Exam 3: Applications of the Derivative

​Suppose the sales S (in billions of dollars per year)for Proctor & Gamble for the years 1997 through 2002 can be modeled by , where t represents the year.During which year were the sales increasing at the lowest rate? ​

​Determine the dimensions of a rectangular solid (with a square base)with maximum volume if its surface area is 441 square meters. ​

​Sketch the graph of the function using any extrema,intercepts,symmetry,and asymptotes. ​

​Use the graph f ' to sketch the graph of ​f​. ​ ​

​Compare dy and Δy for at ​x​ = 1 with dx = -0.01.Give your answers to four decimal places. ​

​Sketch the graph of the function below.Choose a scale that allows all relative extrema and points of inflection to be identified on the graph. ​ ​ ​

The number of people who donated to a certain organization between 1975 and 1992 can be modeled by the equation donors,where t is the number of years after 1975.Find the inflection point(s)from through ,if any exist. ​

Analytically determine the location(s)of any horizontal asymptote(s). ​ ​ ​

​p is in dollars and q is the number of units.Find the elasticity of the demand function at the price ​

​Find the point on the graph of that is closest to the point (8,0.5).Round your answer to two decimal places. ​

​Find the absolute extrema of the function on the interval ​

​Analyze and sketch a graph of the function ​

Use the graph of to identify at which of the indicated points the derivative changes from negative to positive. ​ ​ ​

Find the limit:​ ​ ​ ​

A power station is on one side of a river that is 0.5 mile wide,and a factory is 6 miles downstream on the other side of the river.It costs $14 per foot to run overland power lines and $20 per foot to run underwater power lines.Estimate the value of x that minimizes the cost.The value of is the length of the overland piece of power line. ​

​Sketch a graph of a function f having the following characteristics. ​ ​ ​

The variable cost for the production of a calculator is $13.75 and the initial investment is $210,000.Use differentials to approximate the change in the cost C for a one-unit increase in production when ,where x is the number of units produced. ​

​Production.Suppose that the total number of units produced by a worker in t hours of an 8-hour shift can be modeled by the production function Find the number of hours before the rate of production is maximized.That is,find the point of diminishing returns. ​

​Find all relative extrema of the function Use the Second-Derivative Test when applicable.

​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. ​