Exam 12: Derivatives Continued

Suppose that the monthly revenue and cost (in dollars) for x units of a product are . At what rate per month is the profit changing if the number of units produced and sold is 100 and is increasing at a rate of 10 units per month? ​

Find for the following equation: ​ ​

Find . ​ ​

If , find y'. ​

Find the derivative of the following function.

Write the equation of the line tangent to the curve at the point .

Assume that x and y are differentiable functions of t. Find using the given values. ​ for . ​

At what x-value does the curve defined by have a horizontal tangent? ​

Find for the following equation: ​ ​

Find any relative maxima and minima. Use a graphing utility to check your results. ​ ​

Find . ​ ​

If , find . ​

For many species of fish, the allometric relationship between the weight W and the length L is approximately where k is a constant. For find the percent rate of change of the weight as a corresponding percent rate of change of the length. ​

The pH of a solution is given by , where is the concentration of hydrogen ions (in gram atoms per liter). What is the rate of change of pH with respect to if ? Round your answer to two decimal places.

Suppose that production of a certain agricultural crop is related to the number of hours of labor x, and the number of acres of the crop y, according to Find the rate of change of the number of hours, with respect to the number of acres. ​

Assume that x and y are differentiable functions of t. Find given that , , and . Round your answer to two decimal places. ​ ​

If , find .

The lengths of the edges of a cube are increasing at a rate of 9 ft/min. At what rate is the surface area changing when the edges are 16 ft long? ​

Suppose the proportion P of people affected by a certain disease is described by , where t is the time in months. Find , the rate at which P grows. ​ ​

Find if . ​