(a) Find an equation of the tangent line to the graph of the parabola $f(x)\;=\;\dfrac{1}{4}x^{2}$ at the point $(2,1).$

(b) What is the curvature of the graph of $f$ at the point $(2,1)?$

(c) Find an equation of the tangent line to the graph of $g(x)\;=\;\dfrac{x^{3}+4}{12}$ at the point $(2,1).$

(d) What is the curvature of the graph of $g$ at the point $( 2,1)?$

(e) Graph both curves and their tangent lines.

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Solution (a) $f(x)\;=\;\dfrac{1}{4}x^{2};$ $f^{\prime} (x)\;=\;\dfrac{1}{2}x$. An equation of the tangent line to the graph of $f$ at the point $( 2,1)$ is $$\begin{eqnarray*} y-1 &=&1\,{\cdot}\, (x-2)\qquad \color{#0066A7}{f^{\prime} (2)\;=\;{1}} \\[4pt] y &=&x-1 \end{eqnarray*}$$

(b) $f^{\prime \prime} (x)\;=\;\dfrac{1}{2};\quad f^{\prime} (2)\;=\;1;\quad f^{\prime \prime} (2)\;=\;\dfrac{1}{2}$. We use formula (9) and evaluate $\kappa$ at $x=2$. $$\begin{equation*} \kappa\;=\;\dfrac{\left\vert f^{\prime \prime} ( 2) \right\vert }{ \left( 1+\left[ f^{\prime} ( 2) \right] ^{2}\right) ^{3/2}}=\dfrac{ \dfrac{1}{2}}{( 1+1) ^{3/2}}=\dfrac{1}{2\,{\cdot}\, 2\sqrt{2}}=\dfrac{1 }{4\sqrt{2}}\approx 0.177 \end{equation*}$$

(c) $g(x)\;=\;\dfrac{x^{3}+4}{12};\quad g^{\prime} (x)\;=\;\dfrac{3x^{2}}{12}=\dfrac{x^{2}}{4}$. An equation of the tangent line to the graph of $g$ at the point $(2,1)$ is $$\begin{eqnarray*} y-1 &=&1\,{\cdot}\, ( x-2)\qquad \color{#0066A7}{g^{\prime} (2)\;=\;{1}} \\[4pt] y &=&x-1 \end{eqnarray*}$$

Figure 20

(d) $g^{\prime \prime} (x)\;=\;\dfrac{x}{2};\quad g^{\prime} (2)\;=\;1;\quad g^{\prime \prime} (2)\;=\;1$. The curvature $\kappa$ at the point $(2,1)$ is $$\begin{equation*} \kappa\;=\;\dfrac{\vert g^{\prime \prime} ( 2) \vert }{ ( 1+[ g^{\prime} ( 2) ] ^{2}) ^{3/2}}=\dfrac{1 }{( 1+1) ^{3/2}}=\dfrac{1}{2\sqrt{2}}\approx 0.354 \end{equation*}$$

(e) The functions $f$ and $g$ are graphed in Figure 20. Both graphs have the same tangent line at $(2,1),$ but the tangent line to the graph of $y=\dfrac{1}{4}x^{2}$ at $(2,1)$ turns more slowly $(\kappa \approx 0.177)$ than the tangent line to the graph of $y=\dfrac{x^{3}+4}{12}$ at $(2,1)$ $(\kappa \approx 0.354)$.