Chapter 12 - Vector-Valued Functions - 12.5 Exercises - Page 861: 52

$$\eqalign{ & \left( {\bf{a}} \right)\left( {1,1} \right){\text{ and }}\left( { - 1, - 1} \right) \cr & \left( {\bf{b}} \right)\mathop {\lim }\limits_{x \to \infty } K = 0 \cr} $$

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