Find the derivative of the vector function r(t) = In(10 – t) i+ v10+tj+ 8e-8* k r'(t) = (
Find the derivative of the vector function r(t) = In(10 – t) i+ v10+tj+ 8e-8* k r'(t) = (
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Vector Arithmetic
Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.
Vector Calculus
Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.
Question
![**Objective:** Find the derivative of the vector function.
**Given:**
\[ \mathbf{r}(t) = \ln(10 - t^2) \, \mathbf{i} + \sqrt{10 + t} \, \mathbf{j} + 8e^{-8t} \, \mathbf{k} \]
**Solution:** The derivative of the vector function is expressed as \(\mathbf{r}'(t)\) and is calculated as:
\[ \mathbf{r}'(t) = \left\langle \frac{d}{dt}\left(\ln(10 - t^2)\right), \frac{d}{dt}\left(\sqrt{10 + t}\right), \frac{d}{dt}\left(8e^{-8t}\right) \right\rangle \]
**Step-by-Step Calculation:**
1. **First Component:**
Derivative of \(\ln(10 - t^2)\):
\[
\frac{d}{dt}[\ln(10 - t^2)] = \frac{-2t}{10 - t^2}
\]
2. **Second Component:**
Derivative of \(\sqrt{10 + t}\):
\[
\frac{d}{dt}[\sqrt{10 + t}] = \frac{1}{2\sqrt{10 + t}}
\]
3. **Third Component:**
Derivative of \(8e^{-8t}\):
\[
\frac{d}{dt}[8e^{-8t}] = -64e^{-8t}
\]
**Final Derivative:**
\[ \mathbf{r}'(t) = \left\langle \frac{-2t}{10 - t^2}, \frac{1}{2\sqrt{10 + t}}, -64e^{-8t} \right\rangle \]
This expression provides the rate of change of the vector function \(\mathbf{r}(t)\) with respect to \(t\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc247148b-ac88-4d23-ab16-d846cf9acaa7%2Fab94d241-b1ad-435b-b8ec-08b1af14226e%2Fkzlnxlb_processed.png&w=3840&q=75)
Transcribed Image Text:**Objective:** Find the derivative of the vector function.
**Given:**
\[ \mathbf{r}(t) = \ln(10 - t^2) \, \mathbf{i} + \sqrt{10 + t} \, \mathbf{j} + 8e^{-8t} \, \mathbf{k} \]
**Solution:** The derivative of the vector function is expressed as \(\mathbf{r}'(t)\) and is calculated as:
\[ \mathbf{r}'(t) = \left\langle \frac{d}{dt}\left(\ln(10 - t^2)\right), \frac{d}{dt}\left(\sqrt{10 + t}\right), \frac{d}{dt}\left(8e^{-8t}\right) \right\rangle \]
**Step-by-Step Calculation:**
1. **First Component:**
Derivative of \(\ln(10 - t^2)\):
\[
\frac{d}{dt}[\ln(10 - t^2)] = \frac{-2t}{10 - t^2}
\]
2. **Second Component:**
Derivative of \(\sqrt{10 + t}\):
\[
\frac{d}{dt}[\sqrt{10 + t}] = \frac{1}{2\sqrt{10 + t}}
\]
3. **Third Component:**
Derivative of \(8e^{-8t}\):
\[
\frac{d}{dt}[8e^{-8t}] = -64e^{-8t}
\]
**Final Derivative:**
\[ \mathbf{r}'(t) = \left\langle \frac{-2t}{10 - t^2}, \frac{1}{2\sqrt{10 + t}}, -64e^{-8t} \right\rangle \]
This expression provides the rate of change of the vector function \(\mathbf{r}(t)\) with respect to \(t\).
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
