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9. A point moves along a curve with a(t)= (2,6t, 12t2). Find its position function 7(t) if 7(0) = (0, 1, –1) and v(0) = (1,0, 0). %3D

9. A point moves along a curve with a(t)= (2,6t, 12t2). Find its position function 7(t) if 7(0) = (0, 1, –1) and v(0) = (1,0, 0). %3D

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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Question
### Problem 9 A point moves along a curve with the acceleration function \(\mathbf{a}(t) = \langle 2, 6t, 12t^2 \rangle\). Find its position function \(\mathbf{r}(t)\) given that \(\mathbf{r}(0) = \langle 0, 1, -1 \rangle\) and \(\mathbf{v}(0) = \langle 1, 0, 0 \rangle\). ### Explanation: - **\(\mathbf{a}(t)\): Acceleration Function**: The acceleration vector is given as \(\langle 2, 6t, 12t^2 \rangle\). It represents the rate of change of the velocity function over time. - **\(\mathbf{r}(0)\): Initial Position**: The initial position vector is \(\langle 0, 1, -1 \rangle\). This is the position of the point at time \(t = 0\). - **\(\mathbf{v}(0)\): Initial Velocity**: The initial velocity vector is \(\langle 1, 0, 0 \rangle\). It indicates the velocity of the point at time \(t = 0\). The goal is to derive the position function \(\mathbf{r}(t)\) by integrating the acceleration function to find the velocity function, and using the initial conditions to determine the constants of integration for the position function.
Transcribed Image Text:### Problem 9 A point moves along a curve with the acceleration function \(\mathbf{a}(t) = \langle 2, 6t, 12t^2 \rangle\). Find its position function \(\mathbf{r}(t)\) given that \(\mathbf{r}(0) = \langle 0, 1, -1 \rangle\) and \(\mathbf{v}(0) = \langle 1, 0, 0 \rangle\). ### Explanation: - **\(\mathbf{a}(t)\): Acceleration Function**: The acceleration vector is given as \(\langle 2, 6t, 12t^2 \rangle\). It represents the rate of change of the velocity function over time. - **\(\mathbf{r}(0)\): Initial Position**: The initial position vector is \(\langle 0, 1, -1 \rangle\). This is the position of the point at time \(t = 0\). - **\(\mathbf{v}(0)\): Initial Velocity**: The initial velocity vector is \(\langle 1, 0, 0 \rangle\). It indicates the velocity of the point at time \(t = 0\). The goal is to derive the position function \(\mathbf{r}(t)\) by integrating the acceleration function to find the velocity function, and using the initial conditions to determine the constants of integration for the position function.
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