All Calculus Units Flashcards
slope/intercept/average rate of change for linear equation
y = mx + b
slope formula
m = (y_2 - y_1) / (x_2 - x_1)
standard form for linear equation
y = ax + b
increasing graph slope =
increasing graph point
decreasing graph slope =
decreasing graph point
no change graph slope =
0, max/min point, vertex point
undefined graph slope =
undefined
equation of line
y = m (x-x_1) + y_1
a parabola that is negative slope -> positive slope opens
upwards, local minimum point
a parabola that is positive slope -> negative slope opens
downwards, local maximum point
vertex form
y = a (x-p)^2 + q
vertex form steps
- factor coefficient from a & b
y = -2(x^2-6x)-23 - 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 - factor first three terms
y = -2[(x-3)^2-9]-23 - redistribute factored variable
y = -2(x-3)^2 +18-23 - simplify
y = -2(x-3)^2 -5
vertex = (+3, -5)
quadratic formula
- b +/- root (b^2 -4ac) / 2a
* if quadratic equation -> ax^2 + bx - c = 0
factor theorem
if a numbe entered into an equation results in zero it is a factor.
what is used to find other equation factors, after determing 1 factor?
long division or synthetic division
method of substitution
substituting a linear equation in for a polynomial’s y value to determine what point goes through both lines
secant line
straight line that intersects the curve at 2 points
tangent line
straight line that touches the curve at one point
secant method
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
horizontal limits
lim x-> ∞ (1/x) = 0
lim x-> -∞ (1/x) = 0
vertical limits
lim x-> 0^+ (1/x) = ∞
lim x-> 0^- (1/x) = -∞
does a limit always exist?
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
limit formula
lim f(a+h) - f(a) / h h-> 0
extrema
max/min point on a parabola
global maximum
aka absolute maximum
local maximum
aka critical point aka stationary point
general formula for the slope of any tangent line along the curve
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
derivative
slope of tangent line at specific point aka instantaneous rate of change
f(x) = 2x^2 + 1 -> f’(x) = 4x
polynomial function
f(x) = a_n x^n + a_n-1 x^n-1 + a_2 x^2 + a_0* *n = natural number a= constants
degree
maximum number of directions, maximum number -1 = maximum number of turning points, determined by finite differences
finite differences
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 |
method of first principles
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
power rule
f(x) = x^n , n ≠ 0 -> f’(x) = nx^(n-1)
constant rule
f(x) = c , c = real # -> f’(x) = 0
constant multiple rule
f(x) = kg(x) , k = real # -> f’(x) = kg’(x)
sum rule
F(x) = f(x) + g(x) -> F’(x) = f’(x) + g’(x)
difference rule
F(x) = f(x) - g(x) -> F’(x) = f’(x) - g’(x)
product rule
F(x) = f(x)g(x) -> F’(x) = f’(x)g(x) + f(x)g’(x)
chain rule
F(x) = (f(x))^n -> F’(x) = nf’(x)f(x)^(n-1)
quotient rule
F(x) = f(x) / g(x) -> F'(x) = [f'(x)g(x) - f(x)g'(x)] /[g(x)]^2 F'(x) = (LodeeHI-HIdeeLO)/LOxLO
curve sketching steps
- find original function derivative
- substitute ‘0’ to find min/max
- determine if min or max (left of number = -, min, left of number = +, max)
tangent point equation steps
- find original function derivative
- substitute tangent point for x-value
- substitute tangent slope into original function
- use equation formula (y= m(x-x_1) + y_1
polynomial vs. exponential equations
polynomial; y= x^a
exponential; y= a^x (a > 0 ≠ 1)
decay vs. growth exponential equations
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
how to determine polynomial equation vs. exponential equation
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
method of first principles to determine general derivative exponential formula
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)
inverse of an exponential function
logarithmic function
y = e^x y = log_e x
natural logarithm
y’=lnx y=log_e x
periodic vs. sinusoidal functions
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)
chain rule for exponential functions
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)
derivatives of sinx and cosx
f(x) = sinx -> f'(x) = cosx f(x) = cosx -> f'(x) = -sinx
sin x vs. cos x
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)
rational functions
F(x) = f(x) / g(x) , g(x) ≠ 0
chain rule for sinusoidal functions
f(x) = SIN(2x)
- derivative of type, leave angle -> f’(x) = cos(2x)
- multiply by derivative of angle -> f’(x) = (cos(2x))(2)
- simplify -> f’(x) = 2cos(2x)
second derivative
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
inflection point
- 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
original function’s specification for first and second derivative
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)
equation of function
roots (aka x-intercepts) on a graph
y-intercepts on a graph
y-values of max/min/inflection points
Method A
a
Method B
b
Even function vs odd function
c
Standard form to determine equations
d
rational function
- find asymptotes on graph
- locate x & y intercepts
- determine y values of max/min/inflection points
- create table of values
1st derivatives of rational functions
- locate max/min point
- intervals of increase/decrease
2nd derivatives of rational functions
- locate any inflection points
- find intervals of concavity
- determine if critical number is max/min point
