Chapter 12: Q. 24 (page 964)

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.
$\frac{d y}{d t} w h e n y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$

Short Answer

Expert verified

The value is $\frac{d y}{d t} = t^{2} (\sqrt{t} \cdot \cos \sqrt{t} + 2 \sin \sqrt{t})$

Step by step solution

Step 1. Given Information:

Given:

$y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$

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

Step 2. Solution:

$U s i n g r = t^{3} a n d θ = \sqrt{t} i n y = r \sin θ w e g e t y = t^{3} \sin \sqrt{t} D i f f . w . r . t . t w e g e t \frac{d y}{d t} = t^{3} \frac{d}{d t} \sin \sqrt{t} + \sin \sqrt{t} \frac{d}{d t} t^{3} \frac{d y}{d t} = t^{3} \cdot \cos \sqrt{t} \cdot \frac{1}{2 \sqrt{t}} + \sin \sqrt{t} \cdot 2 t^{2} \frac{d y}{d t} = t^{2} (\sqrt{t} \cdot \cos \sqrt{t} + 2 \sin \sqrt{t})$

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Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.
$\frac{d y}{d t} w h e n y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

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Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.dydtwheny=rsinθ,r=t3,andθ=t

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Solution:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Use Theorem 12.32 to find the indicated derivatives in Exercises 21–26. Express your answers as functions of a single variable.
$\frac{d y}{d t} w h e n y = r \sin θ, r = t^{3}, a n d θ = \sqrt{t}$