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# Derivatives

See attached file for full problem description.

The world's only manufacturer of left-handed widgets has determined that if q left handed widgets are manufactured and sold per year at price p, then the cost function is C = 8000 + 40q and the manufacturer's revenue function is R = pxq. The manufacturer also knows that the demand function for left-handed widgets is q = 2000 - 25p.

a) Using the demand function, write the cost and revenue functions in terms of price p.
b) Write the profit function in terms of price p and draw its graph.
c) For about what price is the profit largest? How many left-handed widgets should be produced at that price? Draw a graph of profit function.

2. Your friend Herman operates a neighborhood lemonade stand. He asks you to be his financial advisor and wants to know how much lemonade he can make with the \$3.27 he happens to have on hand. The only information he can give you is that once last month he spent \$2 and made 19 glasses of lemonade, and another time he spent \$5 and got 83 glasses of lemonade. You decide to use this data to create a cost function, C(q), giving the cost in dollars of making q glasses of lemonade.

a) You first decide to create a linear cost function based on this data. What is the linear cost function, and how much lemonade can Herman make according to this model?
b) You decide to create an exponential cost function. What is the exponential cost function, and how much lemonade can Herman make according to this model?
c) Herman routinely sells his lemonade for 10 cents per glass. In the case of the linear model, what is his break-even point?
d) In the case of the exponential model, what can you tell Herman about maximizing his profit? Again assume a sales price of 10 cents per glass.

3. At a production level of 2000 for a product, marginal revenue is \$4 per unit and marginal cost is \$3.25 per unit. Do you expect maximum profit to occur at a production level above or below 2000? Explain your answer.

4. The marginal cost of extracting iron ore from a mine already producing 10,000 tons of ore is \$15 per ton.

a) In practical terms, what does this statement mean?
b) Give a clear explanation using a difference quotient of why the given marginal cost equals the derivative C' (10,000), where C' (x) is the price in dollars of extracting x tons of ore.

5. Suppose P(t) is the monthly payment , in dollars, on a mortgage which will take t years to pay off. What are the units of P(r)? What is the practical meaning of P(r)? What is its sign?
6. An economist is interested in how the price of a certain items affects its sale. At a price of \$p, a quantity, q, of the item is sold. If q=f(p), explain the meaning of each of the following statement;
a. f(150)=2000
b. f'(150)=-25
7. A company's revenue from car sales, C (in thousand of dollars), is a function of advertising expendure, a, in thousands of dollars, so C=f(a).
a. What does the company hope is true about the sign of f'?
b. What does the statement f'(100)=2 mean in practical terms? How about f'(100)=0.5?
c. Suppose the company plans to spend about \$100,000 on advertising. If f'(100)=2, should the company spend more or less than \$100,000 on advertising? What if f'(100)=0.5?
8. table gives the number of passenger cars, C=f(t), millions, in the US in the year t.
a. Do f'(t)and f''(t) appear to be positive or negative during the period 1940-1980?
b. Estimate f'(1975). Using units, interpret your answer in terms of passenger cars.
T(years) 1940 1950 1960 1970 1980
C(cars, in millions) 27.5 40.3 61.7 89.3 121.6

9. Let P(t) represent the price of a share of stock of a corporation at time t. What does each of the following statements tell us about signs of the first and second derivatives of P(T)?
1. The price of the stock is rising faster and faster
2. The price of the stock is close to bottoming out
10. IBM-Peru uses second derivatives to assess the relative success of various advertising campaigns. They assume that all campaigns produce some increase in sales. If a graph of sales against time shows a positive second derivative during a new advertising campaign, what does this suggest to IBM management? Why? What does a negative second derivative suggest?

11. Investing \$1000 at an annual interest rate of r% compounded continuously, for 10 years give you a balance of \$B, where B=g(r).Give a financial interpretation of the statements.
a. g(5)≈1649

#### Solution Preview

Please see the attached file

The world's only manufacturer of left-handed widgets has determined that if q left handed widgets are manufactured and sold per year at price p, then the cost function is C = 8000 + 40q and the manufacturer's revenue function is R = pxq. The manufacturer also knows that the demand function for left-handed widgets is q = 2000 - 25p.

a) Using the demand function, write the cost and revenue functions in terms of price p.

Revenue =price x demand
= p x (2000 - 25p)

Cost = 8000 + 40 x (2000 - 25p)

b) Write the profit function in terms of price p and draw its graph.

= Revenue - cost
= 2000p - 25p^2 - 8000 + 80,000 - 1000p
= 1000p - 25p^2 +72,000 ----profit eqn

c) For about what price is the profit largest? How many left-handed widgets should be produced at that price? Draw a graph of profit function.

Taking first derivative of profit eqn we have

1000 - 50p = 0
P = 20
Q = 2000 - 25p = 2000 - 25*20 => 1500 units

(note since second derivative <0,it is maximum (see the attached excel for graph)

2. Your friend Herman operates a neighborhood lemonade stand. He asks you to be his financial advisor and wants to know how much lemonade he can make with the \$3.27 he happens to have on hand. The only information he can give you is that once last month he spent \$2 and made 19 glasses of lemonade, and another time he spent \$5 and got 83 glasses of lemonade. You decide to use this data to create a cost function, C(q), giving the cost in dollars of making q glasses of lemonade.

Please see the attached excel file for these answers a-d

a) You first decide to create a linear ...

#### Solution Summary

The derivatives for left-handed widget manufacturing is determined. The expert uses the demand function to write the cost and revenue functions in terms of price. The profits largest price is determined.

\$2.19