Exam 3 Review MATH 140: David Gerberry

ConcepTest • Section 4.1 • Question 5 Given that f′(x) is continuous everywhere and changes from negative to positive at x = a, which of the following statements must be true? (a) a is a critical point of f(x) (b) f(a) is a local maximum (c) f(a) is a local minimum (d) f′(a) is a local maximum (e) f′(a) is a local minimum ConcepTest • Section 4.1 • Question 5 2

ConcepTest • Section 4.1 • Answer 5 (a) and (c). a is a critical point and f(a) is a local minimum. COMMENT: Follow-up Question. What additional information would you need to determine whether f(a) is also a global minimum? Answer. You need to know the values of the function at all local minima. In addition, you need to know the values at the endpoints (if any), or what happens to f(x) as x → ±∞ ConcepTest • Section 4.1 • Answer 5 3

ConcepTest • Section 4.1 • Question 4 If the graph in Figure 4.2 is that of f′(x), which of the following statements is true concerning the function f? (a) The derivative is zero at two values of x, both being local maxima. (b) The derivative is zero at two values of x, one is a local maximum while the other is a local minimum. (c) The derivative is zero at two values of x, one is a local maximum on the interval while the other is neither a local maximum nor a minimum. (d) The derivative is zero at two values of x, one is a local minimum on the interval while the other is neither a local maximum nor a minimum. (e) The derivative is zero only at one value of x where it is a local minimum. Figure 4.2 ConcepTest • Section 4.1 • Question 4

ConcepTest • Section 4.1 • Answer 4 (c). When x ≈ 0.6, the derivative is positive to the left and negative to the right of that point. This gives a local maximum. When x ≈ 3.1, the graph of the derivative is below the axis (negative) on both sides of this point. This is neither a local minimum nor a maximum. COMMENT: You could sketch a different graph and ask the same question. ConcepTest • Section 4.1 • Answer 4 5

ConcepTest • Section 4.1 • Question 1 Concerning the graph of the function in Figure 4.1, which of the following statements is true? (a) The derivative is zero at two values of x, both being local maxima. (b) The derivative is zero at two values of x, one is a local maximum while the other is a local minimum. (c) The derivative is zero at two values of x, one is a local maximum on the interval while the other is neither a local maximum nor a minimum. (d) The derivative is zero at two values of x, one is a local minimum on the interval while the other is neither a local maximum nor a minimum. (e) The derivative is zero only at one value of x where it is a local minimum. Figure 4.1 ConcepTest • Section 4.1 • Question 1

ConcepTest • Section 4.1 • Answer 1 (b). The derivative is zero where it has a horizontal tangent, having a local maximum if it is concave down there, a local minimum if it is concave up. COMMENT: You could sketch a different graph and ask the same question. ConcepTest • Section 4.1 • Answer 1

ConcepTest • Section 4.2 • Question 1 The function y = f(x) is shown in Figure 4.5. How many inflection points does this function have on the interval shown? (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 Figure 4.5 ConcepTest • Section 4.2 • Question 1

ConcepTest • Section 4.2 • Answer 1 (d) The graph is concave down on the far left, then concave up until the origin, then concave down again, then concave up on the far right. The graph changes concavity three times, so there are three inflection points. ConcepTest • Section 4.2 • Answer 1

ConcepTest • Section 4.2 • Question 5 For the first three months of an exercise program, Joan’s muscle mass increased, but at a slower and slower rate. Then there was an inflection point in her muscle mass, as a function of time. What happened after the first three months? (a) Her muscle mass began to decrease. (b) Her muscle mass reached its maximum and remained constant afterward. (c) Her muscle mass continued to increase, but now at a faster and faster rate. (d) The rate of change of her muscle mass changed from positive to negative. ConcepTest • Section 4.2 • Question 5 10

ConcepTest • Section 4.2 • Answer 5 (c) For the first three months, the rate of change of muscle mass was positive and decreasing, so that the graph of muscle mass as a function of time was increasing and concave down. After the inflection point, the rate of change of muscle mass was increasing, so the rate never became negative. After the inflection point the graph of muscle mass was increasing and concave up. COMMENT: Ask the students to sketch a possible graph of Joan’s muscle mass as a function of time. ConcepTest • Section 4.2 • Answer 5 11

ConcepTest • Section 4.2 • Question 2 The graph of the derivative y = f′(x) is shown in Figure 4.6. The function f(x) has an inflection point at approximately (a) x = 0 (b) x = 1 (c) x = 3 (d) x = 5 (e) None of the above Figure 4.6 ConcepTest • Section 4.2 • Question 2

ConcepTest • Section 4.2 • Answer 2 (c) x = 3. The inflection points of f(x) are the critical points of f′(x) which are local extrema. ConcepTest • Section 4.2 • Answer 2

ConcepTest • Section 5.1 • Question 2 Figure 5.1 shows the velocities of two cyclists traveling in the same direction. If initially the two cyclists are alongside each other, when does Cyclist 2 overtake Cyclist 1? (a) Between 0.75 and 1.25 minutes (b) Between 1.25 and 1.75 minutes (c) Between 1.75 and 2.25 minutes Figure 5.1 ConcepTest • Section 5.1 • Question 2

