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math for CS - Coggle Diagram

- - - - definition
        
        limit of function f as x
        approaches c is L if x can get
        close to but never intersect c
        
        notation
        
        lim f(x) = L
        x->c
        
        really means
        
        lim f(x) = L
        x^- ->c
        lim f(x) = L
        x^+ ->c
        
        3 more items...
        
        a limit only exists if
        there are one-sided limits
        for values less than and greater than c that
        share the same value
        
        if f(x) has
        a vertical asymptote at
        x=c
        
        and values of x increase as they approach c,
        
        1 more item...
        
        and values of x decrease as they approach c,
        
        1 more item...
        
        x-> +inf
        
        right half of function on a graph
        
        x-> -inf
        
        left half of function on a graph
        
        lim f(x) = L
        x->+/-inf
        
        means
        
        1 more item...
        
        lim f(x) = inf
        x^- -> c
        lim f(x) = inf
        x^+ -> c
        
        x increases infinitely but never touches c,
        x decreases infinitely but never touches c
        (goes off graph)
      - properties
        
        given limit of f(x) = L and lim g(x) = M, as both x's approach any constant c,
        
        lim(f(x) +/-/x g(x)) = L +/-/x M
        as x ->c
        
        lim(af(x)) = aL
        x->c
        
        lim(f(x))/lim(g(x)) = L/M
        as x-> if M!= 0
        
        and given lim(h(x)) = N
        as x->L
        
        lim(h(f(x)) = N
        as x->c
      - two special limits
        
        lim[sin(x) / x] = 1
        as x->0
        
        lim[cos(x) -1 / x] = 1
        as x->0
      - squeeze theorum
        
        given
        lim f(x) =L
        x->c
        and lim g(x) =L
        x->c,
        and f(x) <= h(x) <= g(x)
        for all x in some interval containing c,
        then lim h(x) = c
        x->c
      - limit of a sequence
        
        3 step
        
        determine the simplest pattern
        
        put in terms of xn
        
        think of values that x can never be
        
        1/2,2/3,3/4,4/5,5/6,6/7,...
        
        xn = n/n+1
  - - - average rate of change of function
        f over interval starting from x = a
        and ending at x = a + h is
        
        [ f(a+h) - f(a) ] / h
        
        if x = a+h
        
        [ f(x) - f(a) ] / x - a
    - - used when h is made smaller so that is approaches zero
      - formulae
        
        f'(a) = lim{ [f(x+a) - f(a)] / h }as h->0
        f'(a) = lim{ [ f(x) - f(a) ] / x - a } as h->0
      - aka
        
        the derivative of f along an interval
        
        meaning
        
        the slope of the tangent line at this point
        
        meaning
        
        limit of the slopes of the secant lines passing through a fixed point on a curve and other points on that curve that get closer to the fixed point
        
        1 more item...
        
        instantaneous rate of change of a function
        
        meaning
        
        limit of the rates of the average rates of change
        between a fixed point and other points on a curve that get closer to the fixed point
        
        1 more item...
        
        f is differentiable at all points along a certain interval
        
        value of a derivative at a certain point
        
        defined as
        
        2 more items...
      - notation
        
        f'(a)
        
        dx/dy
        
        y'
        
        meaning
        
        y prime
    - - it is continuous at a
    - - it may or may not have a derivative at a
    - - sum rule
        
        derivative of the sum of two functions
        equals the sum of their derivatives
      - difference rule
        
        derivative of the difference of two functions
        equals the difference of their derivatives
        (order matters)
      - quotient rule
        
        d/dx[ (f(x) / g(x) ]
        
        [ f'(x) g(x) - f(x) g'(x) ] / { [ g(x) ] ^2 }
        
        the product of the
        [derivative of numerator times denominator] minus the product of the [derivative of denominator times numerator] all over the [square of the denominator]
      - power rule
        
        derivative of a function raised to a power is equal to that function raised to that power after that power
        has been decremented by 1
        
        remember: rules of power
      - constant rule
        
        the derivative of a constant is always 0
      - constant multiple rule
        
        derivative of a product of a function multiplied by a constant
        equals the product of the functions derivative multiplied by that same constant
      - chain rule
        
        the derivative of a function within a function is the product of the [derivative of the outer function with the inner function substituted in for x] times the derivative of the inner function
        
        y = f(u)
        u = g(x)
        
        dy/dx = (dy/du) (du/dx)
        
        y' = {[f'(g(x))] (g(x)}
        
        remember: nested loops, chain links
      - when a value written in terms of y is differentiated, it is multiplied by dy/dx
        
        used for differentiating implicit functions (implicit differentiation)
    - - logarithms
        
        ln x
        
        1/x
        
        e^x
        
        e^x
      - trig ratios
        
        sin x
        
        cos x
        
        cos x
        
        -sin x
        
        tan x
        
        sec^2(x)
        
        cot x
        
        -csc^2(x)
        
        can be derived from
        derivatives of sin and cos
        using product or quotient rule
        
        sec x
        
        sec(x) tan(x)
        
        csc x
        
        -csc(x)cot(x)
- - - - o/h
      - csc
        
        h/o
        
        1/sin
    - - a/h
      - sec
        
        h/a
        
        1/cos
    - - cot
        
        a/o
        
        1/tan
      - o/a
  - - - 30
        
        π/6
        
        trig values
        
        sin
        
        1/2
        
        cos
        
        √3 / 2
        
        tan
        
        √3 / 3
      - 60
        
        π/3
        
        trig values
        
        sin
        
        √3 / 2
        
        cos
        
        1/ 2
        
        tan
        
        √3
      - 90
        
        π/2
        
        trig values
        
        sin
        
        1
        
        cos
        
        0
        
        tan
        
        undef
      - 120
        
        2π/3
        
        trig values
        
        sin
        
        √3 / 2
        
        cos
        
        -1/ 2
        
        tan
        
        -√3
      - 150
        
        5π/6
        
        trig values
        
        sin
        
        1/2
        
        cos
        
        -√3/2
        
        tan
        
        -√3/3
      - 180
        
        π
        
        trig values
        
        sin
        
        0
        
        cos
        
        -1
        
        tan
        
        0
      - 45
        
        π/4
        
        trig values
        
        sin
        
        √2/ 2
        
        cos
        
        √2 / 2
        
        tan
        
        1
      - 0
        
        0
        
        360
        
        2π
        
        trig values
        
        sin
        
        1 more item...
        
