Transcript ConcepTest • Chapter 1

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 25 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 25 in
Section 1 of Chapter 1.
ConcepTest • Section 2.1 • Answer 8