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S. T. Tan, Applied Calculus for the Managerial, Life, and Social Sciences, Thomson, 8th edition, 2011

Section 4.6

1. Average Weight of an infant. The  following  graph shows the weight measurements of the average infant from the time of birth (t = 0) through age 2 (t = 24). By computing the slopes of the respective tangent lines, estimate the rate of change of the average infant’s weight when t = 3 and when t = 18. What is the average rate of change in the average infant’s weight over the first year of life?

2. Forestry. The following graph shows the volume of wood produced in a single-species forest. Here f(t) is measured in cubic meters/hectare and t is measured in years. By computing the slopes of the respective tangent lines, estimate the rate at which the wood grown is changing at the beginning of year 10 and at the beginning of year 30.

3. TV-Viewing patterns. The following graph shows the percent of U.S. households watching television during a 24-hr period on a weekday (t = 0 corresponds to 6 a.m.). By computing the slopes of the respective tangent lines, estimate the rate of change of the percent of households watching television at 4 p.m. and 11 p.m.

4. Crop yield. Productivity and yield of cultivated crops are often reduced by insect pests. The following graph shows the relationship between the yield of a certain crop, f(x), as a function of the density of aphids x. (Aphids are small insects that suck plant juices.) Here, f(x) is measured in kilograms/4000 square meters, and x is measured in hundreds of aphids/bean stem. By computing the slopes of the respective tangent lines, estimate the rate of change of the crop yield with respect to the density of aphids when that density is 200 aphids/bean stem and when it is 800 aphids/bean stem.

5. The position of car A and car B, starting out side by side and traveling along a straight road, is given by s = f(t) and s = t(t), respectively, where s is measured in feet and t is measured in seconds (see the accompanying figure). a. Which car is traveling faster at t1? b. What can you say about the speed of the cars at t2? c. Which car is traveling faster at t3? d. What can you say about the positions of the cars at t3?

6. The velocity of car A and car B, starting out side by side and traveling along a straight road, is given by v = f(t) and v = g(t), respectively, where v is measured in feet/second and t is measured in seconds (see the accompanying figure). a. What can you say about the velocity and acceleration of the two cars at t1? (Acceleration is the rate of change of velocity.) b. What can you say about the velocity and acceleration of the two cars at t2?

7. Effect of a bactericide on bacteria In the following figure, f(t) gives the population P1 of a certain bacteria culture at time t after a portion of bactericide A was introduced into the population at t = 0. The graph of t gives the population P2 of a similar bacteria culture at time t after a portion of bactericide B was introduced into the population at t = 0. a. Which population is decreasing faster at t1? b. Which population is decreasing faster at t2? C. Which bactericide is more effective in reducing the population of bacteria in the short run? In the long run?

8. Market share The following figure shows the devastating effect the opening of a new discount department store had on an established department store in a small town. The revenue of the discount store at time t (in months) is given by f(t) million dollars, whereas the revenue of the established department store at time t is given by t(t) million dollars. Answer the following questions by giving the value of t at which the specified event took place. a. The revenue of the established department store is decreasing at the slowest rate. b. The revenue of the established department store is decreasing at the fastest rate. c. The revenue of the discount store first overtakes that of the established store. d. The revenue of the discount store is increasing at the fastest rate.

In Exercises 9–16, use the four-step process to find the slope of the tangent line to the graph of the given function at any point.
9. f(x) = 13

In Exercises 17–22, find the slope of the tangent line to the graph of each function at the given point and determine an equation of the tangent line.
17. f(x) = 2x + 7 at (2, 11)

23. Let f(x) = 2x2 + 1
a. Find the derivative f′ of f. b. Find an equation of the tangent line to the curve at the point (1, 3). C. Sketch the graph of f.

27. Let y = f(x) = x2 + x. a. Find the average rate of change of y with respect to x in the interval from x = 2 to x = 3, from x = 2 to x = 2.5, and from x = 2 to x = 2.1. b. Find the (instantaneous) rate of change of y at x = 2. c. Compare the results obtained in part (a) with that of part (b).

29. Velocity of a car. Suppose the distance s (in feet) covered by a car moving along a straight road after t sec is given by the function s = f(t)  = 2t2 + 48t. a. Calculate the average velocity of the car over the time intervals [20, 21], [20, 20.1], and [20, 20.01]. b. Calculate the (instantaneous) velocity of the car when t = 20. c. Compare the results of part (a) with that of part (b).

30. Velocity of a ball thrown into the air. A ball is thrown straight up with an initial velocity of 128 ft/sec, so that its height (in feet) after t sec is given by s(t) = 128t - 16t2. a. What is the average velocity of the ball over the time intervals [2, 3], [2, 2.5], and [2, 2.1]? b. What is the instantaneous velocity at time t = 2? c. What is the instantaneous velocity at time t = 5? Is the ball rising or falling at this time? d. When will the ball hit the ground?

31. During the construction of a high-rise building, a worker accidentally dropped his portable electric screwdriver from a height of 400 ft. After t sec, the screwdriver had fallen a distance of s = 16t2 ft. a. How long did it take the screwdriver to reach the ground? b. What was the average velocity of the screwdriver between the time it was dropped and the time it hit the ground? c. What was the velocity of the screwdriver at the time it hit the ground?

32. A hot-air balloon rises vertically from the ground so that its height after t sec is h = 1/2t2 + 1/2t ft (0 ≤ t ≤ 60). a. What is the height of the balloon at the end of 40 sec? b. What is the average velocity of the balloon between t = 0 and t = 40? c. What is the velocity of the balloon at the end of 40 sec?

33. At a temperature of 20 °C, the volume V (in liters) of 1.33 g of O2 is related to its pressure p (in atmospheres) by the  formula V = 1/p. a. What is the average rate of change of V with respect to p as p increases from p = 2 to p = 3? b. What is the rate of change of V with respect to p when p = 2?

34. Cost of producing surfboards. The total cost C(x) (in dollars) incurred by Aloha Company in manufacturing x surfboards a day is given by C(x) = -10x2 + 300x + 130 (0 ≤ x ≤ 15) a. Find C ′(x). b. What  is  the  rate  of  change  of  the  total  cost  when  the level of production is ten surfboards a day?

35. Effect of advertising on profit. The quarterly profit (in thousands of dollars) of Cunningham Realty is given by P(x) = -1/3x2 + 7x + 30 (0 ≤ x ≤ 50) where x (in thousands of dollars) is the amount of money Cunningham spends on advertising per quarter. a. Find P ′(x). b. What is the rate of change of Cunningham’s quarterly profit if the amount it spends on advertising is $10,000/quarter (x = 10) and $30,000/quarter (x = 30)?

36. Demand for tents. The demand function for Sportsman 5 × 7 tents is given by p = f(x) = -0.1x2 – x + 40 where p is measured in dollars and x is measured in units of a thousand. a. Find the average rate of change in the unit price of a tent if the quantity demanded is between 5000 and 5050 tents; between 5000 and 5010 tents. b. What is the rate of change of the unit price if the quantity demanded is 5000?

37. A country’s GDP. The gross domestic product (GDP) of a certain country is projected to be N(t)  = t2 + 2t + 50 (0 ≤ t ≤ 5) billion dollars t yr from now.  What will be the rate of change of the country’s GDP 2 yr and 4 yr from now?

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