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

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

Now putting the value of A,B,C.

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

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Logic and Functions

Pure Maths

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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

Discrete Mathematics

Geometry

Logic and Functions

Pure Maths

Calculus

Probability and Statistics

Study anywhere. Anytime. Across all devices.

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