Advanced Math

Derive e^x

d/dx e^x = e^x
d/dx e^(ax + b) = a x e^(ax + b)
d/dx e^(f(x)) = f'(x) x e^(f(x))

Derive exponetials

d/dx a^(f(x)) = f'(x) x ln(a) x a^(f(x))

Derive Nature Logs

d/dx ln(f(x)) = f'(x)/f(x)

Proof for this

Trig derives

when we derive

cot(f(x)) = −cosec² x F(x)⋅F’(x)

Cosec = −csc(F(x)) x cot(F(x))⋅F’(x)

Sec = sec(F(x)) x tan(F(x)) x F′(x)

Proof for Sin

Proof for cos

Proof for Tan

Further Differentiation

skip

Increasing, decreasing and stationary rules at a point

F’(Φ) > 0, then F(x) is increasing at x=Φ

F’(Φ) < 0, then F(x) is decreasing at x=Φ

F’(Φ) = 0, F(x) is stationary at x=Φ

Graphs to remember Turning points and points of inflection

Turning points

A stationary point is called a turning point if the derivative changes sign around the point

Maximum turning point

The curve goes from increasing to decreasing

Minimum turning point

curve goes from decreasing to increasing

Local Maximum

Where the function reaches a peack in its nearby region. (highest point)

Local minimum

like the local minimum but its at the lowest

Point of inflection and how to determine it

Where the curve changes its shape.

So the point where the tangent crosses the curve making the concavity change

only graph given: find where curve switchs

only equation: find y’' = 0

How to find stationary points

1. find F’(x)

2. equate for x by setting F’(x)=0

3. solve for x

What does it mean when it asks for the nature

its asking for the local minimum or maximum or the point of inflection

To determine its nature we

double derive using this rule

F’’(x) > 0, Local minimum: curves upwards

F’’(x) < 0, Local maximum: curves downwards

F’’(x) = 0, Point of inflection

(if you get x sub in stationary points to determine nature)

Concavity and poitns of inflection

if F’’(Φ) is - then its concave down at x=Φ

if F’’(Φ) is + then it concaves up at x=Φ

What is Global maximum and minimum

It’s basically the highest and lowest point in the graph

how to find the maximum or minimum value of F(x)

So first determine the value of x that maximises F (so do F’(x)=0 and find staionary points if its a parabola if its not then look at the domain and find Global maximum and sub it into F(x) )

NOTE IT IS NOT A CORD

How to do opimisation

1. introduce the two variables “let y be the quantity that is to be maximised and let x be the quantity that can be varied”

2. form equations

3. find global maximum and minimum

4. write conclusion

Using second derivative to test stationary points

F’’(Φ) > 0, concaves up at x=Φ making it the minimum turning point

F’’(Φ) < 0, concaves down at x=Φ making it a maximum turning point

F’’(Φ) = 0, more work needed go back to table

Integration

NOTHING

What is a integral?

reverse differentiation by summing infinitely small quantities. used to find the area under the curve

or

∫ F(x) dx = Area of shaded region

What is a definite integral

The definite integral is defined to be the area of a region under the curve between the boundaries a<x<b

in which

x = a (Lower limit)

x = b (Upper limit)

∫^b_a F(x) dx

Shape areas to remember

Triangle: ½ Bh

Trapizium: ½ (a+b) W

Circle: πr²

Tips for definite integrals

• if y cord is not provided sub x in

• A - (- B) = Base

How to find Area without the graph (fundamental theorem of integrations will touch more on this later)

1. Find primitive function

2. Then use limits of F(B) - F(A)

Integrating functions with a negative value

when it has a negative value, its just saying that the graph is below the x axis

can be show as

Area 1+ Area 2 - Area 3

where Area 1 and 2 are above x axis

Area 3 are below the x axis

Odd functions

It is odd when the area of each shaded hump is the same, hence the whole integral from 2 to -2 is zero because the equal humps above and below the x axis cancel out

(SIN IS ODD BTW)

Show through:

∫^a_-a F(x) dx = 0

Even Functions

Since itll have al line of symmetry in the y axis , the areas to the left and right of the y axis are equal so itll double instead of canceling

