Exam 13: Indefinite Integrals

Newton's law of cooling (and warming) states that the rate of change of temperature of an object is proportional to the temperature difference between the object and its surroundings, where T is the constant temperature of the surroundings. That is, . Suppose an object at 1°C is placed in a room where the temperature is 22°C. If the temperature of the object is 11°C after 1 hour, how long will it take for the object to reach 16°C? Round your answer to one decimal place. ​

Suppose that the marginal cost (in dollars) for a product is and its fixed cost is $380.00. If the marginal revenue (in dollars) for the product is , find the profit or loss from production and sale of 35 units. ​

If consumption is $4 billion when disposable income is 0, and if the marginal propensity to save is (in billions of dollars), find the national consumption function. If it's necessary, round your calculations to two decimal places. ​

Find the general solution to the differential equation . ​

Suppose the rate of change of the percent P of U.S. households with cable TV can be modeled by , where t is the number of years past 1975. If in 1995 65.7% of households had cable TV, predict the percent of households that will have cable TV in 2025. Round your answer to one decimal place. ​

If , find . ​

The marginal revenue for a new calculator is given by , where x represents hundreds of calculators and revenue is in dollars. Find the total revenue function for these calculators. ​

A function f (x) is given below. Find the graph of the member of the family that satisfies . ​ ​

The rate of change of atmospheric pressure P with respect to the altitude above sea level h is proportional to the pressure. That is, . Suppose that the pressure at sea level is denoted by and at 14,000 ft, the pressure is half of what it is at sea level. Find the pressure, as a percent of at 27,000 ft. ​

Evaluate the integral . ​

Evaluate the integral . ​

A new fast-food firm predicts that the number of franchises for its products will grow at the rate , where t is the number of years. If there is one franchise at present , how many franchises are predicted for 26 years from now? ​

Find the general solution to the differential equation . ​

Use integration to find the general solution to the differential equation . ​

Suppose an oil tanker hits a reef and begins to leak. The efforts of the workers repairing the leak cause the rate at which the oil is leaking to decrease. The oil was leaking at a rate of 39 barrels per hour at the end of the first hour after the accident, and the rate is decreasing at a rate of one barrel per hour. How many barrels of oil will leak in the first 6 hours? ​

Suppose the rate of change of the expected life span l at birth of people born in the United States can be modeled by where t is the number of years past 1920. Use integration and the data point for 2007, to find the function that models the life span. If it's necessary, round your calculations to three decimal places, ​ ​

Evaluate the integral . ​

Which of the following integrals cannot be integrated by using the methods studied so far? ​ (a) ​ (b) ​ (c) ​ (d) ​ (e) ​

The total physical output of a number of machines or workers is called physical productivity and is a function of the number of machines or workers. If is the productivity, is the marginal physical productivity. If the marginal physical productivity for bricklayers is , where P is the number of bricks laid per day and x is the number of bricklayers, find the physical productivity of 5 bricklayers. (Note: when .) ​

If consumption is $6.3 billion when disposable income is 0, and if the marginal propensity to consume is (in billions of dollars), find the national consumption function. If it's necessary, round your calculations to three decimal places. ​