Exam 10: Derivatives

For the function in this problem, approximate near the point 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 three decimal places.

The cost per unit of producing a product is dollars, where x represents the number of units produced per week. If the equilibrium price determined by a competitive market is $200, how many units should the firm produce and sell each week to maximize its profit? ​

For the function in this problem, approximate by computing with . ​ ​

Because the derivative of a function represents both the slope of the tangent to the curve and the instantaneous rate of change of the function, it is possible to use information about one to gain information about the other. In this problem, use the graph of the function given in the figure. Over what interval(s) (a) through (d) is the rate of change of f (x) positive?

Find if . ​

Determine whether the given function is continuous. If is not continuous, identify where it is discontinuous and which condition(s) fails to hold. ​

Complete the table and use it to predict the limit, if it exists. ​ ​ ​

Use analytic methods to find any point of discontinuity for the given function. ​ ​

Find if . ​

Suppose the lowest temperature recorded in Indianapolis, Indiana, was If x is the wind speed in miles per hour and , then the wind chill (in degrees Fahrenheit) for an air temperature of can be approximated by the function . At what rate is the wind chill changing when the wind speed is 50 mph? Round your answer to two decimal places.

Find the derivative of the function. ​ ​

Find the indicated derivative and simplify. ​ for ​

The graph of a company's profit function is shown. For the given problem, use the graph to answer the following question about points A, B, and C. Rank from smallest to largest the amounts of profit received at these points. ​ ​

If . ​

Because the derivative of a function represents both the slope of the tangent to the curve and the instantaneous rate of change of the function, it is possible to use information about one to gain information about the other. In this problem, use the graph of the function given in the figure. ​ ​ At what point(s) A through E is the rate of change of f (x) equal to zero? ​

This problem contains a function and its graph, where . Use the graph to determine, as well as you can, . Check your conclusion by using the function to determine analytically. ​ ​ ​

For the given x-value, use the figure to determine whether the function is continuous or discontinuous at that x-value. ​ ​ ​

The data entry speed (in words per minute) of a data clerk trainee is , where x is the number of hours of training he has had. What is the rate at which his speed is changing and what does this rate mean when he has had 15 hours of training? ​

Suppose the revenue (in dollars) from the sale of x units of a product is given by the equation . Find the marginal revenue when 45 units are sold. Round your answer to the nearest dollar. ​

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