Q. 62 Prove Theorem 11.7 for vectors i... [FREE SOLUTION]

Chapter 11: Q. 62 (page 873)

Prove Theorem 11.7 for vectors in R2. That is, let $x (t)$ and $y (t)$ be two scalar functions, each of which is differentiable on an interval I 鈯� R, and let localid="1649578343519" $r (t) = \{x (t), y (t)\}$ be a vector function. Prove that $r^{'} (f) = \lim_{h 鈫� 0} \frac{r (t + h) - r (t)}{h}$ .

Short Answer

Expert verified

Ans:

\lim_{t 鈫� t_{0}} r (t) = \lim_{t 鈫� t_{0}} 鉄� x (t), y (t) 鉄� = \lim_{t 鈫� t_{0}} x (t) i + \lim_{t 鈫� t_{0}} y (t) j = x^{'} (t) i + y^{'} (t) j = r' (t)

Step by step solution

Step 1. Given information:

$r (t) = \{x (t), y (t)\}$ be a vector function.

$r^{'} (f) = \lim_{h 鈫� 0} \frac{r (t + h) - r (t)}{h}$

Step 2. Proving:

The limit of a vector function

Let $r (t) = 鉄� x (t), y (t) 鉄�$ . The limit of $r (t)$ as $t$ approaches $t_{0}$ , denoted by $\lim_{t 鈫� t_{0}} r (t)$ is defined by

$\lim_{t 鈫� t_{0}} r (t) = \lim_{t 鈫� t_{0}} 鉄� x (t), y (t) 鉄� = \lim_{t 鈫� t_{0}} x (t) i + \lim_{t 鈫� t_{0}} y (t) j$

Where $\lim_{t 鈫� t_{0}} x (t)$ and $\lim_{t 鈫� t_{0}} y (t)$ exists.

Consider $\lim_{h 鈫� 0} \frac{r (t + h) - r (t)}{h}$

$= \lim_{h 鈫� 0} \frac{鉄� x (t + h), y (t + h) 鉄� - 鉄� x (t), y (t) 鉄�}{h}$ for $r (t + h)$ , substitute $t + h$ for $t$ in $r (t)$ $= \lim_{h 鈫� 0} \frac{鉄� x (t + h) - x (t), y (t + h) - y (t) 鉄�}{h}$ Difference of two vectors

$= \lim_{h 鈫� 0} <\frac{x (t + h) - x (t)}{h}, \frac{y (t + h) - y (t)}{h}>$ Divide each term by h.

$= \lim_{h 鈫� 0} \frac{x (t + h) - x (t)}{h} i + \lim_{h 鈫� 0} \frac{y (t + h) - y (t)}{h} j$ limit of a vectors function

$= x^{'} (t) i + y^{'} (t) j$

$= <x^{'} (t), y^{'} (t)>$ Derivativeof a scalar fucntion

$= r^{'} (t)$ Derivative of a vector function

thus $r^{'} (t) = \lim_{h 鈫� 0} \frac{r (t + h) - r (t)}{h}$

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

Step 1. Given information:

Step 2. Proving:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Logic and Functions

Discrete Mathematics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

91影视

Prove Theorem 11.7 for vectors in R2. That is, let x(t)and y(t)be two scalar functions, each of which is differentiable on an interval I 鈯� R, and let localid="1649578343519" r(t)=x(t),y(t)be a vector function. Prove that r'(f)=limh鈫�0r(t+h)-r(t)h.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Proving:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Calculus

Logic and Functions

Discrete Mathematics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.