Q. 30 Use Theorem 12.33 to find the in... [FREE SOLUTION]

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Chapter 12: Q. 30 (page 964)

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

Short Answer

Expert verified

The value is $\frac{鈭� z}{鈭� r} = r^{2} e^{r \sin 胃} [(\sin 2 胃 + \sin^{2} 胃 + \cos^{2} 胃 (r \cos 胃 + r \sin 胃 + 1)]$

Step by step solution

Step 1. Given Information:

Given:

$z = (x^{2} + x y) e^{y}, x = r \cos 胃 a n d y = r \sin 胃$

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

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

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

So we have

$\frac{鈭� z}{鈭� x} = e^{y} (2 x + y) \frac{鈭� z}{鈭� y} = e^{y} 路 x^{2} + x e^{y} + x y e^{y} \frac{鈭� x}{鈭� 胃} = - r s i n 胃 \frac{鈭� y}{鈭� 胃} = r \cos 胃$

Use these values in (1) we get

$\frac{鈭� z}{鈭� r} = e^{y} (2 x + y) 路 - r s i n 胃 + x e^{y} (x + 1 + y) 路 r \cos 胃 \frac{鈭� z}{鈭� r} = r e^{y} [(2 x + y) 路 s i n 胃 + x \cos 胃 (x + y + 1)]$

This result is correct, but it is preferable to write the function as a function of just $r a n d 胃$ .

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

$\frac{鈭� z}{鈭� r} = r e^{r \sin 胃} [(2 (r \cos 胃) + r \sin 胃) 路 \sin 胃 + r \cos 胃 \cos 胃 (r \cos 胃 + r \sin 胃 + 1)] \frac{鈭� z}{鈭� r} = r^{2} e^{r \sin 胃} [(2 \sin 胃 \cos 胃 + \sin^{2} 胃 + \cos^{2} 胃 (r \cos 胃 + r \sin 胃 + 1)] \frac{鈭� z}{鈭� r} = r^{2} e^{r \sin 胃} [(\sin 2 胃 + \sin^{2} 胃 + \cos^{2} 胃 (r \cos 胃 + r \sin 胃 + 1)]$

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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.
$\frac{鈭� z}{鈭� 胃} w h e n z = (x^{2} + x y) e^{y}, x = r \cos 胃 a n d y = r \sin 胃$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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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�胃whenz=(x2+xy)ey,x=rcos胃andy=rsin胃

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

Statistics

Theoretical and Mathematical Physics

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Discrete Mathematics

Logic and Functions

Geometry

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.
$\frac{鈭� z}{鈭� 胃} w h e n z = (x^{2} + x y) e^{y}, x = r \cos 胃 a n d y = r \sin 胃$