Worksheet 2 Answers (AC 1.1)
pdf
keyboard_arrow_up
School
Cornell University *
*We aren’t endorsed by this school
Course
1110
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
4
Uploaded by CorporalSardine2209
Average Rates of Change (Active Calculus
1.1) Solutions
Learning Objectives:
You should be able to...
Compute an average rate of change, interpret it in the context of a situation, and represent it
graphically.
Describe the difference between an average rate of change and an instantaneous rate of change.
Represent instantaneous rate of change graphically.
Today we are focusing on the following definition:
The average rate of change
of a function
f
from
x
=
a
to
x
=
b
, denoted
AV
[
a,b
]
, is given by
AV
[
a,b
]
=
f
(
b
)
−
f
(
a
)
b
−
a
.
1.
Warmup.
In 2013, Diana Nyad became the first person to swim the 110-mile Florida Straits unaided.
Other swimmers questioned whether her swim was truly unaided; one of the things they looked at was
her speed along the swim.
1
In response, Nyad released GPS data about her 53-hour swim. Here’s a very
small sample of the data:
Time (hours)
0
5
7
15
31
32
40
Distance (miles)
0
6
.
67
8
.
90
18
.
72
54
.
85
58
.
72
82
.
88
Find Nyad’s average speed between 5 and 7 hours into her swim. Also find her average speed between
31 and 32 hours into her swim. Why might these numbers have raised suspicion?
Answer
.
AV
[5
,
7]
= 1
.
115 miles per hour; AV
[31
,
32]
= 3
.
870 miles per hour
This indicates that Nyad was swimming significantly
faster
later in the swim, which is unlikely when
she should have been much more tired.
Nonetheless, they determined that Nyad was “unaided”: the
increase in speed was due to the tides.
2. (Adapted from
Active Calculus 1.1.2
) Consider the position function
s
(
t
) = 64
−
16(
t
−
1)
2
, where
s
(
t
)
is the height of a ball at time
t
.
(a) Compute the average velocity of the ball on each of the following time intervals:
1
See the New York Times article
“Celebration Gives Way to Questions and Doubts Afer a Record Swim”
Adapted from materials created by the Cornell Active Learning in Math team and Harvard University’s Math 1a.
1/
4
Worksheet 2:
Average Rates of Change (Active Calculus
1.1) Solutions
MATH 1110
Time Interval
Average Velocity
[0
.
4
,
0
.
8]
12.8
[0
.
7
,
0
.
8]
8
[0
.
799
,
0
.
8]
6.416
[0
.
8
,
1
.
2]
0
[0
.
8
,
0
.
9]
4.8
[0
.
8
,
0
.
81]
6.24
(b) On the graph of
s
(
t
) below, draw the line that passes through the points
A
= (0
.
4
, s
(0
.
4)) and
B
= (0
.
8
, s
(0
.
8)). What is the meaning of the slope of this line? In light of this meaning, what is a
geometric way to interpret each of the values computed in part
(a)
?
Answer
.
The slope of the secant line between
A
and
B
is the same as AV
[0
.
4
,
0
.
8]
.
Suppose you are now interested in finding the ball’s
instantaneous
velocity at
t
= 0
.
8.
(c) Would you be able to find the ball’s average velocity between
t
= 0
.
8 and
t
= 0
.
8? Why or why
2/
4
Worksheet 2:
Average Rates of Change (Active Calculus
1.1) Solutions
MATH 1110
not?
Answer
.
No, because we cannot divide by 0.
(d) How could you use average velocities to approximate the ball’s instantaneous velocity at
t
= 0
.
8?
Use your idea to give a good estimate of the instantaneous velocity at
t
= 0
.
8.
Answer
.
We can use points successively closer to
t
= 0
.
8, e.g., AV
[0
.
799
,
0
.
8]
= 6
.
416 is an estimate
of the ball’s instantaneous velocity. However, there are many methods for estimating instantaneous
rates of change!
(e) Based on your answer to part
(d)
, how would you visualize instantaneous velocity on the graph in
part
(b)
?
Answer
.
Slope of the tangent line at
t
= 0
.
8.
3. The maximum swimming speed of salmon depends on the water
temperature. Suppose
S
(
T
) is the maximum swimming speed (in
cm/s) of salmon when the water temperature is
T
degrees Celsius;
the graph of
S
is shown at right. What does the average rate of
change of
S
on the interval [15
,
20] represent? What are its units?
Graphically, how would you represent this average rate of change?
5
10
15
20
25
30
5
10
15
20
25
30
Answer
.
How quickly swim speed is changing as temperature changes.
Units are
cm/s
◦
C
.
Graphically, we draw a secant line; its slope is AV
[15
,
20]
.
4.
Average speed vs. average velocity.
A swimmer is swimming a 100 meter long race, which is one lap in
a 50 meter long pool. Let
s
(
t
) be his distance from the starting position
t
seconds after the start of the
race.
(a) Which of the following is a more reasonable graph for
s
(
t
)? Why?
3/
4
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
- Access to all documents
- Unlimited textbook solutions
- 24/7 expert homework help
Worksheet 2:
Average Rates of Change (Active Calculus
1.1) Solutions
MATH 1110
10
20
30
40
50
20
40
60
80
100
t
s
10
20
30
40
50
10
20
30
40
50
t
s
(A)
(B)
Answer
.
B
.
(b) According to the graph you chose, what was the swimmer’s average speed for the race? Average
velocity for the race?
Answer
.
Average speed is 2 m/s; average velocity is 0 m/s.
(c) What was the swimmer’s average speed over the first 20 seconds of the race? Average velocity?
Answer
.
Average speed is
5
2
m/s; average velocity is
5
2
m/s.
(d) What was the swimmer’s average speed over the last 50 m of the race? Average velocity?
Answer
.
Average speed is
5
3
m/s; average velocity is
−
5
3
m/s.
(e) What is the difference between velocity and speed?
Answer
.
Velocity can be negative, but speed is always positive.
4/
4