(COS IS EVEN)

∫^a_-a F(x) dx= 2∫^a₀ F(x) dx

Dissection of the interval

∫^b_a F(x) dx = ∫^c_a F(x) dx + ∫^b_cF(x) dx

tip ( The c’s across from each other cancel out so use that theory to form this equation when needed)

What if it has zero width

∫^a_a F(x) dx = 0

Reversing the interval

∫^b_a F(x) dx = - ∫^a_b F(x) dx

works with

F(A) - F(B) = - (F(B) - F(A) )

Sums of functions

∫^b_a F(x ) + G(x) dx =∫^b_a F(x) dx + ∫^b_a G(x) dx

Multiple of a integral

∫^b_a F(x) X K dx = K∫^b_a F(x) dx

Proving Fundamental Theorem (Differential Form)

d/dx ∫^x_a F(x) dx = F(x)

If F(x) is continous, then the signed area function for F(x) is the primitive of F(x) of

A’(x) = d/dx ∫^x_a F(T) dT = F(x)

Note

• A’(x)= Orginal founction

• Use A(x) to find area and deriving gives A’(x)

• Equation = Height

Indefinite Integrals

Find the primitive and since there are no bounds we have to add +C at the end of the integral

Compound regions ( no integration)

just find the areas and add them not hard

Compound regions ( with integration)

∫^b_{a (G(x) -F(x) ) dx}

_{F(x) is below G(x)}

• if the graph connects use the formula above

• if it doesn’t but its separated then find both halfs and add

Trapizoidal rule (next 7 slides will be it )

What is the width for it

b-a

F(a)- f(b)/ 2

What is the average of parrallel sides

Area of trapizium in this case

b-a/2 * (f(a) + f(b))

The method is to

Replace the curve by a straight line, by a chord giving us the cords (a,f(a)) and (b, F(b))

if it concaves up it overestimates the integral

if it concaves down it under estimates it

if its linear itll give the exact value

Rules with it

Trapezoidal rule with n subintervals

∫^b_a F(x) dx = b-a/2n (f(a) + f(b) + 2( f(x₁) +…. f(x_n))

How to determine subintervals

look at the table of values. minus the total amount of x’s in the table by 1

Another way to find the subintervals is to

h= b-a/n

n= number of x values - 1

if they give it in terms of function values then to find x1, x2, x3

we do

x1= a+h

x2= a+2h

x3+ a+ 3h

and so on

to determine how many xn’s to use just do n+1

Reverse chain rule

∫ F(x) * F’(x) dx= (F(x)) ^ n+1/ n+1

what are the steps

1. determine u

2. derive u

3. make du= u’ dx

4. sub u into the equation

5. manipulate the equation so you can sub in du

6. integrate

Integration of exponentials

∫e^f(x) dx= 1/F’(x) *e^f(x) +C

integration of exponetials of base a

∫a^f(x) dx = 1/F’(x) Lna * a^F(x) +C

Integration of reciprocal

∫ F’(x)/F(x) dx= Log_e | F(x)| + C

∫ 1/ax+b dx = 1/a [log_e (ax+b) ] + C

1/ xlog_ea

deriving log_ax

a^x= e^xlog_ea (works for a but just remove the x in it)

Tips for this

Trig deriving

each one is mulitpled by 1/F’(x)

∫sin(fx) dx = -cosF(x)+C

∫cos(fx) = sin(F(x))+C

∫ sec²(fx) = tan(F(x))+C

∫cosec(fx) cot(F(x))+C

∫ sec(fx) sec(F(x)) +C

except for these 2

∫ cot fx) = log_e |sinx| +C

∫ tan(fx) = -log_e |cosx|+C

Yr 11 Probability (nothing here)

The basic Formula for Probability

P(Event)= Number of favourable outcomes/ number of possible outcomes

What is sample space

The sample space is the set of all possible outcome

Complement of an Event

The complement of event E, written E′ is the event that E does NOT occur.

.

