015. `query 15

 

Question: `q 2.3.32 P=22t^2+52t+10000, t from 1970; find P at t=0,10,20,25 and explain; find dP/dt; evaluate at given t and explain your results.

Your solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Given Solution:

`a dP/dt=44t + 52 (power function rule on each nonconstant term)

 

When t = 0, 10, 20 and 25 you would have P = 10,000, 12,700, 20,000, 25,000 approx.

 

At these values of t we have dP / dt = 52, 492, 932 and 1152 (these are my mental calculations--check them).

 

dP / dt is the rate of change of the population with respect to time t **

 

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

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Question: `q 2.3.48 demand fn p = 50/`sqrt(x), cost .5x+500. Find marginal profit for x=900,1600,2500,3600

 

Explain how you found the marginal profit, and give your results.

 

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Given Solution:

`a x represents the number of items sold. If x items are sold at price p = 50 / `sqrt(x), then revenue is price of item * number sold = 50 / `sqrt(x) * x = 50 `sqrt(x).

 

The profit is revenue - cost = 50 `sqrt(x) - .5 x - 500.

 

The marginal profit is the derivative of the profit function, which is

 

(50 `sqrt(x) - .5 x - 500 ) ' = 25 / `sqrt(x) - .5.

 

Evaluating the marginal profit at x = 900, 1600, 2500 and 3600 we get values

 

.33..., .125, 0 and -.0833... .

 

This shows us that the marginal profit, which is the limiting value of the increase in profit per additional item manufactured, is positive until x = 2500. This means that it is to the advantage of the producer to produce new items when x = 900 and when x = 1600, but that the advantage disappears as soon as x reaches 2500.

 

So 2500 is the best selling price.

 

When x = 3600 production of additional items reduces profits. **

 

STUDENT COMMENT

 

I can see where I messed up. The only thing that I seemed to have gotten rigth is that Marginal Profit is the derivative of the Profit function. I still have problems taking derivatives with the square root function.

INSTRUCTOR RESPONSE

 

sqrt(x) can be written as x^(1/2).

Using the fact that the derivative of x^n is n * x^(n-1) the derivative of x^(1/2) is 1/2 x^(1/2 - 1) = 1/2 x^(-1/2), or 1/2 * (1 / x^(1/2) ) = 1 / (2 x^(1/2)), alternatively 1 / (2 sqrt(x)).

Self-critique (if necessary):

 

 

 

 

 

 

 

 

 

 

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