Exam 4: Applications of the Derivative

The Mexican subsidiary of ThermoMaster manufactures an indoor-outdoor thermometer. Management estimates that the profit (in dollars) realizable by the company for the manufacture and sale of x units of thermometers each week is . ​ Find the intervals where the profit function P is increasing and the intervals where P is decreasing. ​

The average speed of a vehicle on a stretch of a route between 6 A.M. and 10 A.M. on a typical weekday is approximated by the function ​ , where is measured in miles per hour and is measured in hours, with corresponding to 6 A.M. ​ At what time of the morning commute is the traffic moving at the slowest rate? What is the average speed of a vehicle at that time? ​

Find the relative maxima and relative minima, if any, of the function. Otherwise, answer none. ​ ​ Relative minima: __________ ​ Relative maxima: __________

Determine where the function is concave downward. ​ ​

An efficiency study conducted for a company showed that the number of devices assembled by the average worker t hr after starting work at 7 A.M. is given by ​ ​ At what time during the morning shift is the average worker performing at peak efficiency?​ At __________ A. M.

Find the absolute maximum value and the absolute minimum value, if any, of the given function. ​ ​

Find the horizontal and vertical asymptotes of the graph. ​ ​

Sketch the graph of the function having the given properties. ​ ​ on on , inflection point at ​

The owner of the Rancho Los Feliz has 3,200 yd of fencing material with which to enclose a rectangular piece of grazing land along the straight portion of a river. ​ If fencing is not required along the river, what are the dimensions of the largest area that he can enclose? ​ ​ What is this area? ​

If the graph of f is concave upward on (a, b), then the graph of - f is concave upward on (a, b). ​

Find the relative maxima and relative minima, if any, of the function. ​ ​

Find the relative maxima and relative minima, if any, of the function. ​ ​

The average cost per disc (in dollars) incurred by Herald Records in pressing x DVDs is given by the average cost function. ​ ​ Find the horizontal asymptote of and the limiting value of the average cost. ​

The concentration (in milligrams/cubic centimeter) of a certain drug in a patient's body t hr after injection is given by ​ interval where is decreasing. ​ When is the concentration of the drug increasing and when is it decreasing? ​

If an open box has a square base and a volume of 500 in.³ and is constructed from a tin sheet, find the dimensions of the box, assuming a minimum amount of material is used in its construction. ​

Find the interval(s) where the function is increasing and the interval(s) where it is decreasing. ​ ​

Show that the function has no relative extrema on .

The number of major crimes committed in the city between 1997 and 2004 is approximated by the function ​ ​where denotes the number of crimes committed in year ( corresponds to 1997). Enraged by the dramatic increase in the crime rate, the citizens with the help of the local police organized "Neighborhood Crime Watch" groups in early 2001 to combat this menace. ​When was the growth in the crime rate maximal? ​

Find the absolute maximum value and the absolute minimum value, if any, of the given function. ​ on ​

A truck gets 400/x mpg when driven at a constant speed of x mph (between 40 and 70 mph). If the price of fuel is $1/gallon and the driver is paid $6/hour, at what speed between 40 and 70 mph is it most economical to drive? Round your answer to the nearest whole number. ​ __________ mph