Exam 9: Derivatives

The graph of a company's total cost function is shown. Does this total cost function represent a manufacturing process that is getting more efficient or less efficient as the production level increases? ​ ​

If what is the rate of change of ? ​

Find the second derivative of . ​

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. Find the total revenue from the sale of 200 units. ​

Suppose the revenue (in dollars) from the sale of x units of a product is given by the equation . Find the marginal revenue when 45 units are sold. Round your answer to the nearest dollar. ​

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

If ​ and , find . ​

Write the equation of the tangent line to the graph of the function at the indicated point. Check the reasonableness of your answer by graphing both the function and the tangent line. ​ at ​

Suppose that the average number of minutes M that it takes a new employee to assemble one unit of a product is given by , where t is the number of days on the job. Is this function continuous for all values of t?

For the function given, find all x-values where ​ ​

Suppose that sales (in dollars) are directly related to an advertising campaign according to , where t is the number of weeks of the campaign. Find the rate of change of sales after 8 weeks and interpret your answer. Round your answer to two decimal places. ​

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 , and explain what this value represents. Round your answer to two decimal places. ​

Graph the function using a window with and . What does the graph indicate about ​ ​

Differentiate the given function. ​ ​

Suppose that the cost C in dollars of obtaining water that contains p percent impurities is given by . Find , if it exists. ​

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

Use analytic methods to evaluate the limit and determine which type of asymptote (if any) it represents. You can verify your conclusion by graphing the function with a graphing utility, if one is available. ​ ​

This problem contains a function and its graph, where . Use the graph to determine, as well as you can, the horizontal asymptote. Check your conclusion by using the function to determine the horizontal asymptote analytically. ​ ​ ​

Given the graph of in the figure, determine all x-values from F, G, H, or I for which the function is differentiable. ​ ​

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