|f''(x)| 3/2 expresses the curvature x(x) of a [1 + (f'(x)) ²] ³/2 curvature function of the following curve. Then graph f(x) together with x(x) over the given interval. f(x)=7x², -2≤x≤2 The formula x(x) = - twice-differentiable plane curve y = f(x) as a function of x. Find the
|f''(x)| 3/2 expresses the curvature x(x) of a [1 + (f'(x)) ²] ³/2 curvature function of the following curve. Then graph f(x) together with x(x) over the given interval. f(x)=7x², -2≤x≤2 The formula x(x) = - twice-differentiable plane curve y = f(x) as a function of x. Find the
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 a Plane Curve
The formula for curvature, \(\kappa(x)\), of a twice-differentiable plane curve \(y=f(x)\) as a function of \(x\) is given by:
\[
\kappa(x) = \frac{|f''(x)|}{\left[1 + (f'(x))^2 \right]^{3/2}}
\]
This formula will be used to find the curvature function of a given curve. Here, we are tasked to find the curvature function of the curve:
\[
f(x) = 7x^2, -2 \leq x \leq 2
\]
### Procedure:
1. First, find the first and second derivatives of \(f(x)\):
- \(f'(x)\)
- \(f''(x)\)
2. Substitute these derivatives into the curvature formula to express \(\kappa(x)\).
### Given Curve:
\[
f(x) = 7x^2, \quad -2 \leq x \leq 2
\]
### Solution:
1. Calculate the first derivative \(f'(x)\):
\[
f'(x) = \frac{d}{dx}[7x^2] = 14x
\]
2. Calculate the second derivative \(f''(x)\):
\[
f''(x) = \frac{d}{dx}[14x] = 14
\]
3. Substitute \(f'(x)\) and \(f''(x)\) into the curvature formula:
\[
\kappa(x) = \frac{|14|}{\left[1 + (14x)^2 \right]^{3/2}}
\]
### Final Curvature Function:
\[
\kappa(x) = \frac{14}{\left[1 + 196x^2 \right]^{3/2}}
\]
### Summary:
- The original function \(f(x) = 7x^2\)
- The curvature function \(\kappa(x) = \frac{14}{\left[1 + 196x^2 \right]^{3/2}}\)
**Graphing:** The next step involves graphing the function \(f(x)\) together with the curvature function \(\kappa(x)\) over the given interval \(-2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d576798-e517-451f-b302-9c2c35ed41b2%2Fd6f1e387-e83f-4165-809e-dd8975a8442a%2Fqlt602y_processed.png&w=3840&q=75)
Transcribed Image Text:### Curvature of a Plane Curve
The formula for curvature, \(\kappa(x)\), of a twice-differentiable plane curve \(y=f(x)\) as a function of \(x\) is given by:
\[
\kappa(x) = \frac{|f''(x)|}{\left[1 + (f'(x))^2 \right]^{3/2}}
\]
This formula will be used to find the curvature function of a given curve. Here, we are tasked to find the curvature function of the curve:
\[
f(x) = 7x^2, -2 \leq x \leq 2
\]
### Procedure:
1. First, find the first and second derivatives of \(f(x)\):
- \(f'(x)\)
- \(f''(x)\)
2. Substitute these derivatives into the curvature formula to express \(\kappa(x)\).
### Given Curve:
\[
f(x) = 7x^2, \quad -2 \leq x \leq 2
\]
### Solution:
1. Calculate the first derivative \(f'(x)\):
\[
f'(x) = \frac{d}{dx}[7x^2] = 14x
\]
2. Calculate the second derivative \(f''(x)\):
\[
f''(x) = \frac{d}{dx}[14x] = 14
\]
3. Substitute \(f'(x)\) and \(f''(x)\) into the curvature formula:
\[
\kappa(x) = \frac{|14|}{\left[1 + (14x)^2 \right]^{3/2}}
\]
### Final Curvature Function:
\[
\kappa(x) = \frac{14}{\left[1 + 196x^2 \right]^{3/2}}
\]
### Summary:
- The original function \(f(x) = 7x^2\)
- The curvature function \(\kappa(x) = \frac{14}{\left[1 + 196x^2 \right]^{3/2}}\)
**Graphing:** The next step involves graphing the function \(f(x)\) together with the curvature function \(\kappa(x)\) over the given interval \(-2
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