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Multiple Choice If a smooth curve \(C\) is traced out by the vector function \(\mathbf{r}=\mathbf{r}(t)\), \(a\leq t\leq b,\) then the parameter \(t\) is the arc length if and only if [(a) \(\left\Vert \mathbf{r}(t)\right\Vert\;=1\), (b) \(\left\Vert \mathbf{r}^{\prime}(t)\right\Vert\;=0\), (c) \(\left\Vert \mathbf{r}^{\prime\prime}(t)\right\Vert\;=1\), (d) \(\left\Vert \mathbf{r}^{\prime}(t)\right\Vert\;=1]\) for all \(t.\)

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True or False The curvature \(\kappa\) of a straight line equals the slope of the line.

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True or False The curvature \(\kappa\) of a circle equals the radius of the circle.

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True or False The curvature \(\kappa\) of a twice differentiable smooth curve \(C\) equals the magnitude of the rate of change of the unit tangent vector \(\mathbf{T}\) with respect to arc length.

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True or False The curvature \(\kappa\) of a twice differentiable smooth curve \(C\): \(\mathbf{r}=\mathbf{r}(t)\) is \(\kappa\;=\;\dfrac{\left\Vert \mathbf{r}^{\prime} (t) \times \mathbf{r} ^{\prime \prime} (t) \right\Vert }{\left\Vert \mathbf{r}(t) \right\Vert ^{3}}.\)

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The curvature of a twice differentiable function \(y=f(x)\) is \(\kappa\) = _______.

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The curvature of the circle \(x^{2}+\left( y-2\right) ^{2}=9\) is \(\kappa\) = _______.

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True or False The radius \(\rho\) of the osculating circle at a point \(P\) of a smooth curve \(C\) equals \(\kappa,\) provided \(\kappa \neq 0.\)

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\(\mathbf{r}(t)=4\cos t\mathbf{i}-4\sin t\mathbf{j},\quad 0\leq t\leq 2\pi\)

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\(\mathbf{r}(t)=\sin (3t)\mathbf{i}+\cos (3t)\mathbf{j,} \quad 0\leq t\leq 2\pi\)

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\(\mathbf{r}(t)=t^{2}\mathbf{i}+t\mathbf{j},\quad 0\leq t\leq 4\)

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\(\mathbf{r}(t)=t\mathbf{i}+t^{3}\mathbf{j,}\quad 0\leq t\leq 2\)

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\(\mathbf{r}(t)=(2t+1)\mathbf{i}+(3t-2)\mathbf{j},\quad 0\leq t\leq 5\)

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\(\mathbf{r}(t)=\mathbf{i}+t\mathbf{j},\quad 0\leq t\leq 1\)

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\(\mathbf{r}(t)=\left( \dfrac{2}{\sqrt{13}}t+1\right) \mathbf{i}+\left( \dfrac{3}{\sqrt{13}}t-2\right) \mathbf{j}\), \(0\leq t\leq 5\sqrt{13}\)

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\(\mathbf{r}(t)=\mathbf{i}+t^{2}\mathbf{j},\quad 0\leq t\leq 1\)

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\(\mathbf{r}(t)=\sin t\mathbf{i}+\cos t\mathbf{j}+t\mathbf{k}\)

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\(\mathbf{r}(t)=a\sin t\mathbf{i}+a\cos t\mathbf{j}+\sqrt{1-a^{2}}\kern1pt t\mathbf{k}\)

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\(\mathbf{r}(t)=t^{2}\mathbf{i}+\dfrac{2}{t}\mathbf{j}\)

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\(\mathbf{r}(t)=2t\mathbf{i}+t^{3}\mathbf{j}\)

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\(\mathbf{r}(t)=2\sin t \mathbf{i}+2\cos t\mathbf{j}\)

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\(\mathbf{r}(t)=\cos t\mathbf{i}+2\sin t\mathbf{j}\)

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\(\mathbf{r}(t)=(3t-t^{3})\mathbf{i}+3t^{2} \mathbf{j}\)

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\(\mathbf{r}(t)=(3t-t^{3})\mathbf{i}+(3t+t^{3})\mathbf{j}\)

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\(\mathbf{r}(t)=t\mathbf{i}+2t\mathbf{j}+t\mathbf{k}\)

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\(\mathbf{r}(t)=2t\mathbf{i}+t\mathbf{j}+3t\mathbf{k}\)