P(E′)=1−P(E)

Intersection of sets

A∩B=A and B ‘

A∪B=A or B (or both)

The union of sets

Rules not mutually exclusive

P(A∪B)=P(A)+P(B)−P(A∩B)

For mutually exclusive evnts

P(A∪B)=P(A)+P(B)

Conditional Probaility

Probability of event A given event B has occurred.

P(A|B)=P(B)/P(A∩B)​

if its independent then it just equals to P(A)

P(A∩B) Formulas

P(A∩B)

Probability of A and B

P(A∩B)=P(A)P(B) (InDependent)

P(A∩B)=P(A)P(B∣A) (Dependent)

Product rule

P(A and B)+ P(A|B) x P(B)
if its indepentdent then its just: P(A) x P(B)

What is a discrete probability distribution

A discrete probability distribution assigns probabilities to each possible countable value of a random variable.

What is a random variable

A random variable is a variable that represents a numerical outcome of a random process or experiment.

• the outcomes of a random experiment

what is the expected value

The expected value is the long-term average value of the random variable.

E(X)=∑x P(X=x)

What is the variance

The variance measures the spread of the distribution.

Var(X)=∑(x-mu)² p(x)

∑x² p(x) - mu²

What is standard deviation

Standard deviation is a measure of how far values typically vary from the mean of a distribution.

it is the square root of the variance

YR 12 Stats (nothing)

What is a mode

Mode is the most popular score

What is the range

The range is the difference between the maximum and minimum

What is a cumulative frequency table

its like a normal table but for the cumulative you keep adding the new frequency to the one you already have for example

F= Frequency for this

C= cumulative

F1= 88

C1= 88

F2= 10

C2= 98

etc etc

The formula for Mean

∑xf / n

n= total frequency

Formula for variance

∑x²F/ n - (mean)²

What is the inter quartile range and how to calc it

it is the measure of the spread and its calculated by Q3-Q1

What is the 5 number summary (with its respected formula)

1. minimum

2. Q1 - n+1/4

3. Q2 - n+1/2

4. Q3 - 3(n+1)/4

5. Maxium

What is Bivariate data

Data in the form of ordered pairs

What is interpolation

It is the prediction of results within a range of variables

What is Extrapolation

is the process of estimating values outside the range of known data using an existing trend.

Key points (exam-ready)

• Uses a pattern or model from given data

• Predictions are made beyond the smallest or largest x-values

• Results are less reliable than interpolation

Example

If data shows population growth from 2015–2025, predicting the population in 2035 is extrapolation.

What is Pearsons Correlation Coefficient

Pearson’s correlation coefficient is a numerical measure of the strength and direction of a linear relationship between two variables.

show by

r= ∑ (x- x mean)( y- y mean) / Sqr ∑(x- x mean)² ∑ (y- y mean)²

Pearsons. correlation Regression line formula

m= ∑(x- x mean)( y- y mean) / ∑(x- x mean)²

B= y mean - m(x mean)

What is Relative Frequency

Estimates of probabilities

What is a Discrete probability distribution

A discrete probability distribution lists all possible values of a discrete random variable and their associated probabilities.

for it to be valid P(X=x) must equal to 1

Probability Densitiy function (PDF)

is a function in the domain of [a,b]

it is the basic i definite integral but it will always equal to one

THings to know about PDF

Mode is the global maximum

Probability is the area under the curve

Cumulative distribution function

F(A)=0

F(B)= 1

is the signed area function of the PDF

F(x)= P(X<x)

with bounds x and a of F(t) dt

Tips for in general with PDF and CDF

P(0<X<1) these are youre bounds

P(W<X< Y) = F(Y)- F(W)

How to calc CDF

1. integrate with no bounds

2. make it equal 1

3. find C

Or

use the formula from above

Formula for Mean/ expected vaue using integration

∫_a^b xF(x) dx

Formula for variance using integration

∫ba (x-mu)² F(x) Dx

∫ba x² F(x) dx - mu²

Where is the expected distribution?

It is in the centre

What is Normal Distribution (and its formula)

A normal distribution is a continuous, symmetric probability distribution shaped like a bell curve, centered at the mean.

Φ(z)= e^1/2 x z^2 / sqr 2 pi