Calculus (414 problems)

Evaluate .

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?

Find the Gini index for the given Lorentz curve

Determine the differentialif

A study indicates thatmonths from now the population of a certain town will be increasing at the rate ofpeople per month. By how much will the population of the town increase over the next 8 months?

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.

Use tabular integration to compute

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?

he administrators of a town estimate that the fraction of people who still be residing in the townyears after they arrive is given by the function

If the current population is 20,000 people and new townspeople arrive at the rate of 500 per year, what will the population be 10 years from now?

Compute the average of the functionover the interval [1, 5].

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

Money is transferred continuously into an account at the constant rate of $1000 per year. The account earns interest at the annual rate of 10% compounded continuously. How much will be in the account at the end of ten years?

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