Chapter 13.3, Problem 32E

To find: The curvature of $y=\tan x$ using Formula 11.

Solution: The curvature of $y=\tan x$ is $\underset{\_}{\frac{2{\sec }^{2}x\left|\tan x\right|}{{\left(1+{\sec }^{4}x\right)}^{\frac{3}{2}}}}$.

Given:

Plane equation as $y=\tan x$.

Formula:

Consider a plane curve equation in the form of $y=f\left(x\right)$.

Write the expression for curvature according to Formula 11.

$K\left(x\right)=\frac{\left|f^{″}\left(x\right)\right|}{{\left[1+{\left(f^{\prime }\left(x\right)\right)}^2\right]}^{\frac{3}{2}}}$ (1)

Here,

$f\left(x\right)$ is function of x,

$f^{\prime }\left(x\right)$ is first derivative of function $f\left(x\right)$, and

$f^{″}\left(x\right)$ is second derivative of function $f\left(x\right)$.

Explanation:

The equation $y=\tan x$indicates y is a function of x. Hence consider $y=f\left(x\right)$.

Write the plane equation.

$y=\tan x$

Substitute $f\left(x\right)$ for y.

$f\left(x\right)=\tan x$

Apply differentiation with respect to x on both sides of equation.

$\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}\left(\tan x\right)\\ ={\sec }^{2}t\left\{\because \frac{d}{dt}\left(\tan t\right)={\sec }^{2}t\right\}\end{array}$

Apply differentiation with respect to x on both sides of equation.

$\begin{array}{c}{f}^{″}\left(x\right)=\frac{d}{dx}\left({\sec }^{2}x\right)\\ =\frac{d}{dx}{\left(\sec x\right)}^2\\ =\left(2\sec x\right)\left(\sec x\tan x\right)\left\{\begin{array}{l}\because \frac{d}{dt}\left(t^2\right)=2t\\ \frac{d}{dt}\left(\sec t\right)=\sec t\tan t\end{array}\right\}\\ =2{\sec }^{2}x\tan x\end{array}$

Substitute ${\sec }^{2}x$ for $f^{\prime }\left(x\right)$ and $2{\sec }^{2}x\tan x$ for $f^{″}\left(x\right)$ in equation (1).

$\begin{array}{c}K\left(x\right)=\frac{\left|2{\sec }^{2}x\tan x\right|}{{\left[1+{\left({\sec }^{2}x\right)}^2\right]}^{\frac{3}{2}}}\\ =\frac{2{\sec }^{2}x\left|\tan x\right|}{{\left(1+{\sec }^{4}x\right)}^{\frac{3}{2}}}\end{array}$

Thus, the curvature of $y=\tan x$ is $\underset{\_}{\frac{2{\sec }^{2}x\left|\tan x\right|}{{\left(1+{\sec }^{4}x\right)}^{\frac{3}{2}}}}$.