Tangent and normal line to a cubic

Text

Consider the curve $y=4-3x+\tfrac{1}{2}x^{3}$ and the point $P$ with abscissa $x=2$. Write the equations of the tangent and normal lines at $P$.

Solution

Ordinate: $y(2)=4-6+\tfrac12\cdot8=2$, so $P=(2,2)$. Derivative: $y'(x)=-3+\tfrac{3}{2}x^{2}$, and the tangent slope is $m=y'(2)=3$. Tangent: $y-2=3(x-2)\Rightarrow y=3x-4$ (so $m=3$, $q=-4$). Normal: slope $-\tfrac{1}{3}$, $y-2=-\tfrac13(x-2)\Rightarrow y=-\tfrac{x}{3}+\tfrac{8}{3}$. $\boxed{\text{tangent: } y=3x-4;\qquad \text{normal: } y=-\tfrac{x}{3}+\tfrac{8}{3}}$