MTH405 Wk 3 Tutorial
Basic Operations
1. Find the value of
2. Evaluate
3pq 3 r3
3. Find the sum of
2xy + 3yz xyz ,
when
2
3,
p=
q = 2
3a, 2a, 6a, 5a
x = 2, y = 2
when
and
and
and
r = 1.
4a.
4. Add together
2a + 3b + 4c, 5a 2b + c
5. Add together
3d + 4e,

MTH405 Wk 8 Lect 2
Gradient Function f 0 (x) and Tangent Lines
Denition. Given a function f (x) on real numbers. Then the derivative, f 0 (x), is the gradient function of f (x), which takes input x values and gives as output the gradient (or slope) of the

MTH405 Wk 8 Lect 1
Product Rule
The Product Rule deals with functions multiplied together, i.e.,
y = f (x) g(x).
Theorem. For multiplication of two functions
have
y = f (x) g(x),
we
d
[f (x) g(x)] = [f 0 (x) g(x)] + [g 0 (x) f (x)]
dx
or in a more usable

MTH405 Wk 7 Lect 1
Calculus
Background: Limits of Functions
Denition. (Limit of a function) Let f be a function on the real numbers and
c be an element in R. Then limit of f as x approaches c is dened to be the
value f (x) as x comes very close to c. We d

MTH405 Wk 9 Lect 1
Intervals on Real Numbers R
Given real numbers a and b, we can have the following intervals on
R:
[a, b] = cfw_x R | a x b
This is called a closed interval and includes all points from a
to b on the real line (points a and b are in the

MTH405 Wk 11 Lect 1
Midpoint Approximation
Exercise. Evaluate
3
1
(x2 + 1)dx
where
n=4
using Midpoint Ap-
proximation.
Trapezoidal Approximation
Also known as Trapeziodal Rule. The formula to use are:
width of the subintervals,
b
f (x)dx Tn =
Area functio

MTH405 Wk 10 Tutorial
1.
Integrate the following.
2
a.)
x dx
c.)
1
dx
x2
x dx
f.)
2.
4 cos xdx
x + x2 dx
g.)
i.)
xdx
d.)
1
e.)
x3 dx
b.)
3x6 2x2 + 7x + 1 dx
h.)
t2 2t4
dt
t4
Solve the following initial value problems (IVP).
dy
dx
dy
(b)
dx
dy
(c)
dx
dy
(d

MTH405 Wk 10 Lect 2
Numerical Integration
Numerical Integration is a also another way of nding area under the
graph, but it is very grounded. Most of the time, Numerical Integration is used to nd area for functions which can not be integrated
(Yes, there

MTH405 Wk 11 Tutorial
Numerical Integration Notes
Midpoint Approximation:
b
f (x)dx =
Mn =
a
here
n
m1 , m2 , . . . , mn
ba
n
[a, b]
and
[a, b].
Trapezoidal Approximation:
b
f (x)dx =
Tn =
a
[f (m1 ) + f (m2 ) + . . . + f (mn )]
are midpoints of the subin

MTH405 Wk 10 Lect 1
Integration by Parts
Our primary goal here is to develop a general method of solving
integrals of the form
f (x)g(x)dx.
As a manifestation of the Product Rule in Dierentiation, we have
of the formula called Integration by Parts . This

2.0
BASIC ALGEBRA
2.1
Basic Algebraic Terminology, Notations & Laws
Addition
Like terms can be added.
E.g. a + a + a = 3 x a
= 3a
Unlike terms cannot be added.
E.g. a + b + c = a + b + c
Subtraction
Like terms can be subtracted.
E.g. 3a - a = 2a
Unlike te

Summary of Calculus
Review of Dierentiation
Denition of the Derivative
f 0 (x) = lim
h0
f (x + h) f (x)
h
Properties of Dierentiation
d
d
d
[f (x) g(x)] =
[f (x)]
[g(x)]
dx
dx
dx
d
d
[k f (x)] = k
[f (x)]
dx
dx
d
[k] = 0
dx
Power Rule
Product Rule
Quoti

