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cost function is the sum of what?
fixed costs + variable costs
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revenue function is the product of what?
x(p(x))
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profit function is the subtraction of what?
r(x)-C(x)
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marginal =?
derivative of function
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marginal revenue is?
R'(x)
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Marginal profit is?
P'(x)
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elasticity of demand =?
-p(f'p)) / f(p)
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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
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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
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When is demand elastic?
if E(p) > 1
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When is demand unitary?
if E(p) = 1
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When is demand inelastic?
if E(p) < 1
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what happens to revenue if E(p) > 1
revenue is decreasing as prince increases
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what happens to revenue if E(p) <1
revenue increases
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what happens to revenue if E(p) =1
no change
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how do you determine the domain of a function?
logically figure the numbers that will not =0
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how do you find critical numbers of a function?
take derivative & set it equal to zero
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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
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what does: f'(c) = 0 & f''(c)>0 indicate in relative extrema?
f(c) is relative minimum
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what does: f'(c) =0 & f''(c) <0 indicate in relative extrema?
f(c) is relative maximum
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what does: f'(c) = 0 & f''(c) = 0 indicate in relative extrema?
inconclusive, use first derivative test
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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]
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what value does maximum/minimum refer to?
y value
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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
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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.
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what is an inflection point?
where concavity changes, in sign chart when 2nd derivative = 0
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what does the inflection point mean in terms of revenue?
tells when the return on money is the greatest
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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
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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
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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+1000x-120000
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2. find the marginal revenue function & marginal profit function
- marginal revenue: R'(x) =200-.08x
- marginal profit P'(x)=.000006x^2-.04x+1000
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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
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how do you find the price of the 100th, 200th, 300th, etc. item?
P(x) for the 100th - P(x) for the 99th
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