Exam 10: Derivatives

The graphs of a company's total revenue function and total cost function are shown. From the sale of 100 items, 500 items, and 700 items, rank from smallest to largest the amount of profit received. ​ ​

Find for the given function. ​ ​

Use the graph of and the given c-value to find . ​ ​ ​

The total physical output P of workers is a function of the number of workers, x. The function is called the physical productivity function. Suppose that the physical productivity of x construction workers is given by , find the marginal physical productivity, . ​

The percent p of particulate pollution that can be removed from the smokestacks of an industrial plant by spending C dollars is given by . Find the percent of the pollution that could be removed if spending C were allowed to increase without bound. ​

Use the given table to approximate as accurately as you can. ​ ​

Use properties of limits and algebraic methods to find the limit, if it exists. ​ ​

Find the derivative, but do not simplify your answer. ​ ​

For the function in this problem, approximate near by graphing the function on a graphing utility. Then, zoom in near the point until the graph appears straight, pick two points, and find the slope of the line you see. Round your answer to four decimal places. ​ ​

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

When squares of side x inches are cut from the corners of a 35-inch-square piece of cardboard, an open-top box can be formed by folding up the sides. The volume of this box is given by , find the rate of change of volume with respect to the size of the squares. ​

Find given the equation of the line tangent to the graph of at is . ​

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. ​ ​ ​

The table gives the yearly U.S. federal budget deficit (as a negative value) or surplus (as a positive value) in billions of dollars from 1990 to 2004. ​ Source: Budget of the United States Government Assume the federal budget deficit (or surplus) can be modeled with the function , where D is in billions of dollars and t is the number of years past 1980. Use the data to find an average rate of change that approximates the instantaneous rate in 1996. ​

In this problem, cost, revenue, and profit are in dollars and x is the number of units. Suppose that the total revenue function is given by , and that the total cost function is given by . Find the marginal profit function. ​

Find all x-values for which the slope of the tangent is 0 for the given function. ​ ​

If , what is the rate of change of at ? ​

The amount of photosynthesis that takes place in a certain plant depends on the intensity of light x according to the equation , how fast is the rate of change of photosynthesis with respect to the intensity changing when ? when ? ​

Differentiate the given function. ​ ​

Find the derivative of the given function, and use this to find x-values where the function has horizontal tangent lines. ​ ​