Question

1. find the curvature of the following plane curves: (a) $y = e^{3x}$. (b) $y = x^{2}$.

Explanation:

Step1: Recall the curvature formula for a plane - curve $y = f(x)$

The curvature $\kappa$ of a plane - curve $y = f(x)$ is given by $\kappa=\frac{\vert y''\vert}{(1 + y'^{2})^{\frac{3}{2}}}$.

Step2: For $y = e^{3x}$

First, find the first - derivative:
Using the chain - rule, if $y = e^{3x}$, then $y'=\frac{d}{dx}(e^{3x})=3e^{3x}$.
Second, find the second - derivative:
$y''=\frac{d}{dx}(3e^{3x}) = 9e^{3x}$.
Then, substitute $y'$ and $y''$ into the curvature formula:
$\kappa=\frac{\vert9e^{3x}\vert}{(1+(3e^{3x})^{2})^{\frac{3}{2}}}=\frac{9e^{3x}}{(1 + 9e^{6x})^{\frac{3}{2}}}$.

Step3: For $y=x^{2}$

First, find the first - derivative:
$y'=\frac{d}{dx}(x^{2}) = 2x$.
Second, find the second - derivative:
$y''=\frac{d}{dx}(2x)=2$.
Then, substitute $y'$ and $y''$ into the curvature formula:
$\kappa=\frac{\vert2\vert}{(1+(2x)^{2})^{\frac{3}{2}}}=\frac{2}{(1 + 4x^{2})^{\frac{3}{2}}}$.

Answer:

(a) The curvature of $y = e^{3x}$ is $\frac{9e^{3x}}{(1 + 9e^{6x})^{\frac{3}{2}}}$.
(b) The curvature of $y=x^{2}$ is $\frac{2}{(1 + 4x^{2})^{\frac{3}{2}}}$.