Exam 10: Derivatives

Graph the marginal profit function for the profit function where P(x) is in dollars and x is the number of units. What is the maximum profit? Round your answer to two decimal places. ​

The reaction R to an injection of a drug is related to the dosage x (in milligrams) according to the equation where mg is the maximum dosage. If the rate of reaction with respect to the dosage defines the sensitivity to the drug, find the sensitivity. ​

Find the derivative of the function. ​ ​

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

For the given function, find the instantaneous rate of change of the function at . ​ ​

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

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

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. ​ ​

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

Determine whether the given function is continuous. If it is not, identify where it is discontinuous. You can verify your conclusions by graphing the function with a graphing utility, if one is available. ​ ​

The following table shows the numbers of U.S. cellular subscriberships (in millions) from 1985 to 2002. ​ Source: The CTIA Semi-Annual Wireless Industry Survey Assume that the number of U.S. cellular subscriberships, in millions, can be modeled by the function , where t is the number of years past 1980. Use the second derivative to estimate how fast the rate of increase in subscriberships was changing in 1991 in millions of subscriberships per year. ​

For the function in this problem, approximate by using the numerical derivative feature of a graphing utility. Round your answer to three decimal places. ​ ​

Graph the function with a graphing utility and use it to predict the limit. Check your work either by using the table feature of the graphing utility or by finding the limit algebraically. ​ ​

Find the third derivative. ​ ​

The table gives the yearly U.S. federal budget deficit (as a negative value) or surplus (as a positive value) in billions of dollars from 1990 to 2004. ​ Source: Budget of the United States Government Assume the federal budget deficit (or surplus) can be modeled with the function , where D is in billions of dollars and t is the number of years past 1980. Use the model to find and interpret the instantaneous rate of change of the U.S. federal deficit (or surplus) in 1995. ​

Use the numerical derivative feature of a graphing utility to approximate the given second derivative to two decimal places. Round your answer to two decimal places. ​ ​

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

Differentiate the given function. ​ ​

If and , find . ​

Find the coordinates of points where the graph of has horizontal tangents. As a check, graph and see whether the points you found look as though they have horizontal tangents. ​