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cost function is the sum of what?
fixed costs + variable costs

revenue function is the product of what?
x(p(x))

profit function is the subtraction of what?
r(x)C(x)

marginal =?
derivative of function


marginal revenue is?
R'(x)

Marginal profit is?
P'(x)

elasticity of demand =?
p(f'p)) / f(p)

Elasticity of Demand Ex.:
p= .02x+400
 1. find x > f(p)=50p+20000
 2. find f'(p) > 50
 3. input values in elasticity > (p)(50) / 50p+20000

for E(p) = 50p / 50p+20000 calculate 100 & explain what it means
 1. 50(100) / 50(100)+20000 =.33333
 2. Means when price is $100 a 1% increase in price will cause .33% decrease in sales

When is demand elastic?
if E(p) > 1

When is demand unitary?
if E(p) = 1

When is demand inelastic?
if E(p) < 1

what happens to revenue if E(p) > 1
revenue is decreasing as prince increases

what happens to revenue if E(p) <1
revenue increases

what happens to revenue if E(p) =1
no change

how do you determine the domain of a function?
logically figure the numbers that will not =0

how do you find critical numbers of a function?
take derivative & set it equal to zero

2nd derivative test
take the 2nd derivative of function & set it equal to zero to get relative max & min, then plug max, min & domain into original function

what does: f'(c) = 0 & f''(c)>0 indicate in relative extrema?
f(c) is relative minimum

what does: f'(c) =0 & f''(c) <0 indicate in relative extrema?
f(c) is relative maximum

what does: f'(c) = 0 & f''(c) = 0 indicate in relative extrema?
inconclusive, use first derivative test

extreme value theorem:
if a function f is continuous on a closed interval [a,b] then f has both an absolute max & min value on [a,b]

what value does maximum/minimum refer to?
y value

how do you make first derivative sign chart?
 1. find critical # by solving for DNE or 0
 2. have intervals before & after critical #
 3. pick number in interval & plug into original function, indicate whether + or  which = increasing or decreasing

how do you make second derivative sign chart?
 1. find critical # by solving for DNe or 0
 2. have intervals before & after critical #
 3. pick # in interval & plug into 2nd derivative, indicate whether + or  which = happy or sad face.

what is an inflection point?
where concavity changes, in sign chart when 2nd derivative = 0

what does the inflection point mean in terms of revenue?
tells when the return on money is the greatest

how do you find absolute extrema on open interval w/ 1 critical #?
 1. find the critical # in (a,b)
 2. Use 2nd derivative to see if the ritical # gives a relative max or min > f''(c) >0 = abs min & f''(c) <0 = abs max

what are the steps to solving geometric optimization?
 1. draw pic
 2. assign variables
 3. equation to relate variables
 4. function in 1 variable & interval

weekly demand for photocopying machine is: p=2000.04x on (0<x<50000) p is wholesale unit price in $ & x is quantity demanded. Weekly total cost us: C(x)=.000002x^3.02x^2+100x+120000 where C(x) is total cost incurrend in producing x units
1. find the revenue function & profit function
 1. R(x)= p(x) >(2000.04x)x >2000x.04x^2
 P(x)= R(x)C(x) > 2000x.04x^2(.000002x^3.02x^2+1000x+120000) > .000002x^3.02x^2+1000x120000

2. find the marginal revenue function & marginal profit function
 marginal revenue: R'(x) =200.08x
 marginal profit P'(x)=.000006x^2.04x+1000

3. what is the total profit for 5000 copiers? what is the profit from the 5000th copier?
 P(5000) = $4,130,00
 P(5000) P(4999) = $650.05

how do you find the price of the 100th, 200th, 300th, etc. item?
P(x) for the 100th  P(x) for the 99th

