Exam 11: Applications of Derivatives

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Over what interval in the domain do average costs decrease? ​

The manufacturer of GRIPPER tires modeled its return to sales from television advertising expenditures in two regions, as follows: Region1: Region2: where and are the sales revenue in millions of dollars, and and are millions of dollars of expenditures for television advertising. How much money will be needed to maximize sales revenue in both districts? Round your answer to the nearest million dollars.

For the given function, use the graph to identify x-values for which . You may use the derivative to check your conclusion. ​

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, find the maximum profit.

A company needs 300 items per year. Production costs are $70 to prepare for a production run and $6 for each item produced. Inventory costs are $0.75 per item per year. Find the number of items that should be produced in each run so that the total costs of production and storage are minimized. ​​

Use the graph shown in the figure and identify points from A through I that satisfy the given condition. ​ and

For the given function and graph, estimate the coordinates of the relative minima by observing the graph, where . ​

A vacationer on an island 8 miles offshore from a point that is 48 miles from a town to which the vacationer must travel occasionally. (See the figure.) The vacationer has a boat capable of traveling 30 mph and can go by auto along the coast at 55 mph. At what distance x from the point on shore closest to the vacationer on the island should the boat be left to minimize the time it takes to get to town? Round your answer to one decimal place. ​ Where , ​

Analytically determine the location(s) of any vertical asymptote(s). Round your answer to two decimal places. ​

A travel agency will plan a tour for groups of size 8 or larger. If the group contains exactly 8 people, the price is $800 per person. However, each person's price is reduced by $10 for each additional person above the 8. If the travel agency incurs a price of $100 per person for the tour, what size group will give the agency the maximum profit? ​​

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. Find the critical values of that lie in the domain of the problem. ​ ​

Use the indicated -values on the graph of to find all the intervals where .

The profit function for a commodity is dollars. Find the maximum profit. Round your answer to the nearest dollar. ​​

For the given function, find the horizontal points of inflection. ​ ​

Use the graph shown in the figure and identify points from A through I that satisfy the given condition. ​ and ​

The following figure shows the graph of revenue function and cost function . At which of the four x-values shown is the slope of the tangent to the revenue curve equal to the slope of the tangent to the cost curve? ​

A function and its graph are given. From the graph estimate where has a relative minimum. ​ ​ ​

Analytically determine the location of any vertical asymptotes. Round your answer to two decimal places.

Use the sign diagram for to determine the largest interval on which f (x) increases, where and .

A graph of is given. Use the graph to determine where f (x) is decreasing. ​ ​