r(t) = (t sint, 1 - cost) at the maximum point (x, y) = (1, 2) which occurs when t = . You will need the formula for curvature of a plane curve r(t) = (r(t), y(t)), which is |x"y' - x'y"| k(t) ((x²)² + (y)²) =
r(t) = (t sint, 1 - cost) at the maximum point (x, y) = (1, 2) which occurs when t = . You will need the formula for curvature of a plane curve r(t) = (r(t), y(t)), which is |x"y' - x'y"| k(t) ((x²)² + (y)²) =
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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![**Curvature of Plane Curves**
Given the vector function of a plane curve \( \vec{r}(t) = \langle t - \sin t, 1 - \cos t \rangle \), we are interested in determining properties of this curve at specific points.
**Key Point:**
At the maximum point \((x, y) = (\pi, 2)\), this occurs when \( t = \pi \). To analyze the curvature of the plane curve at this point, we'll use the formula for curvature.
**Curvature Formula:**
The curvature \(\kappa(t)\) of a plane curve \( \vec{r}(t) = \langle x(t), y(t) \rangle \) can be computed using the following formula:
\[
\kappa(t) = \frac{|x'y'' - x''y'|}{((x')^2 + (y')^2)^\frac{3}{2}}
\]
Where:
- \(x'\) and \(y'\) are the first derivatives of \( x(t) \) and \( y(t) \) with respect to \( t \).
- \(x''\) and \(y''\) are the second derivatives of \( x(t) \) and \( y(t) \) with respect to \( t \).
This formula allows us to quantify the "bend" of the curve at any point \( t \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0761ca0b-8a30-47ed-b77e-00687bf0302e%2Fa55f50df-8786-4be0-898a-14ef655ab694%2Fxn1blar_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Curvature of Plane Curves**
Given the vector function of a plane curve \( \vec{r}(t) = \langle t - \sin t, 1 - \cos t \rangle \), we are interested in determining properties of this curve at specific points.
**Key Point:**
At the maximum point \((x, y) = (\pi, 2)\), this occurs when \( t = \pi \). To analyze the curvature of the plane curve at this point, we'll use the formula for curvature.
**Curvature Formula:**
The curvature \(\kappa(t)\) of a plane curve \( \vec{r}(t) = \langle x(t), y(t) \rangle \) can be computed using the following formula:
\[
\kappa(t) = \frac{|x'y'' - x''y'|}{((x')^2 + (y')^2)^\frac{3}{2}}
\]
Where:
- \(x'\) and \(y'\) are the first derivatives of \( x(t) \) and \( y(t) \) with respect to \( t \).
- \(x''\) and \(y''\) are the second derivatives of \( x(t) \) and \( y(t) \) with respect to \( t \).
This formula allows us to quantify the "bend" of the curve at any point \( t \).
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