Mathematics Methods Unit 1&2 Study Notes PDF

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Mathematical Methods Unit 1&2 Notes Daniel Sandvik

VCE Mathematical Methods Units 1&2

Contents Mathematica Commands..................................................................................................................................................................2 Probability & Counting Methods......................................................................................................................................................2 Matrices............................................................................................................................................................................................4 Introduction to Functions.................................................................................................................................................................7 Polynomials.......................................................................................................................................................................................8 Trigonometry & Unit Circle.............................................................................................................................................................10 Exponential and Logarithmic Functions..........................................................................................................................................11 Calculus...........................................................................................................................................................................................12

Mathematica Commands                      

Solve[f(x), x] Simplify[f(x)] Factor[f(x)] Expand[f(x)] Reduce[f(x),x] PolynomialQuotientRemainder[f(x), x, x] Apart[f(x)] f[x_]:= [f(x)] //N //TraditionalForm //MatrixForm Plot[f(x), (x, n1, n2)] Binomial[] Log[b,z] (base, number) Log[z] (natural log) Log10[z] (base 10) D[f(x),x], differentiate. AxesLabel → {x,y} Table[f(x), {x,n}] PlotRange -> {{x1, x2}, {y1, y2}} = (less than or equal to, greater than or equal to) ArcSin/Cos/Tan[a]

Probability & Counting Methods Random Events & Experimental Probability 

Event – a subset of a sample space. o Elementary event – one element of the sample space (e.g. rolling a six, {6}) o Compound event – combined multiple elements (e.g. rolling an even number, {2,4,6})

Relative frequency of event A =

Pr ( A )=

number of ×event A occurs number of trials

n(A ) number of favourable outcomes ∨ n(ε) total number of possible outcomes

Calculating & Representing Probabilities  

A question that states “at least” means 1 – probability that something doesn’t happen. Complementary events – the event that does not occur (A’) o Pr(A’) = 1 – Pr(A) Karnaugh Map (a.k.a. probability table)



Is used to indicate all different areas of a Venn Diagram.

B B’ Pr (A ∩ B) Pr (A ∩ B’) Pr(A) Pr (A’ ∩ B) Pr (A’ ∩ B’) Pr(A’) Pr(B) Pr(B’) 1 The Addition Rule & Mutually Exclusive Events A A’





The Addition Rule o Pr ( A ∪ B ) =Pr ( A ) + Pr (B )−Pr( A ∩ B ) o The probability of the sum of A and B is equal to Pr(A) + Pr(B), minus the intersection of Mutually Exclusive Events o Probability of both occurring = 0; not related. o Pr ( A ∩ B ) =0 and therefore: Pr ( A ∪ B ) =Pr ( A ) + Pr(B)

Conditional Probability

A ∩B .

   

“The probability that this has occurred, given that something else has already occurred.” A question stating “given” or “if” implies that it is a conditional probability question. Pr(A|B) = probability of A given that B has already occurred.

Pr ( A|B )=

Pr( A ∩ B) Pr(B)



If you rearrange the previous formula, you can find the intersection probability as so:



If a question asks to find to find the probability of (A|B’), first use the initial formula, then use:

Pr ( A ∩ B ) =Pr ( A |B )∗ Pr(B) Pr ( A|B' )=

Pr( A ∩ B' ) Pr(B' )

This means that to find A given B hasn’t occurred, find the probability of A only (not intersection) and divide by not B (1 – Pr(B)). These types of problems can be solved either using the formula, or drawing a Venn Diagram. o



Multiplication Rule, The Law of Total Probability & Independent Events 

The Multiplication Rule o Pr ( A ∩ B ) =Pr ( A |B )∗ Pr(B) Tree diagrams can be useful here. When trying to solve conditional probability here, define event A as the first event and B as the second event. If using a tree diagram, show the alternative option to getting A as A’, not B. (e.g. pulling marbles out of a bag, find probability of first one red, second one red) Law of Total Probability ' ' o Pr ( A )=Pr ( A|B ) ∗Pr ( B )+ Pr ( A|B )∗Pr(B ) o o



Easily shown on a tree diagram. This is used to find two different ways that the end result A can be achieved regardless of what the first event was. Independent Events o Two events are independent if the first outcome of the first event does not affect the outcome of the second. o o



Pr ( A|B ) =Pr ( A ) and

o

If two events are independent, the following statements are true:

o

To test if two events are mathematically independent, use the second formula.

