Worksheet 10 Answers (AC 1
.pdf
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School
Cornell University *
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Course
1110
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
4
Uploaded by CorporalSardine2209
Linear Approximation (Active Calculus
1.8) Solutions
Learning Objectives:
You should be able to...
Use the local linearization of a function to make predictions and approximate numerical values.
Describe the limitations of using linear approximation.
1. Thomas is roasting a 14 lb turkey as a test run for Thanksgiving. He starts roasting the turkey at noon.
At 1:00pm, he checks on the turkey using a fancy thermometer; he discovers that it has an internal
temperature of 35
.
6
◦
C,
1
and its temperature is rising at an instantaneous rate of 0
.
25
◦
C per minute.
(a) Approximate the temperature of the turkey at 1:06pm.
Answer
.
37
.
1
◦
C
at 1:06 pm.
(b) Let
I
(
t
) be the turkey’s internal temperature (in
◦
C)
t
minutes after noon. Use function notation
to express what you were told about the turkey.
Answer
.
We were told that
I
(60) = 35
.
6 and
I
′
(60) = 0
.
25.
(c) Here is a graph of
I
(
t
). Use a sketch to explain the approximation you made in
(a)
.
60
120
180
240
t
I
Answer
.
We were pretending that the graph of the turkey’s temperature was a line including
the point (60
, I
(60)) and having slope
I
′
(60). Instead of calculating
I
(66), we estimated using the
t
= 66 on the tangent line.
(d) Based on your sketch, was your approximation too high or too low?
Answer
.
Too high
.
1
According to the USDA, a turkey should be roasted to an internal temperature of 73
◦
C.
Adapted from materials created by the Cornell Active Learning in Math team and Harvard University’s Math Ma, 1a.
1/
4
Worksheet 10:
Linear Approximation (Active Calculus
1.8) Solutions
MATH 1110
(e) Would you be comfortable using the same method to predict the turkey’s temperature at 3 pm?
Why or why not?
Answer
.
No.
Although it seems pretty reasonable to say that the turkey’s temperature changes
at a roughly constant rate for 6 minutes, it seems pretty unlikely that it would change at a constant
rate for 2 hours.
2.
Interpreting rates of change.
Jasmin is interested in understanding how the ra-
dius of a balloon changes as it rises into the air.
She fills a spherical weather balloon with helium
and releases it from sea level (0 meters). As the
balloon rises, its radius can be modeled by a func-
tion
r
(
a
) where
a
is the altitude of the balloon in
meters (see figure to the right).
Note:
As the balloon rises from an altitude of
18
,
000 meters to an altitude of 25
,
000 meters, the
balloon expands from a radius of 2 meters to a
radius of 3 meters.
5000
10000
15000
20000
25000
1
1
.
5
2
2
.
5
3
3
.
5
Altitude (m)
Radius (m)
(a) What is the average rate of change of the balloon’s radius with respect to its altitude from
a
= 18
,
000
to
a
= 25
,
000? Include units in your answer and represent this rate of change on the graph above.
Answer
.
So, between 18,000 and 25,000 meters, the balloon’s radius is changing at an average of
1
7
,
000
meters of radius
meter of altitude
.
(b) After some careful calculations, the scientist determines that
r
(17
,
000) = 1
.
9 and
r
′
(17
,
000) =
1
10
,
000
. Use this data to estimate the radius of the balloon at 22
,
000 meters above sea level.
Answer
.
We expect the radius of the balloon to be around 2
.
4 meters
when the balloon is 22,000
meters above sea level.
(c) Is your answer from part
(b)
an overestimate or an underestimate of the radius of the balloon at
22
,
000 meters above sea level? Explain your reasoning.
Answer
.
The function
r
(
a
) is concave up and so the tangent line to
r
(
a
) at
a
= 17
,
000 is below
2/
4
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