Office Use Only
Monash University
Semester One Examination Period 2012
Faculty Of Science
EXAM CODES:
MAT1830
TITLE OF PAPER:
Discrete Mathematics for Computer Science
EXAM DURATION:
THREE hours writi
Office Use Only
Monash University
Semester One Examination Period 2013
Faculty Of Science
EXAM CODES:
MAT1830
TITLE OF PAPER:
Discrete Mathematics for Computer Science
EXAM DURATION:
3 hours writing t
1.5
Planes in 3-dimensional space
A plane in 3-dimensional space is a flat 2-dimensional surface. The standard
equation for a plane in 3-d is
ax + by + cz = d
where a, b, c and d are some bunch of num
3.5
3.5.1
Parametric curves and dierentiation
Parametric curves
The equation that describes a curve C in the Cartesian xy-plane can sometimes be very complicated. In that case it can be easier to intr
3.6.5
Cubic splines interpolation
Power series (and Taylor series) provide us with a method of approximating
values of a function at some particular point. When we construct Taylor
polynomials we get
3.6.4
Derivation of Taylor polynomials from rst principles
Suppose we do not know the innite power series for a given function f (x)
but wish to derive the rst few Taylor polynomial approximations to
Example 1.3. Given v = (1, 2, 7) draw v , 2v and v .
e
e e
e
Example 1.4. Given v = (1, 2, 7) and w = (3, 4, 5) draw and compute v w.
e
e
e e
1.2
Vector Dot Product
How do we multiply vectors? We have
2.1.2
Some special matrices
The Identity matrix :
I=
1
0
0
0
.
.
0
1
0
0
.
.
0
0
1
0
.
.
0
0
0
1
.
.
.
.
0
0
0
0
.
.
0 0 0 0 1
For any square matrix A we have IA = AI = A.
The Zero matrix : A matrix
3.6.2
Power series
We have seen that a nite geometric series of n terms has the sum
2
3
Sn = a + ar + ar + ar + . . . + ar
n1
=
n1
ark =
k=0
a(1 rn )
1r
Suppose we allow n to become very large.Then
pr
3.3
3.3.1
Differentiating inverse, circular and exponential functions
Inverse functions and their derivatives
The inverse of a function f is the function that reverses the operation done
by f . The in
4.2
Area under the curve
When f (x) is a positive function and a < b then the denite integral
b
f (x)dx
a
gives the area between the graph of the function f (x) and the x - axis. In
other words
b
f (x
Chapter 4
Integration
4.1
4.1.1
Fundamental theorem of calculus
Revision
Computing I = f (x)dx is no dierent from nding a function F (x) such
dF
dF
that
= f (x). Thus
dx = F (x). The function F (x) i
A typical saddle point
A typical case might consist of any number of points like the above. It is
for this reason that each point is referred to as a local maxima or a local
minima.
5.7.2
Notation
Rat
5.6.1
Taylor polynomials of higher degree
In earlier lectures we discovered that the linear approximation function
T1 (x, y) to a function of two variables f (x, y) near the point (a, b) is the
same a
Chapter 1
Vectors, Lines and Planes
1.1
1.1.1
Introduction to Vectors
Notation and definition
Common forms of vector notation are bold symbols (v), arrow notation (~v )
and tilde notation (v ). Throug
3.3
3.3.1
Differentiating inverse, circular and exponential functions
Inverse functions and their derivatives
The inverse of a function f is the function that reverses the operation done
by f . The in
Vector Dot Product - Summary
Let v = (vx , vy , vz ) and w = (wx , wy , wz ). Then the Dot Product
of v eand w is the scalar edefined by
e
e
v w = v x wx + v y wy + v z v z
e e
Consider the angle betw
5.5
Gradient and Directional Derivative
Given any dierentiable function of several variables we can compute each
of its rst partial derivatives. Lets do something out of the square. We
will assemble t
5.4
Chain rule
In a previous lecture we saw how we could compute (partial) derivatives
of functions of several variables. The trick we employed was to reduce the
number of independent variables to jus
4.3
Trapezoidal rule
Sometimes it may not be all that simple to integrate a function. (As an
2
example, try nding the anti-derivative of ex .) When we encounter situations such as this we can again tu
5.3.1
Geometric interpretation
f
of the function of two varix
ables z = f (x, y) is the rate of change of f in the x-direction, keeping y
f
xed. To visualise
as the slope (or gradient) of a straight l
5.2
5.2.1
Partial derivatives
First partial derivatives
We are all familiar with the denition of the derivative of a function of one
variable
df
f (x + x) f (x)
= lim
dx x0
x
The natural question to a
We are going to use equation 5 to give three cubics, y1 (x), y2 (x) and y3 (x).
Recall
yi (x) = yi + ai (x xi ) + bi (x xi )2 + ci (x xi )3
(5)
From the data points we have x1 = 2, x2 = 1, x3 = 1, x4
Chapter 5
Multivariable Calculus
5.1
Functions of several variables
We are all familiar with simple functions such as y = x3 . And we all know
the answers to questions such as
What is the domain and
3.6
Function approximations
We are now going to take a step in an interesting direction, and look at how
to approximate a function by a number of dierent methods.
3.6.1
Introduction to power series
A
3.4
Higher order derivatives
df
If f (x) is a differentiable function, then its derivative f 0 (x) =
is also a
dx
function and so may have a derivative itself. The derivative of a derivative
is called
1.4
Lines in 3-dimensional space
Through any pair of distinct points we can always construct a straight line.
These lines are normally drawn to be infinitely long in both directions.
Example 1.14. Fin