Exam 4: Applications of the Derivative

Bubba, a farmer, wants to fence-in a three-sided area next to his barn, using the side of the barn as the fourth side of the rectangle. He also wants to divide this rectangular area into three identical pens (see figure below). If Bubba has 104 feet of available fencing, then each pen will have what dimensions in order to maximize its area? Round dimensions to nearest tenth.

Determine, by hand, all critical numbers of and use the First Derivative Test to classify each as a local minimum, local maximum or neither.

Find the limit.

A point moves on the graph of the equation so that the x-coordinate changes at a rate of 2 units per second. How fast is the y-coordinate changing at the point

Suppose a snowball melts in such a way that it maintains a spherical shape. If the radius is decreasing at a rate of 1.75 cm per hour when the radius is 6 cm, how fast is the volume of the snowball decreasing at that instant?

Determine, by hand, all critical numbers of and use the First Derivative Test to classify each as a local minimum, local maximum or neither.

Find the limit.

Suppose the size of a particular animal's pupils expand and contract depending on the amount of light available. Let be the size in mm of the pupils at light intensity x. Find the smallest possible size of the pupils by finding

Given , determine the absolute extrema on the interval .

Using the critical numbers of , use the Second Derivative Test to determine all local extrema.

Given the graph of , locate the absolute extrema (if they exist) on the interval .

Find a function whose graph has the given asymptotes. and

Find the intervals where the function is increasing and decreasing on the specified interval. Use this information to determine all local extrema. on

A company's revenue for selling x (thousand) items is given by in millions of dollars. Find the value of x (rounded to the nearest item) that maximizes the revenue and find the maximum revenue (rounded to the nearest dollar).

Given the graph of , locate the absolute extrema (if they exist) on the interval .

Find the linear approximation, L(x), to f(x) at x = x₀. f(x) = e^7x, x₀ = 0

The mass of the first x meters of a thin rod is given by the function grams for Find the linear mass density function for the rod and determine whether the composition of the rod is homogeneous or nonhomogeneous.

Given the graph of , locate the absolute extrema (if they exist) on the interval .

Find the indicated limit.

Determine the level of production that minimizes the average cost if the cost function is . Round answer to nearest whole number.