Question

Use formula 11 to find the curvature

Short answer

\(k(x) = \frac{{{e^x}\left| {x + 2} \right|}}{{{{\left( {1 + {{\left( {x{e^x} + {e^x}} \right)}^2}} \right)}^{\frac{3}{2}}}}}\)

Step-by-step solution

Step 1: Applied the formula 11

By using the formula

\(\begin{aligned}{l}k(x) = \frac{{\left| {f''(x)} \right|}}{{{{\left( {1 + (f'(x)){}^2} \right)}^{\frac{3}{2}}}}}\\y = x{e^x}\\y' = {e^x} + x{e^x}\\y'' = {e^x}(2 + x)\end{aligned}\)

Step 2: Solution of the curvature

Let’s applied the values in formula

\(\begin{aligned}{l}k(x) = \frac{{\left| {{e^x}(2 + x)} \right|}}{{{{\left( {1 + {{\left( {\left( {1 + x} \right){e^x}} \right)}^2}} \right)}^{\frac{3}{2}}}}}\\k(x) = \frac{{{e^x}\left| {x + 2} \right|}}{{{{\left( {1 + {{\left( {x{e^x} + {e^x}} \right)}^2}} \right)}^{\frac{3}{2}}}}}\end{aligned}\)