Maths

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The Trapezium rule

The Trapezium rule

Area = 0.5 h (y0 + y3 + 2 (y1 + y2)

First + last + (2 x rest)

Variable acceleration

Integrate Integrate

S V A

Differentiate Differentiate

Simple random sampling

Takes a random selection from the population to use as a sample. The entire population must be known and identifiable. Each random sample will be different and some will contain bias. The probability of a strongly biased sample is reduced as sample size increases.

Stratified sampling

If the parent population has clear subgroups, it is possible to take a sample. The quick calculation to determine the number for each subgroup to be randomly selected as part of the sample is: (number in subgroup/ number in population) x sample size. This technique requires a lot of information to begin about the population.

Quota sampling

Similar to stratified, though the ‘quota’ of items to be included from different subgroups isn’t necessarily proportional. It also doesn’t require a random sample from within the subgroup, so bias is extremely likely.

Systematic sampling

Midpoints

((X1+X2)/2), ((Y1+Y2)/2)

Distance between two points

√((x_2-x_1)²+(y_2-y_1)²)

Gradient of a chord

Y2-Y1/X2-X1

Parallel lines

The same gradient

Perpendicular lines

Negative reciprocal gradient

Y intercept

When X=0

X intercept

When Y=0

Perpendicular bisector

1) midpoint

2) neg recip gradient

3) Y-Y1= M(X-X1)

4) rearrange into Y=MX+C

Quadratic formula

Discriminant

B²-4ac

One real root

When the discriminant = 0

Two distinct roots

When the discriminant > 0

No real roots

When the discriminant <0

Two rational roots

Perfect square

(X+ 1/2b)² +0

1000= 10³

Log10(1000)=3

LogbB

1

Logb1

0

LogcA+logcB

LogcAB

LogcA-logcB

LogcA/B

Logc(A^B)

BlogcA

Logc√A

1/2logcA

LogeX

lnX

If f(a/b) = 0

Then (bx-a) is a factor

If it isn’t a factor

It is a remainder

The remainder theorum

Y=X

Y=X²

Y=X³

Y= 1/X

Y²+X²=r²

Translation of Y=f(x) by the vector [ab]

Y=f(X-a)+b

Stretch on the x axis by the factor k

Y=f(1/k X)

Stretch on the Y axis by the factor K

Y=KF(X)

Reflection on the x axis

Y= -f(x)

Reflection on the y axis

Y=f(-X)

Reflection on y=x

Swap the x and y

Eg. Y=f(x) ↦ X=f(y)

Finding the centre and radius of a circle

Complete the square

Circle theorem

If <ABC = 90°, then AC is a diameter of the circle with A,B,C on the circumference

Tangent

Line touching a circle at a distinct point

Normal

Line perpendicular to tangent

Given Y= ab^x

Log10Y= log10A+ log10B

Exponential curve

Log10Y and X

1) take logs of both sides

2) simplify

Gradient= log10b

Y intercept = log10a

Y=aX^b

Log10Y= log10A+Blog10X

Polynomial line

Log10Y and log10X

1) take logs of both sides

2) simplify

Gradient= b

Y intercept = log10a

Natural numbers

Positive integers

Integers

All numbers on the numberline

Rational numbers

Fractions

Reals

All numbers you can think of (π, e, 1/3, √2, 500, -2)

Set notation

{x:x ∈ R, a<x<b}

Interval notation

x ∈ (a,b)

( is not including

[ is including

Differentiation in first prinicples

Average speed

Total distance

———————

Time

Average velocity

Total displacement

—-————————

Time

Velocity- time graph into acceleration-time

Acceleration= gradient of velocity

Circle property diameter

The triangle formed from the ends of a diameter has a right angle

Circle property chord

The perpendicular from the centre to a chord bisects the chord

Circle property tangent

A tangent to the circle meets a radius at a right angle