Question

Find the equations of the tangent and normal to the given curve at the indicated point:

$y=x^2$ at $(0,0)$.

Answer 1

We have
$y=x^2$

$\Rightarrow \frac{dy}{dx}=2x$

$\therefore {\left(\frac{dy}{dx}\right)}_{(0,0)}=0$

Thus, the slope of the tangent at $(0,0)$ is $0$. Ergo equation of the tangent is given as,

$(y-0)=0(x-0)\Rightarrow y=0$

The slope of the normal at $(0,0)$ is $\frac{-1}{\text{Slope of the tangent at}(0,0)}=-\infty$.

Therefore, the equation of the normal at $(0,0)$ is given as,
$(y-0)=-\infty (x-0)\Rightarrow x=0$