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PRESENTER: GLENROY PINNOCK MSc. Applied Mathematics; Marine
Engineering Diploma; Mathemusician; PhD. (EDUC)pending
JANUARY 14TH, 2011
1

What is a radical quantity in mathematics? E.g.,√a
Factor for algebraic expression:x 2 - y 2
LAWS OF INDICES
(RECALL SIX (6) OF THEM)
Rationalize 1
2 + √a
Factors for algebraic expressions:x 2 + y 2 , x 3 - y 3 , x 3 + y 3 , x 4 - y 4 , x 4 + y 4
2

(a + b) 0
(a + b) 1
(a + b) 2
(a + b) 3
(a + b) 4
(a + b) 5
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
3

y = ax 2 + bx + c
What is the discriminant to the formula on the left? Clearly, it’s b 2 - 4ac.
The solution x = -b+√ (b 2 - 4ac)
2a
 What happens if this term is negative?
Recall that if the roots of the above quadratic equation is and β.
α
What is the sum of the roots and the product of the roots?
Write down the equation, whose roots are α 3 and β 3 ?
So, we can now discuss the significance of the discriminant.
a. The roots of a quadratic equation are imaginary/complex, if the discriminant is negative.
b. If the discriminant is greater than 0, then the roots are real.
c.
If b 2 = 4ac, then the roots are real and equal.
d. By the way, notice that these roots are basically the solutions of a quadratic equation.
PROVE THIS FORMULA!
4

sine θ = 0.5
Clearly sine θ = ½
Do you remember trig-ratios?
Do you recall the quadrant system?
What does that equation mean for the positive 0.5?
What happens if 0.5 is negative?
Solve:sine 2θ = 0.5, sine 3θ = 0.5
WOW!
What is the difference for the number of solutions, for the equations above?
Solve: sine 2 2θ = 0.25
We are ultimately breaking down
2 nd degree trig equations into a 1 st degree equation or factor.
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2
4 9
- 8
1
This can be written as
9 = 2 (4) + 1.
Conceptually, we can say 9 represents a polynomial, 2 is the quotient, 4 is the divisor, and 1 is the remainder.
Let us now divide x 2 – 1 by x + 1.
Clearly, x 2 – 1≡ Q (x + 1) + 0.
How did I get a zero?
So, generally speaking
P (x) ≡ Q (x + a) + R.
NB: It is advisable to represent a remainder theorem/factor theorem problem in this format.
Remember to use the synthetic rule to reduce tedious long division. Also, you need to know how to solve a pair of simultaneous equations.
6

Always recall the laws of indices and indicial equations whenever you are doing log problems.
What is the logarithm of a number?
Now consider log
10
10 = ?
10 1 = 10
Similarly, log e
100 = y.
Clearly, e y = 100.
Now we can say it’s the definition of a logarithmic quantity.
7

What is the difference between a horizontal line & a slanted line?
Let’s consider the slanted line:
B (x
1
, y
1
)
A (x
2
, y
2
)
What is the gradient of AB?
Clearly, grad AB = y
1 x
1
- y
- x
2
2
 Two parallel lines will have equalgradients.
 When one line is perpendicular to another, the product of their gradients is -1.
Consider the points A(2, 3) and
B(-1, -2). What is the equation of the line?
Recall y = mx + c
*c is the y intercept and m is the gradient.
Also, the gradient of a straight line is also equal to the tangent of inclination (tanθ) of the slanted line.
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h
area of a right-angled triangle
Construct a parallelogram from the points given above, and also find the area of the figure. Prove that the parallel sides of the parallelogram have equal gradients.
A = ½ bh
Construct a rhombus with points of your choice, and find the area of the rhombus.
Prove that the diagonal of the rhombus intersects at 90 0 .
b
Area of a non-right angled triangle
A= ½ ab sin C
This equation also can be written in two other forms. What are these forms?
NB: The coordinates for the point of intersection of two lines is basically the solution of two simultaneous equations.
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 the multiplication law and the addition law
 cumulativefrequency curve (OGIVE)
 upper quartile, lower quartile, median, inter-quartile range, semiinter quartile range
 percentiles
 frequency polygon
 permutation & combination
 probability distribution
 Poisson distribution
 binomial distribution
 orientation of the sample space diagram
 normal distribution
10

Recall gradient (grad) of a straight line:
AB = y x
1
1
-y
-x
2
2
Remember that this represents a straight line.
Concept of a limiting value.
In the case of a curve, the gradient is found by considering the differential expression, namely, dy , f
׳
, f x dx
0
So, what is your interpretation of the limiting value?
δ y
δ y
δ x
δ x
As a novice, we can say dy ≡ δ y dx δ x
As δ x 0 that is the time dy = δ y dx δ x
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DISCUSSION
 differentiation by first principles
 differentiation of a product
 differentiation of a quotient
 differentiation of trigonometrical expressions
 implicit differentiation
 differentiation by approximation
 differentiation express by rate of change
Clearly dy represents the gradient dx which is the tangent of the angle of inclination.
Differentiation by formula y = x n
∴
= nx n-1
Differentiate this equation: y = x 3 – x -2 + x 4
12

B
A
The area under the line AB is ?
B
A
However, the area under the curved AB is found by integration.
Let us consider this curve to be y = x 2 for x ≥ 0.
The area under the curve is ydx.
On the other hand, the volume under the curve is y 2 dx, if the revolution is done about the x axis through 360 0
What would be the formula for the volume of revolution about the y axis?
13

Integration is considered to be anti-differentiation
(anti-derivative)
Given that y = x 2 dy= 2x dx
So, the integral of 2x can be written as 2x
By considering the formula dy= x n dx
∴ y = x n+1 n + 1
Clearly, 2x .dx = x 2
Finally, we can now say integration is to find y whenever dy dx
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DISCUSSION
 Integration of a binomial expression to the n power
 Integration of a trigonometric function
 Integration by parts
IN A NUTSHELL THE INTEGRAL SYMBOL REPRESENTS SIGMA
NOTATION IN INTEGRATION THEORY.
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