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Please help with b. I attached a for reference

Please help with b. I attached a for reference

Advanced Engineering Mathematics
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
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)
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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