( )
3.
(
)
( )
Integrate both sides with respect to x
4.
( )
( )
( )
Perform a substitution (basically cancel the dx on the left hand side)
5.
6.
7.
( )
( )
Evaluate both the integrals (if possible it WILL be on the exam)
Explicitly express as a functio
11
MATH2000 Calculus & Linear Algebra Ii
Second Semester Examination, November, 2008 (continued)
Q4 Use 'Stokes theorem to compute the integral ff curlF dS where F(;z:,y,z) =
xzi + yzj + myk and S is the part of the sphere x2 + y2 + z 5 With 2 > 0 that l
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 3
Semester 1, 2016
SOLUTIONS
(1) Evaluate the integral
1
Z
1
Z
x max(x, y) dy dx,
0
0
where max(x, y) is the maximum value of x and y.
Solution: We have
so
Z 1Z
(
x,
max(x, y) =
y,
1
1
Z
x max(x, y) dy dx =
0
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 5
Semester 1, 2016
Submit your answers - along with this sheet and your cover sheet - by 2:00pm on Tuesday, 31
May, to the designated assignment box in the Priestley Building (67).
You may find some of these p
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 3
Semester 1, 2016
Submit your answers - along with this sheet and your cover sheet - by 2:00pm on Tuesday, 3
May, to the designated assignment box in the Priestley Building (67).
You may find some of these pr
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 4
Semester 1, 2016
Submit your answers - along with this sheet and your cover sheet - by 2:00pm on Tuesday, 3
May, to the designated assignment box in the Priestley Building (67).
You may find some of these pr
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 1
Semester 1, 2016
Submit your answers - along with this sheet and your cover sheet - by 2:00pm on Tuesday, 22
March, to the designated assignment box in the Priestley Building (67).
You may find some of these
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 5 - Solutions
(1)
0
A. Choose r(t) = a cos ti + a sin tj +
ti + a cos tj + 0k and by
0k so r (t) = a sin
2
(0 a sin t)(a sin t) +
Stokes theorem the integral is
F(r(t)r0 (t)dt =
0
2a sin t0a cos t + (a cost
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 1
Semester 1, 2016
SOLUTIONS
(1) Consider the initial-value problem
sin(x y)
dy
sin(x y) = x2 ,
dx
y(1) = 1.
(a) Show that the differential equation is exact.
(b) Find a solution to the initial-value problem,
UQ MATHEMATICS
MATH2000/MATH2001
Revision questions
Z
1
Z
1
3
exp x 2 dxdy.
1. Evaluate the integral
0
y2
2. Find the mass of the portion of the sphere of radius a centred at the origin which lies in
the positive octant (x 0, y 0 and z 0). The density of
DEPARTMENT OF MATHEMATICS
MATH2000
Assignment 2
Semester 1, 2016
SOLUTIONS
(1) Show that
cosh(ln x) + sinh(ln x) = x,
x > 0.
Solution: We have
cosh y + sinh y = ey
so, for y = ln x (where x > 0 and for which eln x = x), we have
cosh(ln x) + sinh(ln x) = x
Summary of integrals in vector calculus
by Michelle Jones (a student in MATH2000) and Tony Roberts
Note: Many details - such as orientation, and outward normals - are not covered below. You
should consult the course notes for a precise statement of the re
Semester One Final Examination, 2016 MATH2000 Calculus and Linear Algebra II
Formula sheet
0 Variation of parameters
W = 31in yiyg
u= dx 1): da:
W W
o Gauss Divergence Theorem
5ij F - 11 d8 = I div(F) dV
3 V
o Stokes Theorem
f(curlF) - 11 d8 = for - T as
If it does, then youve got two values (outputs) for one value (input), which is forbidden by
the definition of a function. Well encounter another line test later, so to memorise this, its
worth looking at the vertical line and counting how many and values
Chapter 3 - Limits
3.1 Definition: Limit
Let
be some function and
. We say
approaches this limit at a value if, as we
approach from both directions (
),
agets arbitrarily close to . Thats a
fancy way of saying that the graph of the function looks as thoug
(
)
Therefore the maximum of (
) is:
(
)
Intro to Ordinary Differential Equations
Lets look at a couple of examples:
Unbounded population growth:
Bounded population growth:
Motion due to gravity:
( )
Spring system:
( )
( )
(
)
( )
Also, the exam probably
Limits
For a 2D function, we say that:
1.
2.
A function has a limit at if the limit where x approaches from the left exists, the limit where x
approaches from the right also exists, and they are the same.
We say the function is continuous if they are both
Chapter 8 Power Series and Taylor Series
8.1 Definition Power Series
A power series is a series of the form:
This is a power series about a (or centred about ). and
are fixed, while is variable. The
ratio test is generally useful for dealing with these. F
Evaluating a line integral of a field
A field is conservative if
Evaluating a line integral of a field around a
closed curve (2D only)
Path independence implies
Evaluating a line integral of a field around a
closed curve (3D)
How might you verify Stokes t
Sub this in:
with
Then you divide both sides by
to get the characteristic equation:
This is an easy quadratic equation. You dont have to derive this each time you CAN skip straight to it. Then
solve, using the quadratic formula:
From here there are three
2.
Plug them in
(
)
(
)
Quadratic Approximation
Weve seen earlier that the value of ( ) near
can we approximated using a straight line:
( )
Remember, this is simply
( )
( )(
)
:
( ) is the y-value of a nearby x-value (to set where the line is)
( ) is the
Where is the CARRYING capacity of the population. When P is at the carrying capacity, no growth occurs!
When P is nowhere near the carrying capacity, our model gets extremely close to the one we had before.
If P is smaller than the carrying capacity, itll
Semester Two Final Examinations, 2014
MATH2000 Calculus and Linear Algebra II
This exam paper must not be removed from the venue
Venue
_
Seat Number
_
Student Number
|_|_|_|_|_|_|_|_|
Family Name
_
First Name
_
School of Mathematics & Physics
EXAMINATION
Semester One Final Examinations, 2014
MATH2000 Calculus and Linear Algebra II
This exam paper must not be removed from the venue
Venue
_
Seat Number
_
Student Number
|_|_|_|_|_|_|_|_|
Family Name
_
First Name
_
School of Mathematics & Physics
EXAMINATION