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Calculus & Vectors > All Calculus Units > Flashcards

All Calculus Units Flashcards

1
Q

slope/intercept/average rate of change for linear equation

A

y = mx + b

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2
Q

slope formula

A

m = (y_2 - y_1) / (x_2 - x_1)

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3
Q

standard form for linear equation

A

y = ax + b

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4
Q

increasing graph slope =

A

increasing graph point

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5
Q

decreasing graph slope =

A

decreasing graph point

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6
Q

no change graph slope =

A

0, max/min point, vertex point

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7
Q

undefined graph slope =

A

undefined

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8
Q

equation of line

A

y = m (x-x_1) + y_1

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9
Q

a parabola that is negative slope -> positive slope opens

A

upwards, local minimum point

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10
Q

a parabola that is positive slope -> negative slope opens

A

downwards, local maximum point

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11
Q

vertex form

A

y = a (x-p)^2 + q

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12
Q

vertex form steps

A
  1. factor coefficient from a & b
    y = -2(x^2-6x)-23
  2. take result ‘b’ coefficient, divide by 2 and square result. Take squared result and add and subtract to brackets
    y = -2(x^2-6x+9-9)-23
  3. factor first three terms
    y = -2[(x-3)^2-9]-23
  4. redistribute factored variable
    y = -2(x-3)^2 +18-23
  5. simplify
    y = -2(x-3)^2 -5

vertex = (+3, -5)

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13
Q

quadratic formula

A
  • b +/- root (b^2 -4ac) / 2a

* if quadratic equation -> ax^2 + bx - c = 0

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14
Q

factor theorem

A

if a numbe entered into an equation results in zero it is a factor.

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15
Q

what is used to find other equation factors, after determing 1 factor?

A

long division or synthetic division

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16
Q

method of substitution

A

substituting a linear equation in for a polynomial’s y value to determine what point goes through both lines

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17
Q

secant line

A

straight line that intersects the curve at 2 points

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18
Q

tangent line

A

straight line that touches the curve at one point

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19
Q

secant method

A

determing the instantaneous slope of a curve by calcualting the slope of a point very close to the curve from a tangent line.

(y_2 - Cy+) / (x_2 -(+0.1x_2) = positive slope
(y_2 - Cy-) / (x_2 -(-0.1x_2) = negative slope

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20
Q

horizontal limits

A

lim x-> ∞ (1/x) = 0

lim x-> -∞ (1/x) = 0

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21
Q

vertical limits

A

lim x-> 0^+ (1/x) = ∞

lim x-> 0^- (1/x) = -∞

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22
Q

does a limit always exist?

A

not if it isn’t the same from the right and left
+∞ = -∞
insert x-value into function, result = limit*
*may need to factor or difference of squares
[g(x) = (rootx) -5 / x-25] ->
[g(x) = {(rootx) -5}{(rootx) +5} / {(rootx) -5}{(rootx) +5}] -> g(x) = 1/(root x) + 5

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23
Q

limit formula

A 
lim   f(a+h) - f(a) / h
h-> 0
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24
Q

extrema

A

max/min point on a parabola

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25
Q

global maximum

A

aka absolute maximum

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26
Q

local maximum

A

aka critical point aka stationary point

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27
Q

general formula for the slope of any tangent line along the curve

A 
f(a) = 4a^2
f(a+h) = 4(a+h)^2 -> 4a^2 + 8ah + 4h^2

lim [(4a^2 + 8ah + 4h^2) - (4a^2)]/(a+h) - (a)
h->0
= 8a

28
Q

derivative

A

slope of tangent line at specific point aka instantaneous rate of change
f(x) = 2x^2 + 1 -> f’(x) = 4x

29
Q

polynomial function

A 
f(x) = a_n x^n + a_n-1 x^n-1 + a_2 x^2 + a_0*
*n = natural number
a= constants
30
Q

degree

A

maximum number of directions, maximum number -1 = maximum number of turning points, determined by finite differences

31
Q

finite differences

A

x | y | 1st dif. | 2nd dif. |
0 | 1 | 2-1 = 1 | 7-1=6 |
1 | 2 | 9-2=7 | 13-7=6 |
2 | 9 | 22-9=13 | 19-13=6 |
3 | 22 | 41-22=19 |
4 | 41 |

32
Q

method of first principles

A

f(x) = 2x^2 + 1

  lim   f(a+h) - f(a) / h
h-> 0

lim (2x^2 + 4xh + 2h^2 +1) - (2x^2 +1) / h
h-> 0

= f’(x) = 4x

33
Q

power rule

A

f(x) = x^n , n ≠ 0 -> f’(x) = nx^(n-1)

34
Q

constant rule

A

f(x) = c , c = real # -> f’(x) = 0

35
Q

constant multiple rule

A

f(x) = kg(x) , k = real # -> f’(x) = kg’(x)

36
Q

sum rule

A

F(x) = f(x) + g(x) -> F’(x) = f’(x) + g’(x)

37
Q

difference rule

A

F(x) = f(x) - g(x) -> F’(x) = f’(x) - g’(x)

38
Q

product rule

A

F(x) = f(x)g(x) -> F’(x) = f’(x)g(x) + f(x)g’(x)

39
Q

chain rule

A

F(x) = (f(x))^n -> F’(x) = nf’(x)f(x)^(n-1)

