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Maths - C4
by Ameer Khan
# Maths - C4

## Differentiation

### Trigonometric functions

### Parametric Functions

### Implicit Functions

## Intergration

### Trigonometric Functions

### Integrals Leading to a Logarithmic Function

### Partial Fractions

### Substitution

### Integration By Parts

### Differential Equations

### Exponential Growth and Decay

## Vectors

### Vector and Scalar Quantites

### Component Form

### Finding the Magnitude of a Vector

### Scalar Product

### Vector Equation of a Line

### Pairs of Lines

## Coordinate Geometry

### Cartesian Form

### Parametric Form

## Algebra And Series

### Rational Expression

### Partial Fractions

### Binomial Expansion For Rational, n

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dy/dx sinx=cosx dy/dx sin(ax+b)=acos(ax+b)

dy/dx cosx=-sinx dy/dx cos(ax+b)=-asin(ax+b)

dy/dx tanx = sec^2x dy/dx tan(ax+b)=asec^2(ax+b)

Chain rule comes in mighty handy with this

If x and y are each expressed in terms of a parameter, t: dy/dx = dy/dt x dt/dx

To find dy/dx when an equation in x and y is given implicitly, differentiate each term in respect to X

d/dx(f(y)) = d/dy(f(y)) x dy/dx

∫cosx dx=sinx+c ∫cos(ax+b) dx=(1/a)sin(ax+b)+c ∫sinx dx=-cosx+c ∫sin(ax+b) dx=-(1/a)cos(ax+b)+c ∫sec^2x dx=tanx +c ∫sec^2(ax+b)=(1/a)tan(ax+b)+c

∫sin^n(x)cosx dx=1/n+1(sin^n+1(x)+c) ∫cos^n(x)sinx dx =-1/n+1(cos^n+1(x)+c)

For odd powers of n, split the function in to even powers or one, then use trig identities to simplify the function into something recognisable and work from there

To integrate even powers use the double angle identities: cos^2(x)=1/2(1+cos2x) sin^2(x)=1/2(1-cos2x)

∫f'(x)/f(x) dx= ln|f(x)|+c

You will need to know this in order to help find the area under a curve

One of the reasons why we learn partial fractions is so that we can integrate them; so we would split the fraction and integrate as usual

Sometimes an integral is made easier by using a substitution.

∫f(x) dx=∫f(x)(dx/du) du

When doing this remember to change the x limits to u limits as soon as possible - it helps

A general result through this is: ∫f'(x)[f(x))]^n dx=(1/n+1)[f(x)]^(n+1) + c

Another way of integrating is integrating through parts

∫(u)(dv/dx)dx=uv-∫(v)(du/dx)dx

Sometimes you won't get the answer straight after putting it in the formula so put it in again.

Always make U the simplest variable or ln(x)

Also under definite integrals the rules are the same, don't act to smart, if you get confused go back to the basics!

When integrating, we get a general solution, sometimes we may get given points in a graph in order to work out c, to get a particular solution

A differential equation can be written as: f(y)(dy/dx)=g(x) this can be solved be separating the variables where: ∫f(y) dy = ∫g(x) dx

You may have to form a differential equation from given information; this is normally exponential growth or decay

Exponential growth: dy/dx = ky so y=Ae^-kt Examples include population growth

Exponential decay: dy/dx = -ky so y=Ae^-kt Examples include the disintegration of radioactive materials and Newton's law of cooling

A scalar quantity has a size (magnitude) and no direction

A vector quantity has both magnitude and a direction

When working in two dimension, it is quite useful to express a vector in terms of i and j. They are unit vector and can be written as r=ai+bj

In three dimension a third vector (k) is used to represent a unit vector in the z-axis

This form can also be used to represent position vectors

If r = ai+bj+ck then |r|= √(a^2+b^2+c^2 )

a.b=|a||b|cosθ

If two vectors are parallel then θ=0 so a.b=|a||b|

If two vectors are perpendicular then θ=90 so a.b=0

R is the position vector of any point on the line B is the direction vector of the line (b-a) A is any point on the line t is the scalar parameter (constant or so to say)

R= A+tB

In 2-D, a pair of lines either are parallel or the intersect

In 3-D, a pair of lines either are parallel, they intersect or they are skew

Parallel = t is a multiple of the other Intersect = there is a unique value for t Skew = no unique solution, nor a multiple, completely unrelated values of t

A curve in Cartesian form is written in terms of x and y

To sketch a graph in Cartesian from: 1. Check for symmetry (if the powers are even) 2. Find where the curves cross the axis (x=0;y=0) 3. Check for Stationary points (dy/dx = 0) 4. Check what happens for large values of x and y 5. Check for any discontinuities

A curve in Parametric form can be expressed in x, y and in a third variable, for example: t.

You will usually be asked to turn it in to Cartesian form (Make t the subject and sub it into the other equation) and then to draw a graph for it

As this is a synoptic unit, they may bring in other topics into this, most likely trigonometry (Pythagoras Identities) So get some practise in using them

Can be in the form of f(x)/g(x)

To simplify, just factorise the two and eliminate like terms

When adding or subtracting, just use the lowest common denominator

Not that hard, but ever so long so don't be lazy and do it!!

This is splitting fractions into two or even three is required; and of course you have to work out the numerator for each new fraction

Only proper fractions can be split, if it is improper, then you must turn it into a mixed fraction before moving forward

(1+x)^n can be written through the binomial expansion formula

The formula is given, but learn how to manipulate the formula algebraically as it will help.