Please help with b. I attached a for reference
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help with b. I attached a for reference
![CHAPTER 10. DERIVATIVES OF MULTIVARIABLE FUNCTIONS 73
10.5 The Chain Rule
Preview Activity 10.5.1 Suppose you are driving around in the xy-plane in
such a way that your position r(t) at time t is given by function
r(t) = (x(t), y(t)) = (2 − t², t³ + 1).
The path taken is shown on the left of Figure 10.5.1.
21
3-
2-
1
Y
by
1
2
X
3
I
Figure 10.5.1 Left: Your position in the plane. Right: The corresponding
temperature.
Suppose, furthermore, that the temperature at a point in the plane is given
Y
T(x, y) 10-1²-²,
=
and note that the surface generated by T is shown on the right of Figure 10.5.1.
Therefore, as time passes, your position (x(t), y(t)) changes, and, as your po-
sition changes, the temperature T(x, y) also changes.
a. The position function r provides a parametrization x = x(t) and y = = y(t)
of the position at time t. By substituting ä(t) for x and y(t) for y in the
formula for T, we can write T = T(x(t), y(t)) as a function of t. Make
these substitutions to write T as a function of t and then use the Chain
Rule from single variable calculus to find d. (Do not do any algebra to
simplify the derivative, either before taking the derivative, nor after.)
dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeeb6393-c916-4741-b6e3-3e61715ee264%2Ffaad7530-c5a6-4e10-aab2-c2c9ca7a6984%2Fvq96xuv_processed.png&w=3840&q=75)
Transcribed Image Text:CHAPTER 10. DERIVATIVES OF MULTIVARIABLE FUNCTIONS 73
10.5 The Chain Rule
Preview Activity 10.5.1 Suppose you are driving around in the xy-plane in
such a way that your position r(t) at time t is given by function
r(t) = (x(t), y(t)) = (2 − t², t³ + 1).
The path taken is shown on the left of Figure 10.5.1.
21
3-
2-
1
Y
by
1
2
X
3
I
Figure 10.5.1 Left: Your position in the plane. Right: The corresponding
temperature.
Suppose, furthermore, that the temperature at a point in the plane is given
Y
T(x, y) 10-1²-²,
=
and note that the surface generated by T is shown on the right of Figure 10.5.1.
Therefore, as time passes, your position (x(t), y(t)) changes, and, as your po-
sition changes, the temperature T(x, y) also changes.
a. The position function r provides a parametrization x = x(t) and y = = y(t)
of the position at time t. By substituting ä(t) for x and y(t) for y in the
formula for T, we can write T = T(x(t), y(t)) as a function of t. Make
these substitutions to write T as a function of t and then use the Chain
Rule from single variable calculus to find d. (Do not do any algebra to
simplify the derivative, either before taking the derivative, nor after.)
dt
![b. Now we want to understand how the result from part (a) can be obtained
from T as a multivariable function. Recall from the previous section that
small changes in x and y produce a change in T that is approximated by
ATTAx+TyAy.
The Chain Rule tells us about the instantaneous rate of change of T, and
this can be found as
AT
lim
lim
At →0 At At →0
(10.5.1)
Use equation (10.5.1) to explain why the instantaneous rate of change of
T that results from a change in t is
dT
dt
=
TxAx+TyAy
At
OT dx
dx dt
+
ᏭᎢ dy
dy dt
(10.5.2)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeeb6393-c916-4741-b6e3-3e61715ee264%2Ffaad7530-c5a6-4e10-aab2-c2c9ca7a6984%2Fvivkbcf_processed.png&w=3840&q=75)
Transcribed Image Text:b. Now we want to understand how the result from part (a) can be obtained
from T as a multivariable function. Recall from the previous section that
small changes in x and y produce a change in T that is approximated by
ATTAx+TyAy.
The Chain Rule tells us about the instantaneous rate of change of T, and
this can be found as
AT
lim
lim
At →0 At At →0
(10.5.1)
Use equation (10.5.1) to explain why the instantaneous rate of change of
T that results from a change in t is
dT
dt
=
TxAx+TyAy
At
OT dx
dx dt
+
ᏭᎢ dy
dy dt
(10.5.2)
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