Exam 10: Derivatives

A rectangular area is to be enclosed and divided into thirds. The family has $800 to spend for the fencing material. The outside fence costs $18 per running foot installed, and the dividers cost $22 per running foot installed. What are the dimensions that will maximize the area enclosed? Round your answer to two decimal places. ​​

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Can 100% of the pollution be removed? ​

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

Find the x-value at which the absolute minimum of f (x) occurs on the interval [a, b]. ,

A firm can produce 100 units per week. If its total cost function is dollars, and its total revenue function is dollars, how many units x should it produce to maximize its 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. ​​

Find any horizontal asymptotes for the given function.

A function and its first and second derivatives are given. Use these to find the relative maxima.

For the given function, classify the critical points as relative maxima, relative minima, or points of inflection. In each case, you may check your conclusions with a graphing utility.

A function and its first and second derivatives are given. Use these to find all relative minima. ​

A function and its graph are given. From the graph, estimate where . ​ ​

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

The graph shows a total cost function. Determine the level of production at which average cost is minimized. ​

A box with an open top and a square base is to be constructed to contain 6,912 cubic inches. Find the dimensions that will require the minimum amount of material to construct the box. ​​

The profit from a grove of orange trees is given by dollars, where x is the number of orange trees per acre. How many trees per acre will maximize the profit? ​​

The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by . Find any C values within the domain of the problem at which the rate of change of p with respect to C does not exist. ​

Use the sign diagram for to determine x-values at which relative minima occur, where and .

The following figure shows the growth of a population as a function of time. ​ If P represents the population and t represents the time, write a mathematical symbol that represents the rate of change (growth rate) of the population with respect to time.

For the given function, find the critical points.

Find all points of inflection of the given function. Round your answer to two decimal places. ​