MTH405 Wk 7 Lect 2
Techniques of Dierentiation
In the last lecture, we discussed the denition of the derivative given
by
f (x + h) f (x)
.
h0
h
f 0 (x) = lim
The formula provides the benchmark for dierentiation but ofcourse,
it is too tedious to use this

MTH405 Wk 7 Tutorial
1. Sketch the vectors with their initial points at the origin.
b.) 5i + 3j
d.) 3i 2j
a.) h2, 5i
c.) h5, 4i
2. Find the components of the vector P1 P2 and sketch them. (Note that
vectors denoted as P1 P2 have starting point as P1 and t

Factorization by Perfect Square
This is one of the six methods to factorize and is the most advanced. It is used to parThe procedure for factorization
by perfect square is given below.
tially or completely factorize quadratic functions.
Given a quadratic

MTH405 Wk 4 Lect 1b
Graphs of Functions
Some of the dierent types of functions include
Linear function
Rational function
Quadratic function
Cubic function
Trigonometric function
Exponential function
Logarithmic function
Absolute value function
Sq

MTH405 Wk 3 Lect 1
Algebra
Denitions
Variable - the letters x, y and z which has the freedom to represent any
number.
Coecient - A number multiplied to a variable.
Term - a group of coecients and variables multiplied or divided to each
other. A term is

MTH405 Wk 4 Lect 1
Change of Base Formula
Let a, b and x be positive real numbers such that a 6= 1 and b 6= 1. Then
loga x =
ln x
.
ln a
Exercise.
1. Use natural logarithms to evaluate log3 5.
2. Use natural logarithms to evaluate log6 2.
Solving Exponent

MTH405 Mid-Trimester Break Lect
2
Determinant of a Matrix
We begin by dening the determinant of a 2 2 matrix below.
Denition. Let A =
of A is given by
a
c
b
be a 2 2 square matrix. Then the
d
a b
= ad bc.
|A| =
c d
determinant
Note that only square mat

MTH405 Mid-Trimester Break Lect
1
Distances in 3-space
Denition. Let P1 = (x1 , y1 ) and P2 = (x2 , y2 ) be points in 2-space.
Then the distance d between P1 and P2 is given by
d=
q
(x2 x1 )2 + (y2 y1 )2 .
For the case of 3-space, given the P1 = (x1 , y1

MTH405 Wk 3 Lect 2
Mathematical Functions
Exponential Functions
Denition. The exponential function f
with base
a is denoted by
y = f (x) = ax
where a > 0, a 6= 1, and x is any real number. The numbers a and x are called
base and exponent, respectively.The

MTH405 Wk 4 Lect 2
Quadratic Function
Functions of the form
or
y = ax2 + bx + c
y = (x + a)2 + b.
The procedure for graphing a quadratic equation is to
nd y -intercept (x = 0)
nd x-intercept (y = 0)
if it is not a factored polynomial, then dierentiate

MTH405 Wk 5 Lect 1
Trigonometric Function
Functions of the form
y = A sin(Bx + C) + D
or
y = A cos(Bx + C) + D.
The importance of A, B, C, D are:
A is the amplitude of the graph
B is the number of cycles that the function has within 360 or 2 radians.
C

MTH405 Wk 6 Tutorial
Exponential Graphs
Sketch the following functions
1.
y = ex
2.
y = 2ex
3.
y = ex + 1
4.
y = e3x
5.
y = 2ex
Logarithmic Graphs
Sketch the following functions
1.
y = 2 ln |x|
2.
y = ln x
3.
y = ln |x|
4.
y = ln(x 3)
5.
y = ln |x 3|
1
Ab

MTH405 Wk 5 Tutorial
Linear Graphs
Sketch the following functions.
1. y = 3x + 5.
2. y = x.
3. y = x + 2.
4. 2y 4x 4 = 0.
5. y = 3, x R.
6. x = 2, y R.
Quadratic Graphs
Sketch the following functions.
1. y = x2 + 3
2. y = (x 4)2 + 2
3. y = x x2
4. y = x2