Q. 22 In Exercises 21鈥�23 you are giv... [FREE SOLUTION]

91影视

Chapter 11: Q. 22 (page 872)

In Exercises 21鈥�23 you are given a vector function $r$ and a scalar function $t = f (蟿)$ . Compute $\frac{d r}{d 蟿}$ in the following two ways:
(a) By using the chain rule $\frac{d r}{d 蟿} = \frac{d r}{d t} \frac{d t}{d 蟿}$ .
(b) By substituting $t = f (蟿)$ into the formula for $r$ . Ensure that your two answers are consistent.

Short Answer

Expert verified

Part (a) The answer is $\frac{d r}{d t} = 3 <- T^{2} \sin T^{3}, T^{2} \cos T^{3}, T^{2}>$

Part (b) The value is $\frac{d r}{d t} = 3 <- T^{2} \sin T^{3}, T^{2} \cos T^{3}, T^{2}>$

The two answers are consistent, that is $\frac{d r}{d 蟿} = 3 <鈭� 蟿^{2} \sin 鈦� 蟿^{3}, 蟿^{2} \cos 鈦� 蟿^{3}, 蟿^{2}>$

Step by step solution

Part (a) Step 1. Given data

The given vector function is $r (t) = 鉄� \cos 鈦� t, \sin 鈦� t, t 鉄�, t = 蟿^{3}$

We have to find $\frac{d r}{d 蟿}$ in two ways,

Part (a) Step 2. Find drdτ

By using the chain rule, $\frac{d r}{d 蟿} = \frac{d r}{d t} 鈰� \frac{d t}{d 蟿}$

$\begin{aligned} \frac{d r}{d 蟿} & = \frac{d}{d t} r (t) 鈰� \frac{d}{d 蟿} (t) \\ = \frac{d}{d t} 鉄� \cos 鈦� t, \sin 鈦� t, t 鉄� 鈰� \frac{d}{d 蟿} (蟿^{3}) \\ = 鉄� 鈭� \sin 鈦� t, \cos 鈦� t, 1 鉄� (3 蟿^{2}) \end{aligned} \begin{aligned} = 鉄� 鈭� \sin 鈦� 蟿^{3}, \cos 鈦� 蟿^{3}, 1 鉄� (3 蟿^{2}) \\ = 3 (鈭� 蟿^{2} \sin 鈦� 蟿^{3}, 蟿^{2} \cos 鈦� 蟿^{3}, 蟿^{2}) \end{aligned}$

Part (b) Step 1. Find drdτ by substituting t=f(τ)

By substituting $t = \sin 鈦� 蟿$ in $r (t)$

$\begin{aligned} r (t) & = 鉄� c o s 鈦� t, s i n 鈦� t, t 鉄� \\ = 鉄� c o s 鈦� 蟿^{3}, s i n 鈦� 蟿^{3}, 蟿^{3} 鉄� \\ \frac{d r}{d 蟿} & = \frac{d}{d 蟿} r (t) \\ = \frac{d}{d 蟿} 鉄� c o s 鈦� 蟿^{3}, s i n 鈦� 蟿^{3}, 蟿^{3} 鉄� \\ = 鉄� 鈭� 3 蟿^{2} s i n 鈦� 蟿^{3}, 3 蟿^{2} c o s 鈦� 蟿^{2}, 3 蟿^{2} 鉄� \\ = 3 鉄� 鈭� 蟿^{2} s i n 鈦� 蟿^{2}, 蟿^{2} c o s 鈦� 蟿^{2}, 蟿^{2} 鉄� \end{aligned}$

Here, results obtained in part (a) and part (b) are equal.

Therefore, the two answers are consistent.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Part (a) Step 1. Given data

Part (a) Step 2. Find drdτ

Part (b) Step 1. Find drdτ by substituting t=f(τ)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 21鈥�23 you are given a vector function rand a scalar function t=f(蟿). Computedrd蟿in the following two ways:(a) By using the chain rule drd蟿=drdtdtd蟿.(b) By substituting t=f(蟿)into the formula for r. Ensure that your two answers are consistent.

Short Answer

Step by step solution

Part (a) Step 1. Given data

Part (a) Step 2. Find drdτ

Part (b) Step 1. Find drdτ by substituting t=f(τ)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Calculus

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.