Equation of a Normal Line - Algebraic Functions

$\mathit{y}\mathbf{=}\frac{{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{4}}{\mathit{x}\mathbf{+}\mathbf{1}}\mathbf{ }\mathbf{a}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{4}\mathbf{)}\mathbf{ }$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{1}\mathbf{:}\mathbf{ }\mathbf{G}\mathbf{e}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{D}\mathbf{e}\mathbf{r}\mathbf{i}\mathbf{v}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{v}\mathbf{e}\mathbf{ }\mathbf{b}\mathbf{y}\mathbf{ }\mathbf{A}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{y}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{ }\mathbf{Q}\mathbf{u}\mathbf{o}\mathbf{t}\mathbf{i}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{R}\mathbf{u}\mathbf{l}\mathbf{e}$

$$\begin{gathered} \text{ \\ ddx(uv)=vdudx-udvdxv2} \end{gathered}$$

$\mathrm{Given}:$

$$\begin{gathered} \text{ \\ u=x2-4} \end{gathered}$$

$$\begin{gathered} \text{ \\ dudx=2x} \end{gathered}$$

$$\begin{gathered} \text{ \\ v=x+1} \end{gathered}$$

$$\begin{gathered} \text{ \\ dvdx=1 \\ } \end{gathered}$$

$\frac{dy}{\mathit{dx}}\mathit{=}\frac{x\mathit{+}\mathit{1}\mathit{\left(}\mathit{2}x\mathit{\right)}\mathit{-}{x}^{\mathit{2}}\mathit{-}\mathit{4}\mathit{\left(}\mathit{1}\mathit{\right)}}{\mathit{(}x\mathit{+}\mathit{1}{\mathit{)}}^{\mathit{2}}}$

$$\begin{gathered} \text{ \\ dydx=2x2+2x-x2-4(x+1)2} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=x2+2x+4(x+1)2 \\ } \end{gathered}$$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{2}\mathbf{:}\mathbf{ }\mathbf{G}\mathbf{e}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{S}\mathbf{l}\mathbf{o}\mathbf{p}\mathbf{e}\mathbf{ }\mathbf{a}\mathbf{t}\mathbf{ }\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{-}\mathbf{4}\mathbf{)}$

$$\begin{gathered} \text{ \\ m=02+2(0)+4(0+1)2} \end{gathered}$$

$$\begin{gathered} \text{ \\ m=4 \\ } \end{gathered}$$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{3}\mathbf{:}\mathbf{ }\mathbf{A}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{y}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{e}\mathbf{q}\mathbf{u}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{ }\mathbf{o}\mathbf{f}\mathbf{ }\mathbf{a}\mathbf{ }\mathbf{N}\mathbf{o}\mathbf{r}\mathbf{m}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{L}\mathbf{i}\mathbf{n}\mathbf{e} \mathrm{at} \mathrm{point} (0, -4)$

$$\begin{gathered} \text{ \\ y-y1=-1m(x-x1)} \end{gathered}$$

$$\begin{gathered} \text{ \\ y+4=-14(x-0)-4} \end{gathered}$$

$$\begin{gathered} \text{ \\ -4y-16=x} \end{gathered}$$

$$\begin{gathered} \text{ \\ x+4y+16=0 \\ } \end{gathered}$$

$\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{A}\mathbf{n}\mathbf{s}\mathbf{w}\mathbf{e}\mathbf{r}\mathbf{:}\mathit{ }\mathit{x}\mathit{+}\mathit{4}\mathit{y}\mathit{+}\mathit{16}\mathit{=}\mathit{0}$

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Normal Line Equation to the curves at the given point of tangency

$\mathit{y}\mathbf{=}\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}{\mathbf{)}}^{\mathbf{2}}\mathbf{ }\mathbf{a}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{ }\mathbf{9}\mathbf{)}\mathbf{ }$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p} 1: \mathbf{G}\mathbf{e}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{D}\mathbf{e}\mathbf{r}\mathbf{i}\mathbf{v}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{v}\mathbf{e}\mathbf{ }\mathbf{b}\mathbf{y}\mathbf{ }\mathbf{A}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{y}\mathbf{i}\mathbf{n}\mathbf{g}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{G}\mathbf{e}\mathbf{n}\mathbf{e}\mathbf{r}\mathbf{a}\mathbf{l} \mathbf{P}\mathbf{o}\mathbf{w}\mathbf{e}\mathbf{r}\mathbf{ }\mathbf{R}\mathbf{u}\mathbf{l}\mathbf{e}$

$$\begin{gathered} \text{ \\ ddx(cun)=cnun-1dudx} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=2x2-12-12x} \end{gathered}$$

$$\begin{gathered} \text{ \\ dydx=4xx2-1 \\ } \end{gathered}$$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{2}\mathbf{:}\mathbf{ }\mathbf{G}\mathbf{e}\mathbf{t}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{s}\mathbf{l}\mathbf{o}\mathbf{p}\mathbf{e}\mathbf{ }\mathbf{a}\mathbf{t}\mathbf{ }\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{ }\mathbf{9}\mathbf{)}$

