Matrices

Rows R       Columns C

Multiplying a

 (3R    2C) by (2R     1C) makes a....

(3R 1C)

Matrices

Rows R       Columns C

Multiplying a

(1R    3C) by (3R    1C) makes a....

(1R 1C) i.e a single number

[image]

Let f(x) = x² + (k+1)x - k - 2, where k is a constant.

Find the value of k for which f(x) = 0 has equal roots

Simplify (a - b)² - (a + b)²

"Difference of Two Squares"

[F-S][F+S]

[(a - b) - (a +b)][(a - b) + (a + b)]

[ -2b ][ 2a ]

-4ab

Alternatively, expand the brackets and gather terms:

[a² - 2ab + b²] - [a² + 2ab + b²] = -4ab

Expand (a + b)³

1a³ + 3a²b + 3ab² + 1b³

^{a goes down in Power, while b goes up,}

^{the coefficients come from Pascal's Triangle,}

^{with all the signs Plus}

1

1           1

1           2            1

1         3         3          1

1        4         6        4          1

1       5       10         10       5       1

...is called ...

...is useful for ...

...and the next row reads ...

Pascal' Triangle

Coeffiecients of Expansions

1      6      15      20       15      6      1     

Without using a calculator, solve the simultaneous equations

  x +  y + z  =  2

2x +  y + z  =  3

x - 2y + 2z = 15

Numerator can be factored,

but it does not have any factors common to Denominator

…so use the Quotient Rule (U over V)

Second by Deriv of First …minus …. First by Deriv of Second

…All Over ….Second Squared

“from First Principles” means use the f(x+h) method 

Steps

Write out f(x)

Write out f(x+h)

Simplify f(x+h) – f(x)

Divide all by h

Get the Limit as ‘h goes to 0’

Solve the Simultaneous Equation

2x - 3y = 1

x² + xy  -4y² = 2

One is Linear, One is Quadratic

Re-arrange the Linear (usually to get x=)

Substitute into Quadratic

Solve Quadratic (to get y=number)

Sub back into Linear (to get x values)

Write the recuring decimal 0.979797... as an infinite geometric series and hence as a fraction

Answer is obviously 97/99 (using 'non-Geometric' method!)

Geometric: .63 + .0063 + .000063 ...

                63/100 + 63/10000 + 63/1000000 ...

 a = 63/100, r = 1/100 (-1<r<1)

Then S(infinity) formula

Formula?

Arithmetic Series

Nth Term

[image]

Formula?

Nth Term

Geometric Series

[image]

Formula?

S_n

of an Arithmetic Series

[image]

Forumula?

S_n

of a Geometric Series

[image]

Difference between Arithmetic and Geometric Series?

Arithmetic: b - a = c - b

Geometric: b/a = c/b

Formula?

Sum to Infinity of a Geometric Series?

[image]

Evaluate

 [image]

[image]

Solve x³ + 3x² – 4x – 12 = 0

This is a Cubic Equation, so use the

Factor Theorem!

Solutions are

x = ±2, -3

Use matrix methods to solve the

simultaneous equations

2x + y = 8

x - 3y = -3

[image]

Prove by induction that...

Prove that is it true for the First Term

(typically n=1 but sometimes n=0)

Assume that the statement is true for P(k)

Use P(k) to prove that P(k+1) is also true

...and make the concluding statements

Prove by Induction ...Types and Tricks

Sum of Series

Sum of Series: e.g.

1² + 2² + 3² ...+ k²                 = [k(k+1)(k+2)]/6

1² + 2² + 3² ...+ k²+(k+1)²= [(k+1)(k+2)(k+3)]/6

Now prove that

[k(k+1)(k+2)]/6 +(k+1)² = [(k+1)(k+2)(k+3)]/6

The equation of a curve is y = 3x⁴ -2x³ -9x² +8

(i) Show that the curve has a local maximum at the point (0, 8).

(ii) Find the coordinates of the two local minimum points on the curve.

(iii) Draw a sketch of the curve.

