Question:
Find the equation of the tangent and the normal to the following curves at the indicated points:
$y=x^{4}-6 x^{3}+13 x^{2}-10 x+5$ at $x=1 y=3$
Solution:
finding slope of the tangent by differentiating the curve
$\frac{d y}{d x}=4 x^{3}-18 x^{2}+26 x-10$
$\mathrm{m}$ (tangent) at $(\mathrm{x}=1)=2$
normal is perpendicular to tangent so, $m_{1} m_{2}=-1$
$\mathrm{m}$ (normal) at $(\mathrm{x}=1)=-\frac{1}{2}$
equation of tangent is given by $y-y_{1}=m($ tangent $)\left(x-x_{1}\right)$
$y-3=2(x-1)$
$y=2 x+1$
equation of normal is given by $y-y_{1}=m($ normal $)\left(x-x_{1}\right)$
$y-3=-\frac{1}{2}(x-1)$
$2 y=7-x$