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\(\mathbf{r}(t)=\sin ( 2t) \mathbf{i}+\cos ( 2t) \mathbf{j}+t\mathbf{k}\)

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\(\mathbf{r}(t)=\sin t\mathbf{i}\,{+}\,\cos t\mathbf{j}\,{+}\,bt\mathbf{k}\)

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\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+\sqrt{2}t\mathbf{k}\)

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\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{2t}\mathbf{j}+e^{-t}\mathbf{k}\)

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\(\mathbf{r}(t)=\cos ^{3}t\mathbf{i}+\sin ^{3}t\mathbf{j}+\mathbf{k}\)

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\(\mathbf{r}(t)=4\cos ^{3}t\mathbf{i}+ 3\mathbf{j} + 4\sin^{3}t\mathbf{k}\)

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\(y=x^{2}\) at \((1,1)\)

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\(y=2x-x^{2}\) at \((1,1)\)

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\(y=x^{2}-x^{3}\) at \((1,0)\)

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\(y=x^{-3/2}\) at \((1,1)\)

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\(y=\sqrt{x}\) at \(\big(2,\sqrt{2}\big)\)

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\(y=\dfrac{1}{\sqrt{x}}\) at \((1,1)\)

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\(4x^{2}+9y^{2}=36\) at \((0,2)\)

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\(y=\sec x-1\) at \(\left(\! \dfrac{\pi}{4}, \sqrt{2}-1\!\right)\)

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\(y=e^{x}\) at \((0,1)\)

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\(y=\ln (x+1)\) at \(( 2,\ln 3)\)

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\(y=x^{3}-6x\) at \((1,-5)\)

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\(y=\dfrac{1}{x^{2}}\) at \((-1,1)\)

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\(y=\sin x\) at \(\left( \dfrac{\pi }{2},1\right)\)

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\(y=e^{-x}\) at \((0,1)\)

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\(x^{2}+xy+y^{2}=3\) at \((1,1)\)

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\(y^{2}-y+x=0\) at \((0,0)\)

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\(y=\ln ( \sec x)\) at \(\left(\dfrac{\pi }{4}, \ln \sqrt{2}\right)\)

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\(y=\cosh x\) at \((0,1)\)

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\(\mathbf{r}(t)=3t^{2}\mathbf{i}+(3t-t^{3})\mathbf{j,}\quad t=1\)

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\(\mathbf{r}(t)=t\mathbf{i}+t^{2}\mathbf{j}+t^{3}\mathbf{k},\quad t=1\)

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\(\mathbf{r}(t)=\sin t\mathbf{i}+\cos ( 2t) \mathbf{j},\quad t=\dfrac{\pi }{4}\)

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\(\mathbf{r}(t)=\sin t\mathbf{i}+\cos t\mathbf{j}+bt\mathbf{k,}\quad b>0,\quad t=\dfrac{\pi }{4}\)

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\(\mathbf{r}(t)=\sin ( 2t) \mathbf{i}+\cos ( 2t) \mathbf{j}+t\mathbf{k,}\quad t=\dfrac{\pi }{4}\)

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\(\mathbf{r}(t)=\cos ^{3}t\mathbf{i}+\sin ^{3}t\mathbf{j}+\mathbf{k,}\quad t=\dfrac{\pi }{3}\)

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\(\mathbf{r}(t)=e^{t}\mathbf{i}+e^{-t}\mathbf{j}+\sqrt{2}\kern1ptt\mathbf{k},\quad t=0\)

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\(\mathbf{r}(t)=a(3t-t^{3})\mathbf{i}+3at^{2}\mathbf{j}+a(3t+t^{3})\mathbf{k,}\quad a>0, \quad t=1\)

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\(\mathbf{r}(t)=4a\cos ^{3}t\mathbf{i}+4a\sin ^{3}t\mathbf{j}+3a\cos ( 2t) \mathbf{k,}\quad a>0,\quad t=\dfrac{\pi }{4}\)

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\(\mathbf{r}(t)=t\mathbf{i}+2t\mathbf{j}+\sqrt{1-5t^{2}}\kern1pt\mathbf{k},\quad -\dfrac{\sqrt{5}}{5} < t< \dfrac{\sqrt{5}}{5},\quad t=0\)

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Radius of Curvature Show that the radius of curvature of the parabola \(y=ax^{2}+bx+c\) is a minimum at its vertex.

