Derivatives (229 problems)

Suppose the demand equation for all-day passes to an amusement park is given by
p = 70 – 0.02q, where p is the price of the pass in dollars and q is the number of people attending at that price.

a. What price corresponds with an attendance of 3000 and what is the total revenue at that price?

b. Write the revenue function as a function of attendance and then determine the attendance that will maximize the revenue.

A manufacturer estimates marginal revenue to be dollars per unit when the level of production is q units. The corresponding marginal cost has been found to be 0.4q dollars per unit. Suppose the manufacturer’s profits is $520 when the level of production is 16 units. What is the manufacturer’s profit when the level of production is 25 units?

A disgruntled employee determines that the company’s profit of the production of x hundred units is dollars. He plans to “get even with the company” by suggesting that it produces the number of items that he knows will minimize their profit. How many units he recommend to the company to purchase?

Money is deposited in a bank offering interest at an annual rate of 6% compounded continuously. Find the percentage rate of change

Suppose that a manufacturer’s profit on x items is dollars.

(a) How many items should be manufactured in order to minimize profit?

(b) What is the maximum profit?

A company’s profit on x item is .

(a) Determine the marginal profit function.

(b) What production level results in a marginal profit of zero.

Solution: (a) The marginal profit is

(b) We set

A new book entitled How to Get Rich in a Hurry costs a bookseller $24 per book. A store can sell 400 copies per month at a price of $32 per book and is considering an increase in the price. Every price increase of $1 reduces sales by twenty books per month. What price should the store set in order to maximize total profits?

Find the rate of change of total profit, when the level of production is 100 items per day.

A manufacturer can supply x items at a price of dollars. How many items should be ordered from this manufacturer in order to minimize the price per item?

Find the solution of the differential equation

given that.

A bicycle producing factory was founded in 1985. The number of bicycles produced t years thereafter is given by:

.

For what year between 1987 [t=2] and 1995 [t=10] (both years included) was the production of bicycles the largest, and for what year was the production the smallest? How many bicycles were produced in each of these years?

Determine the differentialif

After x weeks the number of persons using a new drug was .

(a) At what percentage rate was the use of the drug changing after four weeks?

(b) By how much did the number of persons using this drug actually change during the fourth week?

Assume that the population in year 2008 is given by

where t is measured in years after 2008.

(a) Compute the current population

(b) Compute the population in year 2033

(c) What happens with the population when t is very large.

Express the distance between the point (3, 0) and the point P (x, y) of the parabola y = x² as a function of x.

Consider the following cost function dollars. Determine the maximum marginal cost.