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28 Cards in this Set
 Front
 Back
L'Hopital's Rule

if the derivative is in intederminate form then use which is 0/0 and infinity/infinity
lim x>a f(x)/g(x) = lim x>a f'(x)/g'(x) 

optimization

minimum or maximum problem


critical point

local minimum or maximum


point of inflection

point where concavity of the graph changes
where f"(x) = 0 

y=e^i(xa)^2

a shifts the graph left or right
divide (xa)^2 by b and b makes the graph wider or narrower 

y=a(1e^bx)

a is the asymptote and b is the stretch or shrink of the graph


y=a(xh)^2 + k

a is stretch or shrink
h shifts right and left b shifts up and down 

global max and min

overall maximum and minimum
highest and lowest y values substitute the cp into the f(x) set equal to zero to find y location on the graph to see which is the highest 

Revenue

price x quantity


Profit

Revenue  cost
usually written as pi 

Marginal Cost

C'(q)


Marginal Revenue

R'(q)


maximum profit

Marginal Cost = Marginal revenue
R'(q) = c'(q) on graph where space between is widest 

Modeling Optimization problems

need two equations
the quantity needed to be optimized is the thing you want the derivative of sketches find the cp and ep to evaluate global max and min 

cosh x

e^x + e^x / 2


sinh x

e^x  e^x / 2


d/dx cosh x

sinh x


d/dx sinh x

cosh x


hyperbolic functions to know

cosh(0) = 1
sinh(0) = 0 cosh(x) = cosh x sinh (x) = sinh x cosh^2 x  sinh^2 x = 1 

Extreme Value theorem

if f is continuous then must have a global max and global min on the closed interval


Mean Value theorem

f'(c) = f(b)  f(a) / ba


delta t

ba/n
n is the number of intervals and b and a is the interval 

left and right increments

in a chart begin with the first value but don't take the last for the right hand value
begin with the second value and end with the last value for the left hand sum average the two to find the most acurate measurement of the integral 

sigma notation

definite integral
number at the top tells the number of intervals number at the bottom tells where to begin 

definite integral

integral from a to b f(t) dt
area under the curve total change distance 

definite integral

F(b)  F(a) = integral from b to a F'(t) dt


average value

1/ba integral from b to a f(x) dx


antiderivative

working backwards from the derivative
