Question

Show that the second derivative of the function of y = $x^2$is constant, but its curvature varies with x

Short answer

The second derivative of the function is 2 which is constant.

k depends on x thus the curvature varies with x.

Step-by-step solution

Step 1. Given information.

We have to show that the second derivative of the function of y = $x^2$is constant, but its curvature varies with x

Step 2. Explanation.

Let $y=x^2$

then,

$y^{\mathrm{\prime }}=2x\text{and}{y}^{\prime \prime }=2$

Thus the second derivative of the function is 2 which is constant.

Curvature :

If y=f(x) is a twice-differentiable function then the curvature of f is given by

$k=\frac{\left|f^{\prime \prime }\left(x\right)\right|}{{\left(1+{\left(f^{\mathrm{\prime }}\left(x\right)\right)}^2\right)}^{\frac{3}{2}}}$

Substituting $f^{\mathrm{\prime }}\mathrm{\&}{f}^{\prime \prime }$values in the above formula, we get :

$\begin{array}{r}k=\frac{\left|2\right|}{{\left(1+(2x{)}^2\right)}^{\frac{3}{2}}}\\ k=\frac{2}{{\left(1+4x^2\right)}^{\frac{3}{2}}}\end{array}$

k depends on x thus the curvature varies with x.