For Problems 1-3, assume your car has a broken speedometer. In order to find my average velocity of a trip Tucson to Phoenix, I need (a) The total distance of the trip (b) The highway mile markers (c) The time spent traveling (d) How many stops I made during the trip (e) A friend with a stop watch (f) A working odometer (g) None of the above ConcepTest Section 2.1 Question 1

ANSWER (a) and (c). COMMENT: The choices are intentionally vague. This is meant to provide discussion. Your students may select more than one item. ConcepTest Section 2.1 Answer 1 For Problems 1-3, assume your car has a broken speedometer. In order to find my velocity at the instant I hit a speed trap, I need

(a) The total distance of the trip (b) The highway mile markers (c) The time spent traveling (d) How many stops I made during the trip (e) A friend with a stop watch (f) A working odometer (g) None of the above ConcepTest Section 2.1 Question 2 ANSWER (e) and (f). After I pass the speed trap I can watch my odometer as it increases by

0.1 miles while my friend (simultaneously) records the time it took to travel 0.1 miles. COMMENT: Using (e) and (f) as the odometer increases by 0.1 is a good estimate of the velocity at an instant. It may be beneficial to point out that if the odometer measured in hundredths of a mile, then you could compute an even better estimate of the instantaneous velocity. ConcepTest Section 2.1 Answer 2 For Problems 1-3, assume your car has a broken

speedometer. Regarding the speed trap in Problem 2, when should your friend first start the stopwatch? (a) When the driver of an oncoming vehicle warns you of the speed trap ahead by flashing his/her bright headlights (b) When you spot the cop (c) Either scenario (d) Neither scenario ConcepTest Section 2.1 Question 3 ANSWER

(c). You can use an estimation of the average velocity before (or after) you hit the speed trap to estimate your actual velocity. COMMENT: The focus of this discussion should be on how h can be either positive or negative in order to estimate the derivative. ConcepTest Section 2.1 Answer 3 Which graph represents an object that is slowing down where t is time

and D is distance. Assume the units on each axis is the same for all graphs. ConcepTest Section 2.1 Question 4 ANSWER (c) COMMENT: Students have a good idea of what the words slowing down mean, but try to make the connection between the words and slope of the

tangent line at different points. ConcepTest Section 2.1 Answer 4 At approximately which integer value of x does the graph in Figure 2.1 have each of the following slopes? (a) (b) (c) (d) (e)

2 1 1 2 7 ConcepTest Section 2.1 Question 5 ANSWER (a) x = 1 (b) x = 3 (c) x = 2

(d) x = 4 (e) x = 0 COMMENT: An enlarged version of this figure will make it easier to estimate slopes. Follow-up Question. Put the slopes of the tangent lines occurring at x = 0.5, 1.5, 2.5, and 3.5 in order from smallest to largest. Answer. x = 3.5, x = 0.5, x = 1.5, x = 2.5 ConcepTest Section 2.1 Answer 5 For the graph of y = f (x) in Figure 2.2

arrange the following numbers in ascending order (i.e. smallest to largest). (a) Slope of the graph where x = 0.2 (b) Slope of the graph where x = 1.5 (c) Slope of the graph where x = 1.9 (d) Slope of the line connecting the points on the graph where x = 1.5 and x = 1.9 (e) The number 1 ConcepTest Section 2.1 Question 6

ANSWER (c), (d), (b), (e), (a) COMMENT: This is a good question for an elimination question in a classroom quiz session. One purpose for this question is to note the relationships between the slopes at the points x = 1.5 and x = 1.9 and the slope of the corresponding secant line. ConcepTest Section 2.1 Answer 6 Which of the following graphs

represents the position of an object that is speeding up and then slowing down? ConcepTest Section 2.1 Question 7 ANSWER (b). The instantaneous velocity is the slope of the curve at a point. Speeding up and slowing down requires the slope to increase and then decrease. COMMENT: This question is the same as Problem 24 in Section 1

of Chapter 1, but now you can relate the instantaneous velocity to the slope of the tangent line. ConcepTest Section 2.1 Answer 7 Which of the following graphs represents the position of an object that is slowing down? ConcepTest Section 2.1 Question 8 ANSWER

(b). The instantaneous velocity is the slope of the curve at a point. For an object slowing down this means that the slope decreases and becomes closer to zero as time increases. COMMENT: You could have the students describe possible motion scenarios for the other choices. This question is the same as Problem 24 in Section 1 of Chapter 1. ConcepTest Section 2.1 Answer 8