Pr ( A ∩ B ) =Pr ( A )∗ Pr(B) Counting Methods 







The Addition Principle o Add up all of the different possible outcomes. o n ( A ∪ B )=n ( A ) + n(B) o When seeing “or” in a question, it means to add. The Multiplication Principle o Multiply the different branches of outcomes. o n ( A ∩B )=n ( A )∗n(B) o When seeing “and” in a question, it means to multiply. Permutations with Repetition o A permutation is an ordered set of objects (e.g. 123 is different to 321); the order matters. r o n can be used to calculate the total number of permutations, where ‘n’ equals the number of different possibilities and ‘r’ equals the number of repetitions. Permutations without Repetition o This is where every time one possibility is used up, the total number of subsequent possibilities is reduced. o This can be calculated using factorials (4! = 4 x 3 x 2 x 1), or partial factorials (e.g. 10 x 9 x 8 x 7) o



Partial factorials are technically expressed as:

n! ( n−r ) !

Or: nPr.

Calculating Permutations without a Calculator o

First, write out both entire permutations without repetition:

10 ! 10∗9∗8∗7∗6∗5∗4∗3∗2∗1 = 7∗6∗5∗4∗3∗2∗1 7!

o

Then, cancel out the highest common factorial:

10∗9∗8

Combinations   

n



However, more commonly, the formula is nCr

 

Combinations can also be written in a matrix format. Calculating Combinations without a Calculator o First write out both entire combinations. o Then cancel highest common factorial and any remaining common factors.

A combination is a set of objects where the order does not matter. Cr is used to calculate the number of different combinations. Additionally, nCr = nPr / r!

¿

n! r !( n−r ) !

Discrete Random Variable 





A random variable is a function that assigns a number to each outcome in the sample space ε. o Discrete random variable is one which may take on only a countable number of distinct values o Continuous random variable is one that can take any value in an interval of the real number line Denoted as Pr(X=x) or p(x) X 0 1 2 3 Pr(X=x) 0.125 0.375 0.375 0.125 For any discrete probability function, the following axioms exist: o For every value of X, p(x) is between 0 and 1 inclusively, where p(x) = Pr(X = x)

∑ p (x )=1

o

Sum of all probabilities is 1.

o

To determine the probability that X takes a value in the inclusive interval a to b, values of p(x) from x=a to x=b is x=b

summed. Pr ( a ≤ X ≤ b )=

∑ p(x) x=a

Sampling with Replacement  

“Successive objects are selected from a finite group without being replaced, meaning the probability of success changes after each selection.” Multiplication rule is very useful for this.

Sampling with Replacement: The Binomial Distribution   

Sampling with replacement means that the probability of success remains constant after each selection. There are only two possible outcomes regardless of how many different choices. (e.g. success or failure)

( nx) p (1− p)

Pr ( X =x )=

x

n− x

, where

n! (nx )= x !( n−x )!

PDF = only 1, only 2, only 3, etc. CDF = 1, or 2, or 3, etc.

Matrices 

Representing linear systems with matrices o Matrices have dimensions and are read row x column (i.e. 2x3 matrix = 2 rows, 3 columns) o Any number which takes up a position is called an entry or matrix element and may be denoted as

ar ,c o o

(matrix ‘a’, row r, column c)

“Augmented matrices are a shorthand way of writing systems of equations.” A system of equations can be represented by an augmented matrix. Each row represents an equation and each column represents a variable or constant.

If two equations differ in what variables they have (i.e. eqn 1 has ‘x’ and ‘y’ but eqn 2 has only ‘x’), the equation without the variable will have 0 in the respective entry. o If two equations have their variables written in a different order, rearrange them to fit into the matrix. (i.e. 3x + 2 = 12y -> 3x – 12y = -2) Matrix Row Operations o Switch any two rows – order of equations does not matter.  R1 ↔ R2 means interchange/swap row 1 and row 2. o

















o

Multiply a row by a nonzero constant – multiply both sides by the same nonzero constant to obtain an equivalent equation. Often done to eliminate a variable.  3 R 2 → R2 means replace row 2 with 3x itself.