Get First Deriv,  and solve = 0

Establish that (0,8) is a Turning Point ...i.e. x=0 is a solution for dy/dx=0, and (0,8) is on the curve

Get Second Deriv

"Local Maximum" implies Second Deriv value <0

so sub given value of x into d²y/dx² to see if this is the case

(ii) Use the other two solutions from dy/dx=0;

Sub into y to get co-ords

Sub into d²y/dx² to establish the type of turning point

Solve | x- 3| < 10

 -7       <    x           < 13

Polar Form: z = r(cos θ + i sin θ)

DeMoivre: zⁿ = rⁿ(cos nθ + i sin nθ)

Three numbers are in arithmetic sequence. Their sum is 27 and their product is 704. Find the three numbers

You could use x, x+d, x+2d

…but much better to use x-d, x, x+d

Sum: (x-d)+(x)+(x+d) = 3x= 27 so x= 9

 9(81-d²)=704

   (81-d²)=704/9

        -d² =704/9 - 81

        -d² =704/9 - 9(81)/9

        -d² =  (704 - 729)/9

        -d² =  (704  -729)/9

        -d² =  (-25)/9

The parametric equations of a curve are:

x = cos t + t sin t, y = sin t – t sin t,

α and β are the roots of the equation  

x² – 4x + 6 = 0.

Factorise x³ - y³

"Difference of Two Cubes"

(x - y)(x² - xy - y²)

 Factorise x³ + y³

"Sum of Two Cubes"

(x + y)(x² + xy + y²)

Formula for

Roots of a Quadratic Equation?

-b ± √b² - 4 ac

2a

Simplify √-16

√-16 = 4i, where i² = -1

Use the Quadratic Root formula,

One Root of  4 x² - 2x - 1 = 0 is

1 + √5

4

What is the other root?

 The other root is

1 - √5

4

The other root is the Conjugate ...in the case where the coefficients of the quadratic are Real

k is a real number such that

-1 + i √3   = ki

-4√3 - 4i        

Multiple above and below by the Conjugate of the Denominator:

-1 + i √3   x   -4√3 + 4i  

-4√3 - 4i        -4√3 + 4i 

Factorise a² + b²

Sum of Two Squares

...can only be factored with Complex Numbers

(a + ib)(a - ib),

where i² = -1

Find the General Term in the expansion of

[image]

The General Term found by simplifying

[image]

with the [image]

part being set to some value k

Eq: (y - y₁) = m(x - x₁)

with m = dy/dx and (x₁, y₁) = (2, -4)

Note in this case that the d/dx of 2y² is

(2y)(dy/dx)

is a local maximum point of f(x).

First Derivative: f'(x) = (1/x) - 3

Second Derivative: f''(x) = (-1/x²)

Show that x=1/3 is a solution for f'(x) = 0 and that f'(x)<0

...and that f(1/3) = -1

Find the vlaues of n where

(2n −1)ln 3 < 12 loge 27

(2n −1)ln 3 < 12 loge 27

Newton Raphson method:

[image]

Prove by Induction ...Types and Tricks

DeMoivre's Theorem

Assume P(k): (cos θ + i sin θ)^k = cos kθ + i sin kθ

RTP P(k+1): (cos θ + i sin θ)^k+1 = cos (k+1)θ + i sin (k+1)θ

But (cos θ + i sin θ)^k+1 = (cos θ + i sin θ)^k(cos θ + i sin θ)¹

Use Assumption         = (cos kθ + i sin kθ)(cos θ + i sin θ)

Multiply, gather Real and Imaginary terms

Recognise Expansion Formula from Page 9 Table etc.

The equation of a curve is

y = 3x⁴ - 2x³ - 9x² + 8

Show that the curve has a local maximum at the point (0, 8).

Find the coordinates of the two local minimum points on the curve.

Draw a sketch of the curve.

f'(x) = 0 at x = 0 (where f''(x) <0 Local Max);

f'(x) = 0 at x = ³/₂ (where f''(x) >0 Local Min);

f'(x) = 0 at x = -1 (where f''(x) >0 Local Min).[image]

Find the equations of the Asymptotes

of the graph of f(x) =      1_ 

                                 x+1

(Denominator x+1 → 0)

Therefore Vertical Asymptote  is x = -1

Limit as x → ∞ of 1(x+1) is 0

Therefore Horizontal Asymptote is y = 0

Note this graph has neither Turning Points nor Points of Inflexion

[image]

This FlashCard Set was produced by

David Kearney, (c) 2008 (E&OE!)

It is released under Creative Commons, for free use in not-for-profit classrooms.

Some images were snapped from http://www.examinations.ie/

For this and more ICT advice for Teachers in Irish schools visit

http://homepage.eircom.net/~ictadvisor

Many thanks to the team at Flash Card Machine for making this resource available