/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 75 The function \(f(x)=\sin n x,\) ... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 75

The function \(f(x)=\sin n x,\) where \(n\) is a positive real number, has a local maximum at \(x=\pi /(2 n)\) Compute the curvature \(\kappa\) of \(f\) at this point. How does \(\kappa\) vary (if at all) as \(n\) varies?

Short Answer

Expert verified
Answer: The curvature, \(\kappa\), at the point \(x = \pi/(2n)\) is equal to \(n^2\). As \(n\) increases, the curvature also increases, and as \(n\) decreases, the curvature decreases.

Step by step solution

01

Find the first derivative of \(f(x)\)

To find the first derivative, we use the chain rule: $$\frac{d}{dx}(\sin(nx)) = n\cos(nx).$$ So \(f'(x) = n\cos(nx)\).
02

Find the second derivative of \(f(x)\)

Using the chain rule, we can find the second derivative: $$\frac{d^2}{dx^2}(\sin(nx)) = -n^2\sin(nx).$$ So \(f''(x) = -n^2\sin(nx)\).
03

Evaluate the derivatives at \(x=\pi/(2n)\)

At \(x=\pi/(2n)\), we have: $$f'(\pi/(2n)) = n\cos(n(\pi/(2n))) = n\cos(\pi/2) = 0,$$ and $$f''(\pi/(2n)) = -n^2\sin(n(\pi/(2n))) = -n^2\sin(\pi/2) = -n^2.$$
04

Compute the curvature \(\kappa\)

Plugging the values of the derivatives into the formula for curvature, we have: $$\kappa = \frac{\lvert -n^2 \rvert}{(1 + (0)^2)^{3/2}} = n^2.$$
05

Analyze the variation of \(\kappa\) with respect to \(n\)

From the expression for \(\kappa\), we see that it is equal to \(n^2\). Since \(n\) is a positive real number, as \(n\) increases, the curvature \(\kappa = n^2\) also increases. Similarly, if \(n\) decreases, the curvature also decreases. Thus, as \(n\) varies, the curvature \(\kappa = n^2\) also varies. The curvature increases with increasing \(n\) and decreases with decreasing \(n\).

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Most popular questions from this chapter

Relationship between \(\mathbf{T}, \mathbf{N},\) and a Show that if an object accelerates in the sense that \(d^{2} s / d t^{2}>0\) and \(\kappa \neq 0,\) then the acceleration vector lies between \(\mathbf{T}\) and \(\mathbf{N}\) in the plane of \(\mathbf{T}\) and \(\mathbf{N}\). If an object decelerates in the sense that \(d^{2} s / d t^{2}<0,\) then the acceleration vector lies in the plane of \(\mathbf{T}\) and \(\mathbf{N},\) but not between \(\mathbf{T}\) and \(\mathbf{N}\)

Given a fixed vector \(\mathbf{v},\) there is an infinite set of vectors \(\mathbf{u}\) with the same value of proj\(_{\mathbf{v}} \mathbf{u}\). Let \(\mathbf{v}=\langle 0,0,1\rangle .\) Give a description of all position vectors \(\mathbf{u}\) such that \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}=\operatorname{proj}_{\mathbf{v}}\langle 1,2,3\rangle\).

For the following vectors u and \(\mathbf{v}\) express u as the sum \(\mathbf{u}=\mathbf{p}+\mathbf{n},\) where \(\mathbf{p}\) is parallel to \(\mathbf{v}\) and \(\mathbf{n}\) is orthogonal to \(\mathbf{v}\). \(\mathbf{u}=\langle 4,3\rangle, \mathbf{v}=\langle 1,1\rangle\)

An object moves along a path given by \(\mathbf{r}(t)=\langle a \cos t+b \sin t, c \cos t+d \sin t, e \cos t+f \sin t\rangle\) for \(0 \leq t \leq 2 \pi\) a. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is a circle (in a plane)? b. What conditions on \(a, b, c, d, e,\) and \(f\) guarantee that the path is an ellipse (in a plane)?

A race Two people travel from \(P(4,0)\) to \(Q(-4,0)\) along the paths given by $$ \begin{aligned} \mathbf{r}(t) &=(4 \cos (\pi t / 8), 4 \sin (\pi t / 8)\rangle \text { and } \\\ \mathbf{R}(t) &=\left(4-t,(4-t)^{2}-16\right) \end{aligned} $$ a. Graph both paths between \(P\) and \(Q\) b. Graph the speeds of both people between \(P\) and \(Q\) c. Who arrives at \(Q\) first?
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