Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Calculus of vector-valued functions: Calculate each of the following.
$\frac{d}{d t} (r (t)), where r (t) = 3 \cos^{2} t i + 5 t j + \frac{t}{t^{2} + 1} k$

The value of $\frac{d}{d t} (r (t)) = - 6 \cos t \sin t i + 5 j + \frac{1 - t^{2}}{t^{2} + 1} k$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1.Given Information

Calculus of vector-valued functions: Calculate each of the following.

$\frac{d}{d t} (r (t)), where r (t) = 3 \cos^{2} t i + 5 t j + \frac{t}{t^{2} + 1} k$

Step 2. Now finding the value of ddt(r(t)).

$\frac{d}{d t} (r (t)) = \frac{d}{d t} (3 \cos^{2} t i + 5 t j + \frac{t}{t^{2} + 1} k) \frac{d}{d t} (r (t)) = 3 (\frac{d}{d t} \cos^{2} t) i + 5 (\frac{d}{d t} t) j + (\frac{d}{d t} \frac{t}{t^{2} + 1}) k \frac{d}{d t} (r (t)) = 3 A i + 5 B j + C k$

Step 3. Firstly find the value of A=ddtcos2t

$A = \frac{d}{d t} \cos^{2} t A = 2 \cos t (- \sin t) A = - 2 \cos t \sin t$

Step 4. Now finding the value of B=ddtt

$B = \frac{d}{d t} t B = 1$

Step 5. Now finding the value of C=ddttt2+1

$Applying the quotient rule first, we have C = \frac{d}{d t} \frac{t}{t^{2} + 1} C = \frac{\frac{d}{d t} t \cdot (t^{2} + 1) - t \cdot \frac{d}{d t} (t^{2} + 1)}{t^{2} + 1} C = \frac{1 \cdot (t^{2} + 1) - t \cdot 2 t}{t^{2} + 1} C = \frac{t^{2} + 1 - 2 t^{2}}{t^{2} + 1} C = \frac{1 - t^{2}}{t^{2} + 1}$

Step 6 Now putting the value of A,B,C.

$\frac{d}{d t} (r (t)) = 3 (- 2 \cos t \sin t) i + 5 \times 1 j + \frac{1 - t^{2}}{t^{2} + 1} k \frac{d}{d t} (r (t)) = - 6 \cos t \sin t i + 5 j + \frac{1 - t^{2}}{t^{2} + 1} k$

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Calculus

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Short Answer

Calculus of vector-valued functions: Calculate each of the following.
$\frac{d}{d t} (r (t)), where r (t) = 3 \cos^{2} t i + 5 t j + \frac{t}{t^{2} + 1} k$

Step by Step Solution

Step 1.Given Information

Step 2. Now finding the value of ddt(r(t)).

Step 3. Firstly find the value of A=ddtcos2t

Step 4. Now finding the value of B=ddtt

Step 5. Now finding the value of C=ddttt2+1

Step 6 Now putting the value of A,B,C.

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Calculus

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Short Answer

Calculus of vector-valued functions: Calculate each of the following.ddt(r(t)), where r(t)=3cos2ti+5tj+tt2+1k

Step by Step Solution

Step 1.Given Information

Step 2. Now finding the value of ddt(r(t)).

Step 3. Firstly find the value of A=ddtcos2t

Step 4. Now finding the value of B=ddtt

Step 5. Now finding the value of C=ddttt2+1

Step 6 Now putting the value of A,B,C.

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Calculus of vector-valued functions: Calculate each of the following.
$\frac{d}{d t} (r (t)), where r (t) = 3 \cos^{2} t i + 5 t j + \frac{t}{t^{2} + 1} k$