Problem
The formula $k(t)=\frac{\left|f^{\prime} g^{\prime \prime}-f^{\prime \prime} g^{\prime}\right|}{\left(\left(f^{\prime}\right)^{2}+\left(g^{\prime}\right)^{2}\right)^{3 / 2}}$ expresses the curvature of the curve $r(t)=\langle f(t), g(t)\rangle$, where $f$ and $g$ are twice differentiable and all the derivatives are taken with respect to $t$. Use this formula to find the curvature function of the following curve. \[ r(t)=\left\langle t, 5 t^{2}\right\rangle \] The curvature function is $k(t)=$
Solution
Step 1 ：Given the curve \(r(t)=\langle t, 5 t^{2}\rangle\), we have \(f(t) = t\) and \(g(t) = 5t^2\).
Step 2 ：Find the first and second derivatives of \(f(t)\) and \(g(t)\). We have \(f'(t) = 1\), \(g'(t) = 10t\), \(f''(t) = 0\), and \(g''(t) = 10\).
Step 3 ：Substitute these into the formula for curvature \(k(t)=\frac{|f'g''-f''g'|}{(f'^{2}+g'^{2})^{3 / 2}}\). The numerator is \(10\) and the denominator is \((100t^{2} + 1)^{1.5}\).
Step 4 ：Finally, the curvature function is \(k(t)=\frac{10}{(100t^{2} + 1)^{1.5}}\).
