Core 3 Maths Revision

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Differentiation

Functions

Modulus Functions

Differentiating trigonometry

Logarithms

The exponential function

y=ex

Inverse is lnx which is log base e.

y=ex increases at an ever increasing rate, described as exponential growth

y=e-x exponential decay, approaches the x axis.

Making natural logarithm to solve

N = 2000e 0.1t

Taking natural logs;

5=e 0.1t

ln5=ln(e 0.1t) = 0.1t

t=10ln5 = 16.09

ln(ex)=x

elnx = x

Making something subject using natural logs;

ln(p) -ln(1-p) = t

ln(p/1-p) = t

e ln(p/1-p) = et

(p/1-p) = et

p + pet = et

p(1+et) = et

p - et/1+et

X numbers put in are the domain of the mapping

the y numbers out are the co-domain of the mapping

Mapping types

One to one

Many to one

One to many

Many to many

Inverse functions only work if the function is one to one mapping. Therefore domains have to be placed to make many to one mappings, one to one fora set of values.

Composite functions are functions within functions.

Work from the inside out.

gf(x) so f into g

Inverse functions

Give function f(x)

interchange x and y values

Rearrange to make y the subject

Inverse trigonometry functions

cosec0 = 1/sin0

sec0 = 1/cos0

cot0 = 1/tan0

Even functions have the y axis as a line of symmetry

f(x) = f(-x)

Odd functions have rotational symmetry of 180 about the origin

f(-x) = -f(x)

Periodic functions have a repeating pattern, repeats at regular intervals

f(x +k) = f(x)

Modulus of a function always takes the positive numerical value of x

Log rules

logaN =x => N = a^x

logbM +logbN = logbMN

logbM - logbN = logbM/N

nlogbM = logbM^n

logbB = 1

logb1 = 0

logbB^x = x

logbN = logaN/logaB

Functions are either many to one or one to one.

Z are all integer values

R are all real numbers

Q are all rational numbers

The Chain Rule

differentiate dy/dx

dy/dx= dy/du x du/dx

y=u^2

u= (3x +4)

Product rule

dy/dx= uv' + vu'

differentiate brackets x inside brackets differentiate.

eg.(3x +1)^2

= 6(3x +1)

The factors found identify where the turning points are.

Related rates of change

dA/dx is change in area rate

dA/dx = dA/dr x dr/dt

Inverse functions are reflected in the mirror line of y = x.

Restrictions of the domain

For sin x it is /2For cos x it is 0For tan x it is /2

Quotient rule

dy/dx = vu' - uv'/ v^2

whenever it is y =

Inequalities involving modulus signs require you to take the positive and negative values to solve the equation.

Transformations of functions

f(x - t) + s translation of t/s

af(x) = one way stretch in y axis, scale factor a

f(ax) = one way stretch in x axis, scale factor 1/a

-f(x) = reflection in x axis

f(-x) = reflection in y axis

Differentiating an inverse function

dy/dx is rate of change of y with x

dx/dy is rate of change of x with y

dy/dx = 1/dx/dy

Gradient of f-1x at (a,b) = 1/gradient of f(x) at (b,a)

dy/dx of y =lnx = 1/x

y = e^x dy/dx = e^x

y = e^ax => ae^ax where a is a number

y = ae^x => ae^x

y = e^f(x) => f'(x)e^f(x)

y = alnx => a/x

y = ln(ax) => 1/x

y = ln(f(x)) => f'x/fx

d/dx(sinx) = cos(x)

d/dx(cosx) = -sin(x)

d/dx(tanx) = 1/cos^2x or sec^2x

Chain rule on trig

cos2x

u = 2x y = cosu

u' = 2 y'' = -sinu

Rules of dy/dx on trig

y = cos kx => -ksinkx

y= sinkx => kcoskx

y = tankx => k/cos^2kx

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