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...
Related questions
Question
100%
Write a in the form
a=aT+aN
at the given value of t without finding T and N.
![**Problem Statement:**
Write **a** in the form \( \textbf{a} = a_T \textbf{T} + a_N \textbf{N} \) at the given value of \( t \) without finding \(\textbf{T}\) and \(\textbf{N}\).
\[ \textbf{r}(t) = \left(2t^2 \right) \textbf{i} + \left(2t + \frac{2}{3}t^3 \right) \textbf{j} + \left(2t - \frac{2}{3}t^3 \right)\textbf{k}, \quad t = 0 \]
**Solution:**
\[ \textbf{a}(0) = \left( \frac{\sqrt{8}}{3} \right) \textbf{T} + \left( \sqrt{\frac{160}{9} - \frac{\sqrt{8}}{3}} \right) \textbf{N} \]
*(Type exact answers, using radicals as needed.)*
**Explanation:**
The problem involves expressing the vector \( \textbf{a} \) in terms of tangential and normal components at a specific point without directly finding the tangent (\(\textbf{T}\)) and normal (\(\textbf{N}\)) vectors. The provided expressions use radicals for precise computation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbfc23d67-2878-46d3-b2ec-822f6f1a78a5%2F60faf6ee-60aa-4b4e-9e0c-5a23259ef2fc%2Fbni4a5f_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Write **a** in the form \( \textbf{a} = a_T \textbf{T} + a_N \textbf{N} \) at the given value of \( t \) without finding \(\textbf{T}\) and \(\textbf{N}\).
\[ \textbf{r}(t) = \left(2t^2 \right) \textbf{i} + \left(2t + \frac{2}{3}t^3 \right) \textbf{j} + \left(2t - \frac{2}{3}t^3 \right)\textbf{k}, \quad t = 0 \]
**Solution:**
\[ \textbf{a}(0) = \left( \frac{\sqrt{8}}{3} \right) \textbf{T} + \left( \sqrt{\frac{160}{9} - \frac{\sqrt{8}}{3}} \right) \textbf{N} \]
*(Type exact answers, using radicals as needed.)*
**Explanation:**
The problem involves expressing the vector \( \textbf{a} \) in terms of tangential and normal components at a specific point without directly finding the tangent (\(\textbf{T}\)) and normal (\(\textbf{N}\)) vectors. The provided expressions use radicals for precise computation.
Expert Solution
Step 1
a0=atT+anN
Step by step
Solved in 2 steps with 1 images
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