        cos
        
        1 more item...
        
        tan
        
        1 more item...
      - 270
        
        3π/2
        
        trig values
        
        sin
        
        -1
        
        cos
        
        0
        
        tan
        
        undefined
      - 300
        
        5π/3
        
        trig values
        
        sin
        
        -√3 / 2
        
        cos
        
        1/ 2
        
        tan
        
        -√3
      - 315
        
        7π/4
        
        trig values
        
        sin
        
        -√2/ 2
        
        cos
        
        √2 / 2
        
        tan
        
        -1
      - 330
        
        11π/6
        
        trig values
        
        sin
        
        -1/2
        
        cos
        
        √3 / 2
        
        tan
        
        -√3 / 3
      - 240
        
        4π/3
        
        trig values
        
        sin
        
        -√3 / 2
        
        cos
        
        -1/ 2
        
        tan
        
        √3
      - 210
        
        7π/6
        
        trig values
        
        sin
        
        -1/2
        
        cos
        
        -√3/2
        
        tan
        
        √3/3
      - 135
        
        3π/4
        
        trig values
        
        sin
        
        √2/ 2
        
        cos
        
        -√2 / 2
        
        tan
        
        -1
      - 225
        
        5π/4
        
        trig values
        
        sin
        
        -√2/ 2
        
        cos
        
        -√2 / 2
        
        tan
        
        1
      - trig formulas
        
        sin = y/r == y/sqrt(x^2+y^2)
        cos=x/r == x/sqrt(x^2+y^2)
        tan=y/x
        cot=x/y
        sec=r/x == sqrt(x^2+y^2)/x
        csc=r/y == sqrt(x^2+y^2)/y
    - - =
        
        180 degrees
    - - divide by 180, add a pie
        
        multiply by π/180
    - - multiply by 180, drop the pie
        
        multiply by 180/π
  - - - sin^2+cos^2=1
        tan^2+1=sec^2
        1+cot^2=csc^2
        sin(-theta)= -sin(theta)
        cos(-theta)= cos(theta)
        tan(-theta)= -tan(theta)
        sin(theta+2π)=sin(theta)
        cos(theta+2π)=cos(theta)
        tan(theta+π)=tan(theta)
        sin(a+b) = sinacosb + cosasinb
        sin(a-b) = sinacosb - cosasinb
        cos(a+b) = cosacosb - sinasinb
        cos(a-b) = cosacosb + sinasinb
        tan(a+b) = (tana+tanb )/ (1-tanatanb)
        
        sin(2theta)=2sin(theta)cos(theta)
        cos(2theta)=cos^2(theta)-sin^2(theta) = 2cos^2(theta)-1 = 1-2sin^2(theta)
        sin^2(1/2 theta) = [ 1-cos(theta) ]/2
        cos^2(1/2 theta) = [ 1+cos(theta) ]/2
        
        remember: cos = swish
        tan can be derived
        
        tan(a+b) = sin(a+b)/cos(a+b)
        sinacosb + cosasinb / cosacosb - sinasinb
        tana - 1/tanb + tanb - 1/tana
        tana+tanb -1/(tan + tanb)
- - - - formula
        
        change in y / change in x
        
        remember: rise over run
      - notation
        
        m
      - +
        
        slant up to the right
      - -
        
        slant down to the right
      - 0
        
        hoirzontal line (0/x)
      - slope-intercept form
        
        y=mx+b
        given line has y-intercept (0,b)
        and a slope m
      - undefined
        
        vertical line ( x/0)
      - if two lines have the same slope they are
        
        parallel
        
        L1 || L2 if m1 = m2
        
        any two vertical lines are always parallel
        
        a horizontal line and a vertical line are always perpendicular
        
        perpendicular
        
        L1 ⊥L2 if (m1) (m2) = -1
      - point-slope form
        
        y-y1 = m(x-x1)
        given a line passes through point (x,y) and has a slope m
    - - ax^2 + bx + c,
        given a and b !=0
        
        if a = 0
        
        x=c, vertical line
        
        if b = 0,
        
        y=c, horizontal line
    - - x/a + y/b = 1
        given a line has an x intercept (a,0) and y intercept (0,b)
  - - - set of ordered pairs (x,y) where
        there is only one value of y for each value of x
    - - set of all x values in a function
    - - set of all y values in a function
    - - anytime where y is not defined as a function of x,
        anything in a form other than y = f(x)
- - - - right
        
        subtypes
        
        30 60 90
        
        45 45 90
        
        one ang = 90
      - acute
        
        all angs <90
      - obtuse
        
        one ang >90
    - - a,b,c...
        
        sides
      - A,B,C,...
        
        angles
        
        ex: triangle ABC has angles ABC and sides abc
    - - equilateral
        
        3 equal sides
      - isoceles
        
        1 unequal side, 2 equal
      - scalene
        
        no equal sides