40
Q

quotient rule

A 
F(x) = f(x) / g(x) -> F'(x) = [f'(x)g(x) - f(x)g'(x)] /[g(x)]^2
F'(x) = (LodeeHI-HIdeeLO)/LOxLO
41
Q

curve sketching steps

A
  1. find original function derivative
  2. substitute ‘0’ to find min/max
  3. determine if min or max (left of number = -, min, left of number = +, max)
42
Q

tangent point equation steps

A
  1. find original function derivative
  2. substitute tangent point for x-value
  3. substitute tangent slope into original function
  4. use equation formula (y= m(x-x_1) + y_1
43
Q

polynomial vs. exponential equations

A

polynomial; y= x^a

exponential; y= a^x (a > 0 ≠ 1)

44
Q

decay vs. growth exponential equations

A

both have; no x-intercepts, horizontal asymptote at y=0, y-intercept at (0,1), no max/min points, domain {x element of all real numbers}, range {y element of all real numbers, y>0}

decay; as x approaches positive infinity, y decreases slower
growth; as x approaches negative infinity, y increases faster

45
Q

how to determine polynomial equation vs. exponential equation

A 
polynomial = finite differences
exponential = ratio of 2nd differences

x | y | 1st dif. | 2nd dif. | RATIO*
0 | 1 | 2-1 = 1 | 7-1=6 | 6/6 = 1
1 | 2 | 9-2=7 | 13-7=6 | 6/6 = 1
2 | 9 | 22-9=13 | 19-13=6 |
3 | 22 | 41-22=19 |
4 | 41 |

*aka constant ratio = base (a) of exponential function

46
Q

method of first principles to determine general derivative exponential formula

A

f(x) = 2^x

k = lim (a^h -1)/h
h->0
k = 2^0.0001 -1 / 0.0001 = 0.693

f’(x) = 0.693(2^x)

k = f’(x) / f(x)

47
Q

inverse of an exponential function

A

logarithmic function

y = e^x y = log_e x

48
Q

natural logarithm

A

y’=lnx y=log_e x

49
Q

periodic vs. sinusoidal functions

A

repeats a pattern but can be in pieces, periodic; tanx, cscx, secx, cotx (ripples from a pepple in water)
repeats a pattern and never ends periodic and sinusoidal; cosx, sinx (rope on end of pole up and down)

50
Q

chain rule for exponential functions

A 
f(x) = a^(g(x)) -> f'(x) = a^x (lna)(g'(x))
f(x) = 3^(x^2-5x) -> f'(x) = 3^(x^2-5x)(ln3)(2x-5)
y = e^x -> y' = e^x (lne)(y'(x)) -> y'(x) = e^x(y'(x))
y = e^(3x^2-8x) -> y' = (6x-8)e^(3x^2-8x)
51
Q

derivatives of sinx and cosx

A 
f(x) = sinx -> f'(x) = cosx
f(x) = cosx -> f'(x) = -sinx
52
Q

sin x vs. cos x

A

both have; local max=absolute max at y=1, local min=absolute min at y=-1, zeros (x-intercepts) occur at every π unit, one period every 2π units, domain {x element of all real numbers}, range {y element of all real numbers, -1= y, y= 1}

sinx; y-intercept at (0,0)

cosx; y-intercept at (0,1)

53
Q

rational functions

A

F(x) = f(x) / g(x) , g(x) ≠ 0

54
Q

chain rule for sinusoidal functions

A

f(x) = SIN(2x)

  1. derivative of type, leave angle -> f’(x) = cos(2x)
  2. multiply by derivative of angle -> f’(x) = (cos(2x))(2)
  3. simplify -> f’(x) = 2cos(2x)
55
Q

second derivative

A

f(x) = 2x^3 - 6x^2 + x + 14
f’(x) = 6x^2 -12x + 1
f’‘(x) = 12x - 12
-determines inflection point
-determines if other points beside inflection point are concave up or down
-2 degrees less than original function i.e. f(x) = x^3 f’‘(x) = x^1

56
Q

inflection point

A
  • where the slope is zero because the graph changes from positive or negative to the opposite (negative or positive)
  • enter 0 into second derivative to determine inflection point(s)
  • can have more than one
57
Q

original function’s specification for first and second derivative

A 
f(x) = increasing -> f'(x) = positive
f(x) = decreasing -> f'(x) = negative

f(x) = 3rd degree (x^3) -> f’(x) = 2nd degree (x^2) -> f’‘(x) = 1st degree (x^1 aka x)

58
Q

equation of function

A

roots (aka x-intercepts) on a graph
y-intercepts on a graph
y-values of max/min/inflection points

59
Q

Method A

A

a

60
Q

Method B

A

b

61
Q

Even function vs odd function

A

c

62
Q

Standard form to determine equations

A

d

63
Q

rational function

A
  • find asymptotes on graph
  • locate x & y intercepts
  • determine y values of max/min/inflection points
  • create table of values
64
Q

1st derivatives of rational functions

A
  • locate max/min point

- intervals of increase/decrease

65
Q

2nd derivatives of rational functions

A
  • locate any inflection points
  • find intervals of concavity
  • determine if critical number is max/min point

Calculus & Vectors (1 decks)

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