o

Add one row to another  R1 +R2 → R2

Adding and Subtracting Matrices o Two matrices of the same dimensions will have both corresponding entries added together. o Two matrices of the same dimensions will have both corresponding entries subtracted. o Two matrices of different dimensions will give an undefined result. Multiplying Matrices by Scalars o A multiplier applied to a matrix is known as a scalar. o A scalar (multiplier) will apply to each entry. o Repeated addition of a matrix (i.e. A + A + A) is the same thing as multiplying (i.e. 3A) o Solving a matrix with a predefined scalar requires division or multiplication (i.e. 1/3A * 3 = A or 3A / 3 = A) Zero Matrices o A zero matrix is where all entries are 0. o Indicated by ‘O’ and a subscript where dimensions can be added (e.g. O 2 ×3 ) o Adding opposite matrices creates a zero matrix. o Multiplying by scalar 0 creates a zero matrix. Multiplying Matrices by Matrices o N-tuples and the dot product  N-tuples are ordered pairs, triples, etc. (2,3) * (2,3) = 2*2 + 3*3 = 4 + 9 = 13  Often indicated by a variable with an overhead arrow (e.g.  a =(3,1,8 ) ) o Matrices and n-tuples  When multiplying matrices, assign each row and column as an n-tuple. o Matrices with different dimensions can be multiplied as long as the first one’s columns is equal to the second one’s rows. o The dimensions of the product matrix are the two outer numbers. o E.g. 2x3 * 3x1 = 2x1 matrix. o Matrices are not commutative. AB ≠ BA o Matrices are associative (AB)C = A(BC). o Matrices are also distributive A(B+C) = (AB + BC) and (B+C)A = (BA + CA) Defined matrix operations o Two matrices are defined if the number of columns in the first matrix equals the number of rows in the second matrix. o The reason being is when multiplying matrices, the order matters. Matrix multiplication dimensions o The first matrix’s number of rows by the second matrix’s number of columns are the dimensions of the new matrix (e.g. 2x3 and 4x4 = 2x4) Identity Matrix o n x n identity matrix ( I n ) has each entry from top left to bottom right diagonally equal to 1 and all other o



means replace row 2 with the sum of row 1 and row 2.

entries 0. A matrix multiplied by its respective identity matrix equals the original matrix (i.e. I n∗ A= A )

o All identity matrices will be square. Determinant of 2x2 matrices



o Product of the diagonal top left to bottom right minus the product of diagonal top right to bottom left. (ad-bc) o Also denoted as |A| Matrix Inverses o A−1 A=I

[ ]

A= a b c d 1 d −b o A−1= ad −bc −c a ¿ A∨¿∗adj( A) o Or: 1 −1 A = ¿ o

o o

[

]

Remember that ab-bc is the determinant. The adjugate of a matrix is having a and d swapped, and having b and c inverting their signs. (e.g.

[ ac db] →[ −cd −ba ] 



Determining Invertible Matrices o If the determinant of a matrix is anything but a 0, then it is invertible. o If and only if (iff) |A|=0 , A−1 is undefined/singular/degenerate. Solving Equations with Inverse Matrices o Matrices can be used to solve systems of equations, such as 2 x +3 y=−1 and x+2 y =−1 . o These two equations can be expressed in matrix forms, where the coefficients fit into a 2x2 matrix of numbers

[ 21 32]

and the variables fit into a column vector

x . The constants also fit into a column vector of y

−1 −1  X

and

The column vectors can be represented by

o

To solve for an unknown column vector, synthesise the matrices into an equation. In this case, Then multiply both sides by the inverse matrix (which is

A 

 B .

o

−1

A=I

1 det(A )

)

A

−1

A X= A

−1

 B . Remember that

(inverse matrix) so it effectively cancels out. Then multiply the inverse matrix to

final column vector which defines the variables. Representing transformations with matrices o Represent the coordinate pair in a cartesian plane as a 2x1 column matrix. o To multiply most transformations, a 2x2 matrix known as a transformation matrix is used.

X=  A B .