$$\begin{gathered} \text{ \\ m=4(-2)-22-1} \end{gathered}$$

$$\begin{gathered} \text{ \\ m=-8(3)} \end{gathered}$$

$$\begin{gathered} \text{ \\ m=-24 \\ } \end{gathered}$$

$\mathbf{S}\mathbf{t}\mathbf{e}\mathbf{p}\mathbf{ }\mathbf{3}\mathbf{:}\mathbf{ }\mathbf{A}\mathbf{p}\mathbf{p}\mathbf{l}\mathbf{y}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{E}\mathbf{q}\mathbf{u}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{ }\mathbf{o}\mathbf{f}\mathbf{ }\mathbf{N}\mathbf{o}\mathbf{r}\mathbf{m}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{L}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{ }\mathbf{a}\mathbf{t}\mathbf{ }\mathbf{p}\mathbf{o}\mathbf{i}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{ }\mathbf{9}\mathbf{)}$

$$\begin{gathered} \text{ \\ y-y1=-1m(x-x1)} \end{gathered}$$

$$\begin{gathered} \text{ \\ y-9=--124(x--2)} \end{gathered}$$

$$\begin{gathered} \text{ \\ y-9=124(x+2)24} \end{gathered}$$

$$\begin{gathered} \text{ \\ 24} \end{gathered}$$y-216=x+2

24y-216-x-2=0

$$\begin{gathered} \text{ \\ x-24y+218=0 \\ } \end{gathered}$$

$\mathbf{F}\mathbf{i}\mathbf{n}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{A}\mathbf{n}\mathbf{s}\mathbf{w}\mathbf{e}\mathbf{r}\mathbf{:}\mathbf{ }\mathit{x}\mathbf{-}\mathbf{24}\mathit{y}\mathbf{+}\mathbf{218}\mathbf{=}\mathbf{0}$

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Find the equation of the tangent line and normal line

$\mathit{F}\mathit{i}\mathit{n}\mathit{d}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{ }\mathit{o}\mathit{f}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathit{g}\mathit{e}\mathit{n}\mathit{t}\mathbf{ }\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{n}\mathit{o}\mathit{r}\mathit{m}\mathit{a}\mathit{l}\mathbf{ }\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathbf{ }$

$$\begin{gathered} \text{ \\ to the graph of y=4x+1 at 1, 2} \end{gathered}$$

$$\begin{gathered} \text{ \\ } \end{gathered}$$

$\mathit{S}\mathit{t}\mathit{e}\mathit{p}\mathbf{ }\mathbf{1}\mathbf{:}\mathbf{ }\mathit{U}\mathit{s}\mathit{e}\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{d}\mathit{i}\mathit{f}\mathit{f}\mathit{e}\mathit{r}\mathit{e}\mathit{n}\mathit{t}\mathit{i}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathbf{ }\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{u}\mathit{l}\mathit{a}\mathbf{ }$

$$\begin{gathered} \text{ \\ ddxcu=-cu2ddxu } \end{gathered}$$

$$\begin{gathered} \text{ \\ y'=-4x+12 (1) \\ } \end{gathered}$$

$\mathit{:}\mathit{ }\mathit{A}\mathit{t}\mathit{ }\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t}\mathit{ }\mathit{(}\mathit{1}\mathit{,}\mathit{ }\mathit{2}\mathit{)}$

$$\begin{gathered} \text{ \\ dydx=-41+12 =-1 \\ } \end{gathered}$$

$\mathrm{Hence}, \mathrm{slope} \mathrm{of} \mathrm{the} \mathrm{tangent} \mathrm{line} \mathrm{at} (1, 2) \mathrm{equals} -1$

$\mathrm{while} \mathrm{slope} \mathrm{of} \mathrm{the} \mathrm{normal} \mathrm{line} \mathrm{at} \mathrm{the} \mathrm{point} \mathrm{is} 1.$

$\mathit{Step\; 2:\; U}\mathit{s}\mathit{i}\mathit{n}\mathit{g}\mathit{ }\mathit{t}\mathit{h}\mathit{e}\mathit{ }\mathit{p}\mathit{o}\mathit{i}\mathit{n}\mathit{t-}\mathit{s}\mathit{l}\mathit{o}\mathit{p}\mathit{e}\mathit{ }\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{ }\mathit{o}\mathit{f}\mathit{ }\mathit{t}\mathit{h}\mathit{e}\mathit{ }\mathit{e}\mathit{q}\mathit{u}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{ }\mathit{o}\mathit{f}\mathit{ }\mathit{t}\mathit{h}\mathit{e}\mathit{ }\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{.}$

$$\begin{gathered} \text{ \\ y-y1=m(x-x1) \\ } \end{gathered}$$

$\mathbf{E}\mathbf{q}\mathbf{u}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{ }\mathbf{o}\mathbf{f}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{g}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{ }\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{:}$

$$\begin{gathered} \text{ \\ y-2=-1(x-1)} \end{gathered}$$

$$\begin{gathered} \text{ \\ y-2=-x+1} \end{gathered}$$

$$\begin{gathered} \text{ \\ x+y-3=0 \\ } \end{gathered}$$

$\mathbf{E}\mathbf{q}\mathbf{u}\mathbf{a}\mathbf{t}\mathbf{i}\mathbf{o}\mathbf{n}\mathbf{ }\mathbf{o}\mathbf{f}\mathbf{ }\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{ }\mathbf{n}\mathbf{o}\mathbf{r}\mathbf{m}\mathbf{a}\mathbf{l}\mathbf{ }\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{:}$

$$\begin{gathered} \text{ \\ y-2=1(x-1)} \end{gathered}$$

$$\begin{gathered} \text{ \\ y-x-1=0} \end{gathered}$$

$$\begin{gathered} \text{ \\ x-y+1=0 \\ } \end{gathered}$$

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