Maths AP

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Logs and Algebra

logX cannot be

ln cannot be

therefore look for asymptotes in this regard with graphs

ln graph is similar to log graph, e graph is similar to exp.

Basic log rules

adding two logs is muliplying their 'X' values

subtracting is division

Partial Fractions

USe A and B over seperate denominators

Add both fractions under normal rules

Normal equation then

Remember grouping rules, like complex numbers, equate form of term to form of a/b/c

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Basic function rules

Always look for asymptotes (see log rules also)

f'(x) value will tell you whether graph is increasing or decreasing (see if 0)

For domain and range always look at asymptotes and impossible values

Conitnuity

Normal cicrumstances: f(x) = lim f(x)

Removable discontinuity will occur if above is not true

in different sector functions, f(x), lim x- and limx+ must be equal

if f(x) is not equal to limits, but limits are equal (i.e they exist) then it is also removable

Jump discontinuity: no limit exists due to:

asymptote at that point

unequal limits at - and +

Differentiable

not contonous functions are also not differentiable

f'(x)- must equal f'(x)+

Asymptotes

Vertical-denominators

Horizontal- lim of infinity

for H , numerator must not have higher power than denominator

Turning points

Minimum- f''(x) > 0

Reverse for maximum

Newton's Method

a2 = a1 - (f(a))/(f'(a))

Remember guess rule

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Mathematical induction

First substitute 1

Then k, that's all

Then k+1

Either make RHS simple and k+1 a valid replacement for n

OR substitute n=k by LHS and make it like RHS

Riemann Sums

State width -----> (b-a)/n

State xi------> width * i (plus a if a is actual value)

Therefore approximate area = 'E' of function where xi is substitued by x and function is timed by width

use rule of i, remember everything else is before 'E' for each term

values with no i get (* n)

use lim to infinity and find answer of values

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Circles and radians

1 radian is equal to 180/pi

1 degree is equal to pi/180

length of arc = radius * radian angle

area of a sector = .5 * radius^2 * radian angle

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i and complex numbers

for solving a+bi, simplify terms to similar form

remember one root of i also correpsonds to another with different sign

always get rid of i in deonimantor

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Diffrentiation

Product Rule

T'1.T2 + T'2.T1

First Principles

Remember if complicated root, then use surd/denominator rule

Qoutient Rule

(T'1.T2 - T'2.T1)/T2^2

Implicit

Always diffrentiate x and y, and if together than do like product rule

Always include dy/dx with y terms

Trigs

First power, then ratio, then bracket

Example squared becomes two with single exp. ratio, then ratio changes with same bracket, then normal term diffrentiation

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Integration

With root x in denominator, always use simple u/du rule

With trigs, first try normal double angle identities

With two unrelated terms, use Sudv = uv -Svdu

Generating solid

pi * S(f(x)^2)

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Population

Ligistic Model

delta P = r.P(1- P/K )

K is horizontal asymptote, carrying capacity, equilibrium point

r is vertical asymptote, instristic growth rate

Next yr population is Pcurrent plus delta P

Delta P is actual growth rate!!

Predator Prey model

Prey: R2 = R1 + a.R1(1- R1/K ) - b.R1.F1

T2 is equal to reproductive rate

a is intristic rate

T3 is number of deaths

b is attack rate

Predator: F2 = F1 + f.b.F1.R1 - c.F1

T2 is reproductive rate

f is efficiency of turning resources into growth

c is death rate

Simple PP Model

Prey (lacking predators) : R2 = R1(1+a)

Predator (lacking prey) : F2 = F1(1-c)