Q. 4 Calculus of vector-valued functi... [FREE SOLUTION]

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Chapter 14: Q. 4 (page 1095)

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

Short Answer

Expert verified

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

Step by step solution

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

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

Now putting the value of A,B,C.

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

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91影视

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

Short Answer

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

Now putting the value of A,B,C.

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91影视

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

Short Answer

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

Now putting the value of A,B,C.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Statistics

Theoretical and Mathematical Physics

Probability and Statistics

Decision Maths

Study anywhere. Anytime. Across all devices.

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