Exam 10: Derivatives

A small business has weekly average costs (in dollars) of , where x is the number of units produced each week. The competitive market price for this business's product is $40 per unit. If production is limited to 443 units per week, find the level of production that yields maximum profit. ​​

Use the graph of to identify at which of the indicated points the derivative changes from positive to negative. ​

Use the sign diagram for to determine the largest interval on which f (x) increases, where and .

For the given function, find ​ ​

Suppose the average costs of a mining operation depend on the number of machines used, and average costs, in dollars, are given by , , where x is the number of machines used. What is the minimum average cost? Round your answer to the nearest dollar. ​

A function and its first and second derivatives are given. Use these to find all points of inflection.

A function and its first and second derivatives are given. Use these to find all critical values.

Analytically determine the location(s) of any vertical asymptote(s). Round your answer to two decimal places. ​

Suppose that the monthly cost in dollars of mining a certain ore is related to the number of pieces of equipment used, according to , , where x is the number of pieces of equipment used. Using how many pieces of equipment will minimize the cost? ​​

A ball thrown into the air from a building 140 ft high travels along a path described by where y is the height of the ball in feet and x is the horizontal distance of the ball from the building in feet. What is the maximum height the ball will reach? Round your answer to one decimal place. ​

In this problem, and its graph are given. Use the graph of to determine where is concave up. ​ ​

Both a function and its derivative are given. Use them to find all critical values.

A function and its graph are given. Use the graph to find the vertical asymptotes, if they exist. Confirm your results analytically. ​ ​

The following figure shows the growth of a population as a function of time. Which of A, B, and C correspond(s) to the point(s) at which the growth rate attains its maximum? ​

If the total cost function for a product is dollars, determine how many units x should be produced to minimize the average cost per unit? ​​

Assume that the total daily cost, in dollars, of producing plastic rafts for swimming pools is given by where x is the number of rafts produced per day, then the average cost per raft produced id given by , for . Find the level of production that minimizes average cost. ​​

A travel agency will plan a tour for groups of size 8 or larger. If the group contains exactly 8 people, the price is $800 per person. However, each person's price is reduced by $10 for each additional person above the 8. If the travel agency incurs a price of $100 per person for the tour, what size group will give the agency the maximum profit? ​​

Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill is given by . Find how long it will be before the oxygen level reaches its minimum. ​​

For the given function and graph, estimate the coordinates of the relative maxima by observing the graph, where . ​ ​ ​

Make a sign diagram for the function and determine all x-values at which relative minima occur. ​