Exam 9: Derivatives

The total physical output P of workers is a function of the number of workers, x. The function is called the physical productivity function. Suppose that the physical productivity of x construction workers is given by , find the marginal physical productivity, . ​

Use the graph of and the given c-value to find . ​ ​ ​

In this problem, cost, revenue, and profit are in dollars and x is the number of units. Suppose that the total revenue function is given by , and that the total cost function is given by . Find at , and explain what it predicts. Round your answer to two decimal places. ​

Find given the equation of the line tangent to the graph of at is . ​

The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable. This problem explores this relationship. Write the equation of the tangent line to the curve at the indicated point. To check your answer, use a graphing utility to graph both the function and its tangent line, and repeatedly zoom in on the point of tangency until the function and the tangent line cannot be distinguished. ​ ​

Find the slope of the tangent at ​ ​

Suppose that the demand for a product depends on the price p according to , where p is in dollars. Find and explain the meaning of the instantaneous rate of change of demand with respect to price when . ​

Total revenue is in dollars and x is the number of units. Suppose that in a monopoly market, the demand function for a product is given by , where x is the number of units and p is the price in dollars. Is more revenue expected from the 101st unit sold or from the 401st unit sold? ​

For the function given, find all x-value(s) where ​ ​

In this problem, cost is in dollars and x is the number of units. Find the marginal cost function for the given cost function. ​ ​

Find . ​

Suppose that the cost function for a commodity is dollars. Find the marginal cost at units and tell what this predicts about the cost of producing 1 additional unit. Round your answer to two decimal places. ​

The daily sales S (in thousands of dollars) attributed to an advertising campaign are given by where t is the number of weeks the campaign runs. What is the rate of change of sales at ? Round your answer to five decimal places. ​

Find the derivative of the given function. Simplify and express the answer using positive exponents only. ​ ​

Use the graph of and the given c-value to find . ​ ​ ​

Find the derivative, but do not simplify your answer. ​ ​

A graph of is shown and a c-value is given. For this problem, use the graph to find . ​ ​ ​

For the function given, find all x-values where f (x) has a local minimum point. ​ ​

Find the derivative of the function. ​ ​

Find the derivative of the given function. Simplify and express the answer using positive exponents only. ​ ​