basic alg - based on books of Кочетковых(Kochetkovs'), Глейзера(Gleiser's)

Numbers

N - natural numbers 1,2,3,4…

Z - whole numbers -2,-1,0,1,2,...

Q - rational numbers -2,½, ¾, 0, 1,...

*N,Z,Q - R, real numbers

P - irrational numbers sqr(2), PI, e,...

Equation and Inequation

Laws

a + b = b + a

(a + b) + c = a + (b + c)

a * b = b * a

a * (b * c) = (a * b) * c

a * (b + c) =a * b + a * c

if a > b then b < a

if a > b & b > c then a > c

If a > b then a (+/-) c > b (+/-) c

Dependencies

Direct dependency

y = k*x

for example: 

*always through (0,0)

**if k>0 function is increasing

   if k<0 function is decreasing

Linear dependency

y=k*x+b

*b is shifting function to up/down

Inverse proportional dependency

y=k/x

*hyperbola, x != 0

k1*x = k2*y - Proportional dependency

Systems of equations

Form

| x = k*y + b (1)

| y = k*x + b (2)

| … 

Solving

This graphics can be crossed each other(points of crossing are “solving” the system):

-if (1) is crossing (2) then 1 solution

-if (1) = (2) then infinite solution

-if (1) do not crossing (2) then 0 solutions

Methods:

1. Reducing:

possible to sum (1) and (2) and subtract them. It is useful when one of undefined variables can be “reducing” after “arithmetic” between equations, it can show a simple equation for the second undefined variable.

For example: 

| 2*y  = 2*x + 3 (1)

|-2*y = 4*x + 6  (2)

Let’s sum (1) + (2):

0 = 6*x + 9 => x = -9/6

*then calculate y…

2. Replacement:

possible to replace an undefined variable through dependency to second, and after it may be solved like a regular equation.

For example:

| 2*y = 2*x + 3 (1)

| 4*y = 6*x       (2)

Let’s find y in (2):

y = (6*x)/4 

Let’s replace y in (1):

2*((6*x)/4) = 2*x+3

12*x/4 = 2*x+3

3*x = 2*x+3 => x = 3

*then calculate y…

3. Graphical - build graphics of equations and find solutions

4. *matrix methods

Quadratic equation

Form

y=x^2

y = a*x^2+b*x+c( *if a=1,b=0,c=0 => y = x^2)

Solving

*crossing with y = 0

Methods:

1. Through discriminant:

d = b^2 - 4*a*c

x(1,2)=(-b(+/-)sqr(D))/2*a

*if d > 0 then 2 roots

 if d < 0 then 0 roots

 if d = 0 then 1 root

2. Through theorem Vietta:

|x1 + x2 = -b

|x1 * x2  = c

3. Graphical

4. With system:

| y1 = a*x^2

| y2 = -b*x - c 

*peak point of graphic:

x0 = -b/(2*a) 

y0 = - d/(4*a)

Square root

y=sqrt(x)

*x>=0

** sqrt(x) = x^½, sqrt(x) = 2_rt(x), 3_rt(x) = x ^1/3

Inequation

Solving

1. Find zeroes(cross y = 0)

2. Check intervals relation with y=0

For example:

5-x>0

1)5-x=0 => x = 5

2)

if x = 2 then 5 - 2 = 3  &  3 > 0 => +

If x = 7 then 5 - 6 = -2 & -2 < 0 => -

Answer: (-infinity,5)

*for 5-x>=0 answer: (-infinity,5]

For systems of inequations, combine each conditions in common

Module

|x|

if x >= 0 then x = x

if x < 0   then x = -x

Exponentation

Examples

Laws

a^m * a^n = a^(m+n)

(a^m)^n = a^(m*n)

(a*b)^n = a^n * b^n

(a/b)^n = (a^n)/(b^n)

(a^m)/(a^n)=a^(m-n)

a^n = 1/(a^(-n))

a^(m/n)=n_rt(a^m)

Logarithm

log a b = c => a ^ c = b 

log a (x1*x2) = log a x1 + log a x2

log a (x1/x2) = log a x1 - log a x2

log a (x^b) = b*log a x

log a x = log b x / log b a

 

(a^x)' = a^x ln a  

(log a x) = 1 /(x * ln a)

*log a = log 2 a

*ln a = log 10 a 

Analysis of function

y=f(x)

D - all possible x in function

E - all possible y in function

Possible to set function as a:

• analytic function ( y = f(x) )

• like a table:

x  1  2  3

y  2  4  6

*y = 2*x

• graphic

All coefficients in function while changing changes behavior of function - as we saw before(in graphics of dependencies)

Function limit

f(x) -> b (infinity moving to b, but not b) when x -> a

lim f(x) = b 

x->a

Properties

• If lim f(x) = b

x -> a 

then only one limit

• lim ( f(x) +/-/*// g(x) ) = lim f(x) +/-/*// lim g(x)

x -> a                           x -> a        x -> a

Delta

delta ( △ ) for x : △x = x1 - x0

△f(x) = f(x+△x)

Function derivative

f’(x) = lim( △f(x) / △x)

           ~△x->0

*~△x->0 because from definition:

△f(x) = f’(x)*△x + q

**where q is insignificant error

Laws

All functions have tables for converting derivative functions to equal regular functions…

https://en.wikibooks.org/wiki/Calculus/Tables_of_Derivatives

Tangent

Derivative’s functions is tangents for their root functions

*in quadratic functions their derivatives have linear form and equal to angle’s tangent to axis x 

**in regular functions tangent’s equation: y - y0 = f’(x0)*(x-x0) 

Geometrical and physical understanding

△x/△y - angle of tangent =>

=> y’(x) speed of changing graphics in certain moment

**acceleration is y’’(x)

Maximums and minimums

*criticals point, extremums:

where f’(x) = 0 

if f’(x) < 0 then root function is decreasing

if f’(x) > 0 then root function is increasing