MATH 2130 Tutorial 5
1.
2.
3.
4.
5.
6.
7.
8.
In questions 17, determine whether the limit exists. If it does not exist, give reasons for its
nonexistence.
x2 xy + y 2
lim
(x,y)(0,0)
x2 + 2y 2
x2 y
lim
(x,y)(0,0) x4 + y 2
x2 4x y 2 6y 5
lim
(x,y)(2,3) x2 4
Absolute Maxima and Minima
Section 12.11
Definition
(a) A set in the plane is called a closed set if it contains all of its boundary points.
(b) A set in the plane is called a bounded set if it is contained in a disk (no matter how large the disk
is).
1
E
Triple Integrals and Triple Iterated Integrals
section 13.8
The Fubinis Method to evaluate triple integrals is this: We first walk on a line parallel to one of the
x-axis, y-axis, or z-axis. This calculates the most inner integral. Then, parallel to the a
Equation of tangent plane:
for implicitly defined surfaces
section 12.9
Some surfaces are defined implicitly, such as the sphere x2 + y2 + z2 = 1. In general an implicitly
defined surface has the equation F(x, y, z) = 0. Consider a point P = (x0 , y0 , z0
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Directional Derivatives
section 12.8
Consider a function f(x, y) defined for the points close to a point P = (x0 , y0 ). Consider a unit vector
u = a , b that starts from the point P = (x0 , y0 ). The parametric representation of the line through P
with d
Relative Maxima and Minima
Section 12.10
Definition By a critical point for a function f(x, y) we mean a point at which either f does not exists
(i.e., at least one of fx or fy does not exist) or at that point we have f = 0
f(a, b) does not exist
critical
Volume of Solids of Revolution
from section 13.3
Consider a region R in the xy-plane. Take any point (x, y) of the region. If we rotate this point about
the x-axis , it generates a circle whose radius is |y| and therefore the perimeter of the circle is 2
MATH 2130 Summer Evening 2013 Problem Workshop 7 Solutions
1. Even though z = 9 x2 y 2 would work well in cylindrical coordinates, the parabolic
cylinder z = x2 would not. Hence well start in cartesian coordinates.
The paraboloid z = 9 x2 y 2 has a z-coor
1.
Consider the two surfaces z = 2 x2 + y 2 and 0 = x2 + y 2 + 4y + z.
(a) Identify the curves.
4 Marks
(b) Find the projection of the intersection of the two surfaces in the yz-plane.
4 Marks
For the rst curve, it can be re-written in the form
z 2 = 4x2
MATH 2130 Summer Evening 2013 Problem Workshop 6 Solutions
1. The region is question is given below
(a) The formula for mass is
dA.
R
Since density is constant, this is the same as
dA = (area of R).
R
To nd the area of R, we can note that it is just one-e
THE UNIVERSITY OF MANITOBA
DATE: December 14, 2012 FINAL EXAMINATION
DEPARTMENT 84 NO: MATH213O TIME: 3 hours
EXAMINATION: Engineering Mathematical Analysis 1 EXAMINER: M. Davidson, D. Trim
PAGE NO: 1 of 13
INSTRUCTIONS:
. No aids permitted.
.
MATH 2130 Summer Evening 2013 Problem Workshop 2 Solutions
1. Let P = (3, 1, 5). There are a few ways to do this question. All of which require the
vector on the line which is v = 3, 2, 1 and a point on the line. Using t = 0 we get
the point is Q = (2, 1,
MATH 2130 Problem Workshop 10
1. Evaluate the double iterated integral
0
6
2
ey dydx.
2
3x
2. Evaluate the double integral
R
1
dA
y1
where R is the region bounded by the curves y = 2x, y = x, x = 2, x = 3.
3. Find the volumes of the solids of revolution w
MATH 2130 Tutorial 4
1. Find all unit tangent vectors to the curve x2 + z 2 = 4, x + y = 1 at the point ( 2, 1 2, 2).
2. Find the unit tangent vector to the curve x = t2 , y = 3t3 , z = 3t2 at the origin.
3. Find the angle between the tangent vectors to
MATH 2130 Tutorial 3
1. The three lines below dene a triangle. Find its area.
x = 11 + 5s,
y = s,
z = 2 + 2s;
x = 1 + 2u,
y = 1 u,
z = 2 4u;
x = 2 + 3t,
y = 1 + 2t,
z = 8 + 6t.
2. The vertices of the triangle in question 1 are three vertices of a parallel
MATH 2130 Tutorial 2
1. Find the equation of the plane that contains
2x + 3y + 4z = 6,
x 2y + z = 3
and
2x 1
y+2
1z
=
=
.
22
2
7
2. Find equations for the line perpendicular to the plane x + 5y 2z = 6 and through the point of
intersection of the lines
x =
UNIVERSITY OF MANITOBA
DATE: November 5, 2014
EXAMINATION: Engineering Mathematical Analysis 1
COURSE: MATH 2130
TERM TEST 2
PAGE: 1 of 6
TIME: 70 minutes
EXAMINER: various
1. Evaluate the limit, if it exists. Justify your answer.
[3]
(a)
lim
(x,y)(0,0)
[
MATH 2130 Problem Workshop 3
1. The following three lines dene a triangle
x = 4 + s,
x = 3 u,
x = 1 + t,
y = 1 s,
y = 6 2u,
y = 2 + t,
z = s
z =1+u
z =5t
Find the area of the triangle.
2. The vertices of the triangle in question 1 are three vertices of a
MATH 2130 Problem Workshop 1 Solutions
In questions 1-12, draw the surface dened by the question. In questions 13-16, draw the
curve and nd the projections in the xy, yz and xz-coordinate planes.
1. x = 2y 2 + z 2
This is an elliptic paraboloid opening in
MATH 2130 Problem Workshop 7
1. The equations
x2 + y + 3s2 + s = 2t 1,
y 2 x4 + 2st + 7 = 6s2 t2 ,
s
when s = 0 and t = 1. Assume x > 0.
x
dene s and t as functions of x and y. Find
2. The equations
x3 y 2 + uv = x + y + 2,
xy y(u2 + v 2 ) = 3u + 3,
dene
MATH 2130 Problem Workshop 8
1. Find the rate of change of the function f (x, y, z) = sin(xy) z 3 at the point (2, 0, 3)
in the direction of the upward normal to the surface xz 2 x2 z = 6.
2. Find equations for the tangent line to the following curve at t
MATH 2130 Problem Workshop 2
1. Find the equation of the plane that contains
2x + 3y + 4z = 6,
x 2y + z = 3 and
2x 1
y+2
1z
=
=
.
22
2
7
2. Find equation for the line perpendicular to the plane x + 5y 2z = 6 and through the
point of intersection of the li