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Radius of Curvature Show that the radii of curvature at the ends of the axes of the ellipse \(b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}\) are \(\dfrac{b^{2}}{a}\) and \(\dfrac{a^{2}}{b},\quad a>0, b>0.\)

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Maximum Curvature Find the point on the curve \(y=\ln x\) at which the curvature is maximum.

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Maximum Curvature Find the point on the curve \(y=e^{x}\) at which the curvature is maximum.

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Maximum Curvature Find the point(s) on the curve \(y=\dfrac{1}{3}x^{3}\) at which the curvature is maximum.

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Maximum Curvature Find the point(s) on the curve \(y=\sin x\) at which the curvature is maximum.

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Maximum Curvature Find \(\alpha >0\) so that \(\mathbf{r} (t)= \alpha \cos t\mathbf{i} + \alpha \sin t\mathbf{j}+t\mathbf{k}\) has maximum curvature.

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Curvature of a Plane Curve What is the curvature at a point of inflection of a plane curve?

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Curvature of a Catenary Show that the curvature of the catenary \(y=a\cosh \dfrac{x}{a}\), \(a>0\), at any point \((x,y)\) is \(\dfrac{a}{y^{2}}\).

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Curvature of a Cissoid Find the curvature of the cissoid \(y^{2}(2-x)=x^{3}\) at the point \((1,1)\). See the figure.

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Curvature of a Cycloid Find the curvature of the cycloid \(x( \theta)\;=\;\theta -\sin \theta\) and \(y( \theta )\;=\;1-\cos \theta\) at the highest point of an arch. See the figure.

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1. Find the curvature of the curve \(\mathbf{r} (t)=(1-t^{3})\mathbf{i}+t^{2}\mathbf{j}\).
2. Graph \(\mathbf{r}=\mathbf{r}(t)\). Where is the curvature undefined? Do you see any geometric reason for this?

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Curvature of a Spiral Find the curvature of the spiral \(\mathbf{r}(t)=e^{-t}\cos t\mathbf{i}+e^{-t}\sin t\mathbf{j}\) shown in the figure. How does the curvature behave when \(t\rightarrow \infty\)? Do you see any geometric reason for this?

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Curvature Find the curvature \(\kappa\) of the curve \(\mathbf{r}(t)=2a\cos t\mathbf{i}+2a\sin t\mathbf{j}+bt^{2}\mathbf{k},\qquad a>0, b>0.\)

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Curvature Show that the curvature of an ellipse \(\dfrac{x^{2}}{ a^{2}}+\dfrac{y^{2}}{b^{2}}=1,\) \(a> b,\) is a maximum at the points \( \left( \pm a,0\right)\) and is a minimum at the points \(\left( 0,\pm b\right).\)

Source: Contributed by the students at the University of Missouri.

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Curvature Show that the circular helix \(\mathbf{r}( t)\;=\;a\cos t\mathbf{i}+a\sin t\mathbf{j}+t\mathbf{k}\) has constant curvature.

Source: Contributed by the students at the University of Missouri.

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Write a vector equation for the curve \(C\) traced out by \( \mathbf{r}(t)=2t\mathbf{i}+(2t-1)\mathbf{j}+t\mathbf{k}\), \(0\leq t\leq 2\), using arc length \(s\) as the parameter. [Hint: For each \(t\), \(0\leq t\leq 2\), calculate the length \(s(t)\) of the curve from \(0\) to \(t\).]

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Compare the solutions of Problems 13 and 15. Then show how to change the parameter \(t\) of the line \(\mathbf{r}(t)=(at+b)\mathbf{i}+(ct+d) \mathbf{j}\), where either \(a\neq 0\) or \(c\neq 0\), to one that is arc length as measured along the line.

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Suppose a smooth curve \(C\) is traced out by a twice differentiable vector function \(\mathbf{r}=\mathbf{r}(t)\), \(a\leq t\leq b\). If the curvature \(\kappa \neq 0\) at a point \(P\) on \(C\), show that the position vector \(\mathbf{C}\) of the center of the osculating circle at \(P\) is given by \begin{equation*} \mathbf{C}(t)=\mathbf{r}(t)+\rho \mathbf{N}(t) \end{equation*} where \(\mathbf{N}\) is the principal unit normal vector to \(C\) at \(P\), and \(\rho\;=\;\dfrac{1}{\kappa }\).

