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< 1,1,0>, v(0) = <0,0,1 > and Find the position function, F(1), given that F(0) = a(t) = < t, e', e¯t >.

< 1,1,0>, v(0) = <0,0,1 > and Find the position function, F(1), given that F(0) = a(t) = < t, e', e¯t >.

Calculus: Early Transcendentals
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 the position function, \(\vec{r}(t)\), given that \(\vec{r}(0) = \langle 1,1,0 \rangle\), \(\vec{v}(0) = \langle 0,0,1 \rangle\) and \(\vec{a}(t) = \langle t, e^t, e^{-t} \rangle\). **Explanation:** This problem involves finding the position function \(\vec{r}(t)\) from the given initial conditions and the acceleration function. Let's break down the information provided: - \(\vec{r}(0) = \langle 1,1,0 \rangle\): This is the initial position vector. - \(\vec{v}(0) = \langle 0,0,1 \rangle\): This is the initial velocity vector. - \(\vec{a}(t) = \langle t, e^t, e^{-t} \rangle\): This is the acceleration vector as a function of time. To find the position function, we need to integrate the acceleration function twice. First, we find the velocity function \(\vec{v}(t)\) by integrating \(\vec{a}(t)\) with respect to \(t\). Then, we integrate \(\vec{v}(t)\) to find the position function \(\vec{r}(t)\). Constants of integration will be determined using the initial conditions provided. In more specific steps: 1. **Integrate the acceleration function** \(\vec{a}(t) = \langle t, e^t, e^{-t} \rangle\) to find the velocity function \(\vec{v}(t)\). 2. **Determine the constant of integration** for the velocity function using \(\vec{v}(0) = \langle 0,0,1 \rangle\). 3. **Integrate the resulting velocity function** to find the position function \(\vec{r}(t)\). 4. **Determine the constant of integration** for the position function using \(\vec{r}(0) = \langle 1,1,0 \rangle\). This systematic approach will yield the position vector function \(\vec{r}(t)\). This transcription should help users understand the process
Transcribed Image Text:**Problem Statement:** Find the position function, \(\vec{r}(t)\), given that \(\vec{r}(0) = \langle 1,1,0 \rangle\), \(\vec{v}(0) = \langle 0,0,1 \rangle\) and \(\vec{a}(t) = \langle t, e^t, e^{-t} \rangle\). **Explanation:** This problem involves finding the position function \(\vec{r}(t)\) from the given initial conditions and the acceleration function. Let's break down the information provided: - \(\vec{r}(0) = \langle 1,1,0 \rangle\): This is the initial position vector. - \(\vec{v}(0) = \langle 0,0,1 \rangle\): This is the initial velocity vector. - \(\vec{a}(t) = \langle t, e^t, e^{-t} \rangle\): This is the acceleration vector as a function of time. To find the position function, we need to integrate the acceleration function twice. First, we find the velocity function \(\vec{v}(t)\) by integrating \(\vec{a}(t)\) with respect to \(t\). Then, we integrate \(\vec{v}(t)\) to find the position function \(\vec{r}(t)\). Constants of integration will be determined using the initial conditions provided. In more specific steps: 1. **Integrate the acceleration function** \(\vec{a}(t) = \langle t, e^t, e^{-t} \rangle\) to find the velocity function \(\vec{v}(t)\). 2. **Determine the constant of integration** for the velocity function using \(\vec{v}(0) = \langle 0,0,1 \rangle\). 3. **Integrate the resulting velocity function** to find the position function \(\vec{r}(t)\). 4. **Determine the constant of integration** for the position function using \(\vec{r}(0) = \langle 1,1,0 \rangle\). This systematic approach will yield the position vector function \(\vec{r}(t)\). This transcription should help users understand the process
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