Question

Use formula 11 to find the curvature

y = tanx

Short answer

\(k(x) = \frac{{2{{\sec }^2}x\tan x}}{{{{\left( {1 + \left( {{{\sec }^4}x} \right)} \right)}^{\frac{3}{2}}}}}\)

Step-by-step solution

Step 1: Applied the formula 11

By using the formula

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

Step 2: Solution of the curvature

Let’s applied the values in formula

\(\begin{aligned}{l}k(x) = \frac{{\left| {2{{\sec }^2}x\tan x} \right|}}{{{{\left( {1 + {{\left( {{{\sec }^2}x} \right)}^2}} \right)}^{\frac{3}{2}}}}}\\k(x) = \frac{{2{{\sec }^2}x\tan x}}{{{{\left( {1 + \left( {{{\sec }^4}x} \right)} \right)}^{\frac{3}{2}}}}}\end{aligned}\)