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Use the result of Problem 81 to find the center and radius of the osculating circle for the helix \(\mathbf{r}(t)=a\sin t\mathbf{i}+a\cos t\mathbf{j}+a^{2}t\mathbf{k}\), \(a>0;\) \begin{equation*} {\bf (a)}\enspace at\enspace t=\dfrac{\pi }{2}.\qquad {\bf (b)}\enspace at\enspace t=\pi. \end{equation*}

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Show that the three vectors \(\mathbf{T}, \mathbf{N}\), and \(\mathbf{B}\) form a collection of mutually orthogonal unit vectors at each point on \(C\).

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Show that \(\dfrac{d\mathbf{T}}{ds}=\kappa (s)\mathbf{N}(s)\).

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Show that \(\dfrac{d\mathbf{B}}{ds}\) is orthogonal to both \(\mathbf{B}(s)\) and \(\mathbf{T}(s).\)

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If the torsion \(\tau (s)\) of \(C\) is defined by the equation \(\dfrac{d\mathbf{B}}{ds}=-\tau \mathbf{N}\), show that \(\dfrac{d\mathbf{N}}{ds}=\tau \mathbf{B}-\kappa \mathbf{T}\).

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Find \(\kappa, \mathbf{T}, \mathbf{N}\), and \(\mathbf{B}\) for \(\mathbf{r}(s)=\dfrac{1}{\sqrt{2}}\left[ \sin s\mathbf{i}+\cos s\mathbf{j}+s\mathbf{k}\right].\)

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Curvature of a Polar Curve Show that the formula for the curvature of a polar curve \(r=f(\theta )\) is \begin{eqnarray*} \\[-24pt] \kappa\;=\;\frac{\left\vert r^{2}+2\!\left( \dfrac{dr}{d\theta }\right) ^{2}-r\left( \dfrac{d^{2}r}{d\theta ^{2}}\right) \right\vert }{\left[ r^{2}+\left( \dfrac{dr}{d\theta }\right) ^{2}\right] ^{3/2}} \end{eqnarray*}

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\(r=2\cos ( 2\theta )\) at \(\theta\;=\;\dfrac{\pi }{12}\)

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\(r=e^{a\theta }\) at \(\theta\;=\;\dfrac{\pi }{2}, a>0\)

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\(r=a\theta\) at \(\theta\;=\;1\) and \(a>0\)

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\(r=1-\cos \theta\) at \(\theta\;=\;0\)

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\(r=3-2\sin \theta\) at \(\theta\;=\;\dfrac{\pi }{6}\)

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\(r=2+3\cos \theta\) at \(\theta\;=\;\dfrac{\pi }{3}\)

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Use the figure below to show that the coordinates \((h,k)\) of the center of curvature of \(y=f(x)\) are \begin{equation*} h=x-\rho \sin \phi\qquad k=y+\rho \cos \phi \end{equation*} where \(\rho \) is the radius of curvature. Show that \begin{equation*} \sin \phi\;=\;\frac{y^{\prime} }{\sqrt{1+( y^{\prime} ) ^{2}}}\qquad \hbox{and}\qquad \cos \phi\;=\;\frac{1}{\sqrt{1+( y^{\prime} ) ^{2}}} \end{equation*} so \begin{equation*} h=x-\frac{y^{\prime} \left[ 1+( y^{\prime} ) ^{2}\right] }{y^{\prime \prime}}\qquad \hbox{and}\qquad k=y+\frac{1+( y^{\prime} ) ^{2}}{y^{\prime \prime} } \end{equation*}

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\(y=x^{2}\) at \(x=1\)

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\(y=\sin x\) at \(x=\dfrac{\pi }{2}\)

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\(y=\dfrac{x}{x+1}\) at \((0, 0)\)

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\(x^{3}+y^{3}=4xy\) at \((2, 2)\)

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\(xy=4\) at \(x=2\)

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As a point \(P\) moves along a curve \(C\), the center of curvature corresponding to \(P\) traces out a curve \(C_{1}\) called the evolute of \(C\); conversely, \(C\) is the involute of \(C_{1}\). Show that parametric equations of the evolute of \(y=\dfrac{1}{2}x^{2}\) are \(h=-x^{3}\), \(k=\dfrac{3}{2}x^{2}+1\). Then eliminate the parameter \(x\) to obtain \begin{equation*} h^{2}=\frac{8}{27}\left( k-1\right) ^{3} \end{equation*}

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Refer to Problem 101. Find parametric equations and a rectangular equation for the evolute of \(\mathbf{r}( t)\;=\;a\cos t\mathbf{i}+b\sin t\mathbf{j}\), \(a>0, b>0, 0\leq t\leq 2\pi \).