Chapter 12 - Vector-Valued Functions - 12.5 Exercises - Page 861: 49

$$\eqalign{ & \left( {\bf{a}} \right)\left( { \pm \frac{1}{{\root 4 \of {45} }}, \pm \frac{1}{{{{\left( {45} \right)}^{3/4}}}}} \right) \cr & \left( {\bf{b}} \right)\mathop {\lim }\limits_{x \to \infty } K = 0 \cr} $$

$$\eqalign{ & y = {x^3} \cr & {\text{Calculate the curvature, use }}K = \frac{{\left| {y''\left( x \right)} \right|}}{{{{\left( {1 + {{\left[ {y'\left( x \right)} \right]}^2}} \right)}^{3/2}}}} \cr & {\text{Find }}y'\left( x \right){\text{ and }}y''\left( x \right),{\text{ }} \cr & y'\left( x \right) = \frac{d}{{dx}}\left[ {{x^3}} \right] \cr & y'\left( x \right) = 3{x^2} \cr & y''\left( x \right) = \frac{d}{{dx}}\left[ {3{x^2}} \right] \cr & y''\left( x \right) = 6x \cr & \underbrace {K = \frac{{\left| {y''\left( x \right)} \right|}}{{{{\left( {1 + {{\left[ {y'\left( x \right)} \right]}^2}} \right)}^{3/2}}}}}_ \Downarrow \cr & K = \frac{{\left| {6x} \right|}}{{{{\left( {1 + {{\left( {3{x^2}} \right)}^2}} \right)}^{3/2}}}} \cr & K = \frac{{6x}}{{{{\left( {1 + 9{x^4}} \right)}^{3/2}}}} \cr & \left( {\bf{a}} \right){\text{ Find the point where }}K{\text{ is maximum}} \cr & K' = \frac{{{{\left( {1 + 9{x^4}} \right)}^{3/2}}\left( 6 \right) - 9x{{\left( {1 + 9{x^4}} \right)}^{1/2}}\left( {36{x^3}} \right)}}{{{{\left( {1 + 9{x^4}} \right)}^3}}} \cr & K' = \frac{{\left( {1 + 9{x^4}} \right)\left( 6 \right) - 9x\left( {36{x^3}} \right)}}{{{{\left( {1 + 9{x^4}} \right)}^3}}} \cr & K' = \frac{{6 + 54{x^4} - 324{x^4}}}{{{{\left( {1 + 36{x^4}} \right)}^{5/2}}}} \cr & K' = \frac{{6 - 270{x^4}}}{{{{\left( {1 + 36{x^4}} \right)}^{5/2}}}} \cr & K' = 0 \cr & 6 - 270{x^4} = 0 \cr & {x^4} = \frac{1}{{45}} \cr & x = \pm \frac{1}{{\root 4 \of {45} }} \cr & y = {x^3} \cr & y = {\left( { \pm \frac{1}{{\root 4 \of {45} }}} \right)^3} = \pm \frac{1}{{{{\left( {45} \right)}^{3/4}}}} \cr & K{\text{ is maximum when }}x = \pm \root 4 \of {45} ,{\text{ at the points}} \cr & \left( { \pm \frac{1}{{\root 4 \of {45} }}, \pm \frac{1}{{{{\left( {45} \right)}^{3/4}}}}} \right) \cr & \cr & \left( {\bf{b}} \right){\text{ Find }}\mathop {\lim }\limits_{x \to \infty } K \cr & \mathop {\lim }\limits_{x \to \infty } \frac{{6x}}{{{{\left( {1 + 9{x^4}} \right)}^{3/2}}}} = 0 \cr & \cr & {\text{Summary}} \cr & \left( {\bf{a}} \right)\left( { \pm \frac{1}{{\root 4 \of {45} }}, \pm \frac{1}{{{{\left( {45} \right)}^{3/4}}}}} \right) \cr & \left( {\bf{b}} \right)\mathop {\lim }\limits_{x \to \infty } K = 0 \cr} $$