3. Find the tangential and normal components of the acceleration vector at the given point. (a) r(t) = ln ti + (t² + 3t)j + 4√tk; (0,4,4) 1 1 1 3); (1,1,1) (b) r(t) = t2' 13
3. Find the tangential and normal components of the acceleration vector at the given point. (a) r(t) = ln ti + (t² + 3t)j + 4√tk; (0,4,4) 1 1 1 3); (1,1,1) (b) r(t) = t2' 13
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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Can you help me with these parts and can you show it step by step
![### Problem 3: Tangential and Normal Components of the Acceleration Vector
Find the tangential and normal components of the acceleration vector at the given point.
#### Part (a)
Given:
\[ \mathbf{r}(t) = \ln(t)\mathbf{i} + (t^2 + 3t)\mathbf{j} + 4\sqrt{t}\mathbf{k} \]
Evaluate at the point \((0, 4, 4)\).
#### Part (b)
Given:
\[ \mathbf{r}(t) = \left\langle \frac{1}{t}, \frac{1}{t^2}, \frac{1}{t^3} \right\rangle \]
Evaluate at the point \((1, 1, 1)\).
To solve these problems, you would follow these general steps:
1. Compute the first derivative \(\mathbf{r}'(t)\) to find the velocity vector.
2. Compute the second derivative \(\mathbf{r}''(t)\) to find the acceleration vector.
3. Find the tangential component of the acceleration by projecting the acceleration onto the unit tangent vector.
4. Find the normal component of the acceleration by determining the component perpendicular to the tangential component.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c51d254-4763-40fe-b7bf-7e1edfa53148%2F86adcc2d-7f88-4d00-8946-8f181c80fc46%2Flj3gl89_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 3: Tangential and Normal Components of the Acceleration Vector
Find the tangential and normal components of the acceleration vector at the given point.
#### Part (a)
Given:
\[ \mathbf{r}(t) = \ln(t)\mathbf{i} + (t^2 + 3t)\mathbf{j} + 4\sqrt{t}\mathbf{k} \]
Evaluate at the point \((0, 4, 4)\).
#### Part (b)
Given:
\[ \mathbf{r}(t) = \left\langle \frac{1}{t}, \frac{1}{t^2}, \frac{1}{t^3} \right\rangle \]
Evaluate at the point \((1, 1, 1)\).
To solve these problems, you would follow these general steps:
1. Compute the first derivative \(\mathbf{r}'(t)\) to find the velocity vector.
2. Compute the second derivative \(\mathbf{r}''(t)\) to find the acceleration vector.
3. Find the tangential component of the acceleration by projecting the acceleration onto the unit tangent vector.
4. Find the normal component of the acceleration by determining the component perpendicular to the tangential component.
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