Q. 28 Use Theorem 12.33 to find the in... [FREE SOLUTION]

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Chapter 12: Q. 28 (page 964)

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.
$\frac{鈭� z}{鈭� t} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Short Answer

Expert verified

The value is $\frac{鈭� z}{鈭� t} = s^{2} t 路 \sin^{2} s 路 \cos^{2} t (2 s \cos t + 3 t \sin t)$

Step by step solution

Step 1. Given Information:

Given:

$z = x^{2} y^{3}, x = t \sin s a n d y = s \cos t$

We have to find the indicated derivatives and express your answers as functions of a single variable.

Step 2. Solution:

By Theorem 12.33, we have

$\frac{鈭� z}{鈭� t} = \frac{鈭� z}{鈭� x} 路 \frac{鈭� x}{鈭� t} + \frac{鈭� z}{鈭� y} 路 \frac{鈭� y}{鈭� t} - - - (1)$

So first we find $\frac{鈭� z}{鈭� x}, \frac{鈭� x}{鈭� t}, \frac{鈭� z}{鈭� y} 路 \frac{鈭� y}{鈭� t}$

So we have

$\frac{鈭� z}{鈭� x} = 2 x y^{3} \frac{鈭� z}{鈭� y} = 3 x^{2} y^{2} \frac{鈭� x}{鈭� t} = \sin s \frac{鈭� y}{鈭� t} = - s \sin t$

Use these values in (1) we get

role="math" localid="1649695474446" $\frac{鈭� z}{鈭� t} = 2 x y^{3} 路 \sin s + 3 x^{2} y^{2} 路 (- s \sin t) \frac{鈭� z}{鈭� t} = 2 x y^{3} 路 \sin s - 3 x^{2} y^{2} s 路 \sin t$

This result is correct, but it is preferable to write the function as a function of just s and t.

We use $x = t \sin s a n d y = s \cos t$ to do so:

$\frac{鈭� z}{鈭� t} = 2 (t \sin s) {(s \cos t)}^{3} 路 \sin s + 3 {(t \sin s)}^{2} {(s \cos t)}^{2} s 路 \sin t \frac{鈭� z}{鈭� t} = 2 s^{3} t 路 \sin^{2} s 路 \cos^{3} t + 3 s^{2} t^{2} \sin^{2} s 路 \cos^{2} t 路 \sin t \frac{鈭� z}{鈭� t} = s^{2} t 路 \sin^{2} s 路 \cos^{2} t (2 s \cos t + 3 t \sin t)$

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91影视

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.
$\frac{鈭� z}{鈭� t} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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91影视

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.鈭�z鈭�twhenz=x2y3,x=tsins,andy=scost

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Probability and Statistics

Decision Maths

Logic and Functions

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.
$\frac{鈭� z}{鈭� t} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$