Exam 9: Derivatives

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

McRobert's TV Shop sells 400 sets per month, at a price of $600 per unit. Market research indicates that the shop can sell one additional set for each $1 it reduces the price. The total revenue (in dollars) is given by , where x is the number of additional sets beyond the original 400. What is the marginal revenue if 300 more units are sold? ​

A certain calling card costs 6.1 cents per minute to make a call, rounded up to the nearest minute. However, when the card is used at a pay phone, there is an additional 61 cent connection fee. If is the charge from a pay phone for a call lasting t minutes, create a table of charges for calls lasting close to 1 minute and use it to find the following limit, if it exists. Round your answer to the nearest cent. ​ ​

If the cost function for a commodity is dollars, find the marginal cost at units, and tell what this predicts about the cost of producing 2 additional units. Round your answer to two decimal places. ​

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. ​ ​

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. ​

For the given function, find the slope of the tangent line at . A graphing utility's numerical derivative feature may be used to check your work. ​ ​

Differentiate the given function. ​ ​

If ​ and , find . ​

Suppose that for selected years from 1970 to 2006, the following table shows the percent of total U.S. workers who are female. ​ Assume these data can be modeled with the function where p(t) is the percent of the U.S. workforce that is female and t is the number of years past 1970. Find the function that models the instantaneous rate of change of the percent of U.S. workers who are female. ​

For the function in this problem, find the slope of the tangent line at the given value. ​ ; ​

Suppose that the revenue function for a certain product is given by , where x is in thousands of units and R is in thousands of dollars. Find the marginal revenue (in thousands of dollars per unit) when 4,000 units are sold. Round your answer to two decimal places. ​

Write the equation of the line tangent to the graph of the given function at the indicated point. As a check, graph both the function and the tangent line you found to see whether it looks correct. ​ ​

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

Total revenue is in dollars and x is the number of units. Suppose that the total revenue function for a commodity is Find the marginal revenue function. ​

Differentiate the given function. ​ ​

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

Suppose that the proportion P of voters who recognize a candidate's name t months after the start of the campaign is given by . Find the rate of change of P when and explain its meaning. ​

Suppose the graph shows a manufacturer's total revenue, in thousands of dollars, from the sale of x cellular telephones to dealers. Decide whether the sale of the 100st cell phone or the 700st brings in more revenue. ​ ​