Wize University Calculus 1 Textbook > Applications of Differentiation for Business & Econ

Assume that the price ppp and the demand qqq of a certain good are related by the equation
q=2e1−p+4p+1q = 2e^{1- p} + \frac{4}{p + 1}q=2e1−p+p+14​
a) Find the price elasticity ε(p)\varepsilon(p)ε(p) as a function of ppp .

b) If p=1p = 1p=1 , does increasing the price slightly increase the revenue?

c) Find the second order Taylor polynomial for the revenue function R(p)R(p)R(p) centered at p=1p = 1p=1 .

d) If the demand is increasing at a rate of 24 units per week when the price is one dollar, find the rate of change of the price.

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The price and quantity of certain commodity are related by the following expression:
2p2+3q2=3602p^2 + 3q^2 = 3602p2+3q2=360

a) Find the price elasticity of demand, and express it as a function of

p

only.

b) If the price of the commodity is decreased by 2% when

p=10

, will the revenue increase or decrease? (increase/decrease)

c) At what price will the revenue be maximized? (

p=

)

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The profit of a company measured in dollars is modelled by the function P(x)=400x−x2P(x)=400x-x^2P(x)=400x−x2, where xx
x is the amount of product sold in a week.
a) Find the profit if 10 products are sold in a week.
b) Find the rate of change of the profit if 100 products are sold per week.

a) Enter the profit in dollars

b) Enter the rate of change in dollars/product sold

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The monthly profit, in hundreds of dollars, of a company is given by P(x)=100x2e−x−300e−xP(x)=100x^2e^{-x}-300e^{-x}P(x)=100x2e−x−300e−x, where xxx is the units of goods sold, measured in hundreds.

If the maximum production capacity of this company is 500 units per month, determine the number of units to produce in order to maximize profit

What is the maximum monthly profit, rounded to the nearest dollar?

How many units of goods should the company product to maximize proft?

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A company manufactures and sells xxx electric drills per month. The monthly cost is defined as C(x)=68000+40xC(x)=68000+40xC(x)=68000+40x, and the price-demand relationship is given as p=190−x20p=190-\dfrac{x}{20}p=190−20x​. Find the production level that results in the maximum profit.

Answer

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The selling price for xxx objects is given by the function p(x)=1005+xp(x)=\dfrac{100}{5+x}p(x)=5+x100​ . Find the marginal revenue.

500(5+x)\dfrac{500}{(5+x)}(5+x)500​

500(5+x)2\dfrac{500}{(5+x)^2}(5+x)2500​

100x(5+x)\dfrac{100x}{(5+x)}(5+x)100x​

100(5+x)2\dfrac{100}{(5+x)^2}(5+x)2100​

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A very cautious individual stores his money both in the bank, and in a safe hidden underneath the floorboards below his bed. He keeps $5000 underneath his bed for a rainy day while investing the other $15000 into a savings account that grows 2% annually that is compounded continuously. Write an expression for the expected value of your total savings after 10 years.

Total savings after 10 years = (do not include units)

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Find the annual interest rate such that a principle of 10,000 dollars will quadruple in 20 years given that it is continuously compounded.

r=

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