Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
Find \(\frac{dX}{dt}\) for the given matrix function.
**Given Matrix Function:**
\[
X(t) =
\begin{bmatrix}
\sin 5t & \cos 5t & e^{-5t} \\
-\sin 5t & 8 \cos 5t & 3e^{-5t} \\
3 \sin 5t & \cos 5t & e^{-5t}
\end{bmatrix}
\]
**Solution:**
Compute the derivative \(\frac{dX}{dt} = \boxed{\phantom{answer}}\)
---
**Explanation:**
To solve this problem, differentiate each element of the matrix \(X(t)\) with respect to \(t\). The elements involve trigonometric and exponential functions.
- For elements with \(\sin 5t\), use the derivative \(\frac{d}{dt}[\sin 5t] = 5\cos 5t\).
- For elements with \(\cos 5t\), use the derivative \(\frac{d}{dt}[\cos 5t] = -5\sin 5t\).
- For elements with \(e^{-5t}\), use the derivative \(\frac{d}{dt}[e^{-5t}] = -5e^{-5t}\).
Substitute these derivatives into the corresponding positions in the matrix to find \(\frac{dX}{dt}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf84dde2-f77b-4203-b744-f3414691451c%2F50366fd5-8179-4452-b82e-88598f1f1130%2F6l77vhc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find \(\frac{dX}{dt}\) for the given matrix function.
**Given Matrix Function:**
\[
X(t) =
\begin{bmatrix}
\sin 5t & \cos 5t & e^{-5t} \\
-\sin 5t & 8 \cos 5t & 3e^{-5t} \\
3 \sin 5t & \cos 5t & e^{-5t}
\end{bmatrix}
\]
**Solution:**
Compute the derivative \(\frac{dX}{dt} = \boxed{\phantom{answer}}\)
---
**Explanation:**
To solve this problem, differentiate each element of the matrix \(X(t)\) with respect to \(t\). The elements involve trigonometric and exponential functions.
- For elements with \(\sin 5t\), use the derivative \(\frac{d}{dt}[\sin 5t] = 5\cos 5t\).
- For elements with \(\cos 5t\), use the derivative \(\frac{d}{dt}[\cos 5t] = -5\sin 5t\).
- For elements with \(e^{-5t}\), use the derivative \(\frac{d}{dt}[e^{-5t}] = -5e^{-5t}\).
Substitute these derivatives into the corresponding positions in the matrix to find \(\frac{dX}{dt}\).
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