# Calculus used to find marginal profit and critical values

Please see the attachment for the full question.

1. Given the graph of the Marginal Profit function, explain what points A and B imply about the profit function.

2. Given the graph of P(x), locate the point of diminishing returns.

3. Suppose that the total number of units produced by a worker in t hours of an 8 hour day can be modeled by
Use calculus to find the point of diminishing returns. Explain what this point means in terms of units produced by a worker.

4. The weekly sales S of a product during an advertising campaign are given by
a. Use calculus to find the critical values that lie in the domain of the problem.
b. Over what interval in the domain do the sales decrease?
c. Over what interval in the domain do the sales increase?
d. When will the sales be a maximum?
e. What will the maximum sales be?
f. Graph the function and label the critical values on the graph. (You may want to this after part a.)

5. For the cost function given by
a. Find the average cost function.
b. Use calculus to find the minimum of the average cost function.
c. Graph the average cost function.

6. A manufacturer estimates that x units of its product can be produced at a total cost of
C(x) = 120x + x2 + 35
and the manufacturers' total revenue is
R(x) = 168x  0.2x2.
Use calculus to find the profit equation and determine the level of production that will maximize profit. Find the maximum profit.

7. A company needs 400,000 items per year. Production costs are \$250 to prepare for a production run and \$7 for each item produced. It also costs are \$1.75 to store an item for up to one year. Find the number of items that should be produced in each run so that the total costs of production and storage are minimized.

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