 B to get a

Introduction to Functions Linear and Quadratic Functions

y 2− y 1 rise or m= run x 2−x 1



y=mx + c where



m 1=m 2 if parallel (same gradient) −1 m1= if perpendicular (inverse gradient) or m 1 m 2=−1 (product of both gradients = -1) m2 x1 + x2 y1 + y2 =midpoint , 2 2 2 2 d= √ (x2 −x1 ) +( y 2− y 1 ) … distance formula. Remember to use each combination of vertex points, as in for a

  

m=

triangle with three states vertices, use the distance formula three times for each combination of coordinates. Add all three to find the perimeter. Remember: this is Pythagoras’ Theorem.  

−1

θ=tan (m) m=tan(θ)

 

Co-linear = same gradient, points are on the same line. Find the rule of a graph, use: y − y 1=m(x −x1 ) ; gradient and a point.

 

To express a quadratic equation in standard form to the y =a( x −e )( x −f ) form, first remove any denominators from the standard form, use DOPS or the ac/b method. To find the vertex of a parabola, find the x-intercepts by establishing the average distance between the two x-intercepts, apply whether the graph will be maximum or minimum and then substitute the x-coordinate into the original equation to find the y-coordinate. Finally, you will have an ordered pair.



Completing the square:

2

2

a x + bx+ c=0 → a(x +d ) + e=0

b 2a

o

d=

o

e =c −

b2 4a

Functions and Relations 

    



A relation is a set of ordered pairs, usually expressed as a rule. (e.g. A = {(0,1), (2,3)} or y = 5x – 1) A function is a relation if for every x value there is a different y value. Anything that can be mapped on a cartesian plane using a rule is a relation. Functions can be one-to-one or many-to-one. All functions are relations, but not all relations are functions. To test if a relation is a function, we use the vertical line test. If placing a vertical line across the graph and it only touches the graph once, it is a function. To test if a function is one-to-one or many-to-one, we use the horizontal line test. Draw a horizontal line and if it contacts the line once, it is one-to-one, if it contacts >1 time it is many-to-one.

Function and set notation 

A rational number is any number that can be written as a fraction in some way (i.e.



a ) b

An irrational number is a number that cannot be expressed into a fraction; no patterns (e.g. surd, Pi)

    

Use [ ] and ( ) to indicate in interval notation whether a number is inclusive or exclusive. (e.g. On number lines open circle = exclusive, closed circle = inclusive. {x: x} means “x such that…” Interval notation on a number line can have multiple points on it. If ‘x’ is part of all real numbers, it is an element of R. (e.g. x ∈ R )

{ x : x ≥0 }=¿

Domain and Range    

Domain of a relation is the set of x-values Range of a relation is the set of y-values If the domain is unspecified, the largest subset of R possible. A.k.a. ‘implied’ or ‘maximal domain’. Co-domain in Math Methods is always R. It will never change in this subject. It means “we are only using real numbers” 2



y= x



y=x 2

   

x ∈ R we write f : R → R , f ( x )= x 2 where x ∈ ¿ we write f :¿ → R , f ( x )=x2 where

For x-intercepts, we solve f(x) = 0 (sub y=0) For y-intercepts, we evaluate f(0) (sub x=0) f(0) means substitute x=0 into the function to find the y-value. With interval notation always start with the lowest number.

Polynomials Expansion of Polynomials    

(a+b)6 – to find the coefficients of each term, use Pascal’s triangle. The ‘a’s start at powers of 6, 5, 4, 3, 2, 1, 0 The ‘b’s start at powers of 0, 1, 2, 3, 4, 5, 6 6

5

4

2

3

3

2

4

5

a +6 a b+15 a b + 20 a b +15 a b +6 a b + b

6

Division of Polynomials     

Use long division. (DMSB) First term will always cancel every time you follow the cycle. Put in 0x or 0 in to keep it neat. If dividing a polynomial and it results in a remainder, put it into a fraction. Or use synthetic division.

Remainder Theorem    

P(x ) −α =P( ) βx +α β If a polynomial is being divided by a divisor (e.g. 2x 32x2+3x+1 / x-2) Substitute the opposite value of -2 (+2) into P(x), making P(2) Then solve the expression substituting x = 2.

Factor Theorem/Factorising a Polynomial 

( −αβ )=0 −α =0, βx +α∨P ( x ) , P( β )

βx +α∨ P ( x ), P



Conversely,

 

Note: '∨' in this case means “a factor of…” A cubic function can be factorised using the factor theorem to find the first linear factor and then using polynomial division or the method of equating coefficients to complete the process. 1. Use the ...