x, y) = ex ey
2
A

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial V
Thomas Section 14.5, 14.6. Friday, October 14, 2016.
A1
You are racing in a hang-glider, and your nemesis Madame Vole-Vite is way ahead of you. You
know theres a strong tailwind in the direction high abo

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial I
Solutions
Thomas Section 12.1-12.3. Friday, September 16, 2016.
A1
1. x2 + z 2 = 4 along with y = 0 represents a circle of radius 2 in the xzplane.
2. x2 + z 2 = 4 represents an infinite cylinder of radi

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial IV
Solutions
A1
f
x
f
y
1
= z ln(y)
ln(x)2
z2
1
=
,
ln(y)
y
2
1
,
x
f
z
=
2z ln(y)
.
ln(x)
f
x
= 2 ex ey ex = 2e2x 2exy ,
f
y
= 2 ex ey ey = 2exy + 2e2y ,
A2
2f
x2
= 4e2x 2exy ,
2f
y 2
= 2exy + 4e2y .

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial II
Solutions
A1
(1) Calculating
i
a b = 3
1
the cross-products:
j
k
2 1 = h2, 4, 2i
0 1
and
i j
k
b a = 1 0 1 = h2, 4, 2i = a b
3 2 1
This can be observed by either looking at the formula or by simply do

Math 202
Dr. Ibrahim
Fall, 2016
Math 202 Tutorial III
Solutions
A1
Lines intersect planes unless they are parallel, that means that the direction vector
of the line is perpendicular to the normal vector of the plane. Lets check!
~v ~n = h1, 4, 2i h2, 0, 1

'Name: g 0/ (_/L l (0V1 e StudentNumber:
Math 202 A01
Midterm 1
February 15th, 2016
Version 1
This exam contains 5 pages and 5 problems. Please check that you have a complete exam.
NO textbooks, notes, or cell phones permitted during the exam. The only ca

Math 202 A01
Tutorial 8
1. Determine whether the differential equation
(3x2 y + ey )dx + (x3 + xey 2y)dy = 0
is exact. If it is exact, solve it.
2. Recall the logistic equation dP
= kP (c P ), where c is the carrying capacity and k is a
dt
growth constant

Math 202 A01
Tutorial 10
1. Consider the differential equation
(6xy 3 + cos y)dx + (2kx2 y 2 x sin y)dy = 0.
(a) Find the value of k so that the differential equation is exact. (b) Solve it.
2. Consider the differential equation:
y 00 + 2y 0 + y = 0.
(a)

Math 202 A01
Tutorial 8
1. Determine whether the differential equation
(3x2 y + ey )dx + (x3 + xey 2y)dy = 0
is exact. If it is exact, solve it.
2. Recall the logistic equation dP
= kP (c P ), where c is the carrying capacity and k is a
dt
growth constant

Math 202 A01
Solutions to Assignment 1
To be handed in at the beginning of class on Sept. 25th. Late assignments will not be
accepted for marking. Each student must hand in their own original work. Part marks will be
deducted from solutions which are mess

Name :
Student Number:
Math 202 A01
Midterm 1
This exam contains 5 pages and 5 problems. Please check that you have a complete exam.
NO textbooks, notes, formula sheets or cell phones during the exam.
You may reach a total of 40 marks.
1. [4 marks] Consid

Solutions to Assignment 3 for Math 202 [A01]
Due Day: Friday November 6, 2015 at the class, late assignments will not
be accepted
There are four problems for you to solve and hand in. Illegible disorganized or
missing solutions will receive no credit. Ple

Math 202 A01
Tutorial 1
1. Let ~a = h0, 2, 0i and vector ~b = h0, 0, 3i.
(a) Draw ~a, ~b, ~a + ~b and ~a ~b.
(b) Is ~a perpendicular to ~b? Why?
(c) The points P (~a), Q(~b) and R(~a + ~b) form a triangle. What is its area?
(d) Show that |~a + ~b| |~a| +

Math 202 A01
Tutorial 2
1. Decompose the vector ~v = h5, 1, 3i into components v| and v , where
~v = v| + v and v| is parallel to w
~ = h 2, 2, 1i, and v is perpendicular to w.
~
2. Find an equation of the plane that passes through the point R(1, 0, 1) an

Name :
Student Number:
Solutions to Math 202 A01
Midterm 2
This exam contains 5 pages and 5 problems. Please check that you have a complete exam.
NO textbooks, notes, formula sheets or cell phones during the exam. The only calculator that
may be used for

Math 202 A01
Tutorial 4
1. Find all the local maxima, local minima, and saddle points of the function f (x, y) = x3 +
y 3 + 3x2 3y 2 8. Do the absolute extrema exist?
2. Find all the local maxima, local minima, and saddle points of the function f (x, y) =

Math 202 A01
Tutorial 5
1. Find the extreme values of f (x, y) = 3x y + 6 subject to the constraint x2 + y 2 = 4.
2. Your firm has been asked to design a storage tank for liquid petroleum gas. The customers
specifications call for a cylindrical tank with

Math 202 A01
Tutorial 3
1. Suppose elevation above sea level is described by
z = f (x, y) = 150 30x2 20y 2 .
(a) If you are standing at the point P (1, 2), which direction would you walk to experience
the greatest intial change in elevation? (b) Compute t

Extra review questions for Math 202
Please note that these question are extra. In your review, please go over all the homework
questions assigned and the extra questions for the tests. Also please check the solutions and
correct the mistakes that you made

Math 202 A01
Tutorial 6
1. Determine the critical points of the function
f (x, y) = 6xy 2 2x3 3y 4 .
For each critical point use the second derivative test to determine whether it corresponds to
a local maximum, to a local minimum, or to a saddle point. N

Math 202 A01
Tutorial 7
1. Consider the ordinary differential equation
tan(x)y 0 + sin(x)y = tan(x) cos(x).
(a) Determine if the ode is linear, if so, put it into standard form y 0 + P (x)y = f (x). Find
the interval of definition for the general solution

Math 202 A01
Tutorial 9
1. Consider the initial value problem
y0
2
+ tan(x)y = ex ln(x3 ),
y +
ln x
00
with y
4
= 2 and y
0
4
= 6.
Determine the largest interval x (a, b) where we can gaurentee existence and uniqueness
of a solution.
2. Show that the

Math 202 A01
Tutorial 9
1. Consider the initial value problem
y0
2
+ tan(x)y = ex ln(x3 ),
y +
ln x
00
with y
4
= 2 and y
0
4
= 6.
Determine the largest interval x (a, b) where we can gaurentee existence and uniqueness
of a solution.
2. Show that the