Tangent

$y\=f\prime \left(a\right) \left(x-a\right)\+f\left(a\right)$

Normal

$y\=-\frac{1}{f\prime \left(a\right)} \left(x-a\right)\+f\left(a\right)$

Table 3.1.1   Equations for tangent and normal lines

Example 3.1.1

Example 3.1.2

Which point on the graph of $g\left(x\right)\=7 x\+9-x^2$ is closest to the point $\left(3\,10\right)$?

What is that minimal distance?

Example 3.1.3

Let $C_1$ be the ellipse defined by $x\=u\left(\mathrm{\θ}\right)\=5 \cos \left(\mathrm{\θ}\right)\,y\=v\left(\mathrm{\θ}\right)\=3 \sin \left(\mathrm{\θ}\right)\,0\le \mathrm{\θ}\le 2 \mathrm{\π}$, and let $C_2$ be the curve defined parametrically by $x\=U\left(\mathrm{\theta }\right)\=\frac{16}{5}{\cos }^3\left(\mathrm{\theta }\right)$, $y\=V\left(\mathrm{\theta }\right)\=-\frac{16}{3}{\sin }^3\left(\mathrm{\theta }\right)$,  $0\le \mathrm{\θ}\le 2 \mathrm{\π}$.