ConcepTest • Section 5.1 • Answer 2 (b). Between 1.25 and 1.75 minutes, because the area under the two curves is about equal at some time during this interval. COMMENT: You could ask the students to answer the question if either, or both, of the statements “traveling in the same direction” or “If initially the two cyclists were alongside each other” are removed. You could also use this question to review the concepts of derivative. Follow-up Question. When the cyclists pass each other, which one is accelerating more? Answer. Cyclist 2 is accelerating more because between 1.25 and 1.75 minutes the slope of the velocity graph at every point in that interval is steeper than the slope corresponding to Cyclist 1. ConcepTest • Section 5.1 • Answer 2

ConcepTest • Section 5.1 • Question 1 A drug leaves the body at the rate of 20 ng/min at the start of a 10-minute period, and the rate of decrease continually declines until it is 8 ng/min at the end of the time period. The total decrease in the amount of the drug in the body during this time period might be: (a) 150 ng (b) 12 ng (c) 200 ng/min (d) 300 ng (e) 15 ng/min ConcepTest • Section 5.1 • Question 1

ConcepTest • Section 5.1 • Answer 1 (a). An upper estimate for the total decrease is (20 ng/min)·(10 min) = 200 ng (nanograms), and a lower estimate is (8 ng/min)·(10 min) = 80 ng. The total decrease must be between 80 ng and 200 ng. Notice also that the units of the total change are ng, not the units of the rate of change, ng/min. ConcepTest • Section 5.1 • Answer 1

ConcepTest • Section 5.2 • Question 1 Using Figure 5.2, which of the following is the best estimate of (a) 13 (b) 30 (c) 3 (d) 18 (e) 9 Figure 5.2 ConcepTest • Section 5.2 • Question 1

ConcepTest • Section 5.2 • Answer 1 (d). The integral gives the area under the curve from t = 0 to t = 9. Since the highest point on the curve on this interval is at 3, the area of the entire rectangle is 9·3 = 27. Since the curve fits within this rectangle, the value of the integral must be less than 27. The area under the curve is more than half of the area of the rectangle, so the value of the integral is greater than 13.5. The answer must be (d), 18. ConcepTest • Section 5.2 • Answer 1

ConcepTest • Section 5.3 • Question 1 Which of the following gives the area of the shaded region in Figure 5.7? Figure 5.7 ConcepTest • Section 5.3 • Question 1

ConcepTest • Section 5.3 • Answer 1 The shaded area is under the graph of the function f(t) = 2t3 − 25 + 3, not under f′(t) = 6t2 − 2, therefore (b) and (d) are incorrect. The shaded region lies between t = −1 and t = 12 so the correct answer is (c). ConcepTest • Section 5.3 • Answer 1

ConcepTest • Section 5.3 • Question 2 Use the graph in Figure 5.8 to determine which of the following is equal to (a) 19 (b) 11 (c) -19 (d) -11 Figure 5.8 ConcepTest • Section 5.3 • Question 2

ConcepTest • Section 5.3 • Answer 2 We can compute as the area of two triangles, one of which has a negative contribution: So the answer is (b). ConcepTest • Section 5.3 • Answer 2

ConcepTest • Section 5.3 • Question 3 The graphs of five functions are shown in Figure 5.9. The scales on the axes are the same for all five. Which function has the value of its integral closest to zero? Figure 5.9 ConcepTest • Section 5.3 • Question 3

ConcepTest • Section 5.3 • Answer 3 (c). Recall that area above the x-axis counts positively and area below the x-axis counts negatively in an integral. Therefore, the integrals for the functions in (a), (d), and (e) are all positive, and the integral for (b) is negative. In (c), it appears that the area above the axis is approximately equal to the area below the axis, so the integral for the function in (c) will be very close to zero. ConcepTest • Section 5.3 • Answer 3

ConcepTest • Section 5.4 • Question 2 The function f(t) gives the number of gallons of fuel used per minute by a jet plane t minutes into R a flight. The integral represents: (a) The average fuel consumption during the first half-hour of the trip. (b) The average fuel consumption during any 30-minute period on the trip. (c) The total fuel consumption during the first 30 minutes of the trip. (d) The total time it takes to use up the first 30 gallons of fuel. (e) The average rate of fuel consumption during the time it takes to use up the first 30 gallons. t ConcepTest • Section 5.4 • Question 2

ConcepTest • Section 5.4 • Answer 2 (c). Since f(t) is the rate at which fuel is used, the integral gives the total quantity of fuel used during a particular time interval, here 0 to 30 minutes. ConcepTest • Section 5.4 • Answer 2

ConcepTest • Section 5.4 • Question 3 The rate of change of the amount of water in a tank is given by f(t) gallons per hour. The tank holds 30 gallons of water at noon, denoted by t = 0. If , which of the following statements are true? (a) At 10 am there are 44 gallons of water in the tank. (b) At 10 am there are 16 gallons of water in the tank. (c) At 2 pm there are 44 gallons of water in the tank. (d) At 2 pm there are 16 gallons of water in the tank. (e) 14 hours after noon there are 2 gallons of water in the tank. (f) 14 hours before noon there were 2 gallons of water in the tank. ConcepTest • Section 5.4 • Question 3

ConcepTest • Section 5.4 • Answer 3 The units of f are gallons per hour and time is measured in hours, so the units of the definite integral are gallons. Therefore the units of −14 are gallons, and (e) and (f) are incorrect. The definite integral measures the net change in the amount of water in the tank between t = 0, noon, and t = 2, 2pm so (a) and (b) are incorrect. Finally, the negative sign indicates that the net change is a loss of water, so (d) is the correct answer. ConcepTest • Section 5.4 • Answer 3