$0$

$\=\left(\mathbf{R}-\mathbf{P}\right)·\mathbf{N}$

$\=\left(\left[\begin{array}{c}x\\ y\\ z\end{array}\right]-\left[\begin{array}{c}2\\ 3\\ -17\end{array}\right]\right)·\left[\begin{array}{c}-27\\ 32\\ 1\end{array}\right]$

$\= -27\(x-2\+32\left(y-3\right)\+z\+17$

$\= -27 x\+32 y\+z\+54-96\+17$

$\= -27 x\+32 y\+z-25$

$f$

$\doteq -17\+27\left(x-2\right)-32\left(y-3\right)$

$\=27 x-32 y\+\left(96-54-17\right)$

$\=27 x-32 y\+25$

$z$

$\doteq f\left(2\,3\right)\+\left(f_x\left(2\,3\right)\cdot \mathrm{dx}\+f_y\left(2\,3\right)\cdot \mathrm{dy}\right)$

$\=-17\+\left(27\left(x-2\right)-32\left(y-3\right)\right)$

$\=27 x-32 y\+\left(96-54-17\right)$

$\=27 x-32 y\+25$

Initialize

Loading Student:-MultivariateCalculus

$f\left(x\,y\right)\=3 x^2\+5 x y-7 y^2\+4$$\stackrel{\text{assign as function}}{\to }$$f$

Obtain the tangent plane from first principles

$\left[2\,3\,f\left(2\,3\right)\right]\,\genfrac{}{}{0ex}{}{⟨-\frac{\partial }{\partial x} f\left(x\,y\right)\,-\frac{\partial }{\partial y} f\left(x\,y\right)\,1⟩}{\phantom{x\=a}}|\genfrac{}{}{0ex}{}{\phantom{\mathrm{f(x)}}}{x\=2\,y\=3}$

$\stackrel{\text{make plane}}{\to }$

$\mathrm{<<\; Plane\; 2\; >>}$

$\stackrel{\text{representation}}{\to }$

$-27x\&plus;32y\&plus;z\&equals;25$

Obtain the first-degree Taylor polynomial

Obtain the first-degree Taylor polynomial from first principles

$\mathrm{f\_\_x}\left(x\&comma;y\right)\&equals;\frac{\partial }{\partial x} f\left(x\&comma;y\right)$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_x}$

$\mathrm{f\_\_y}\left(x\&comma;y\right)\&equals;\frac{\partial }{\partial y} f\left(x\&comma;y\right)$$\stackrel{\text{assign as function}}{\to }$$\mathrm{f\_\_y}$

$f\left(2\&comma;3\right)\&plus;\left(\mathrm{f\_\_x}\left(2\&comma;3\right)\cdot \left(x-2\right)\&plus;\mathrm{f\_\_y}\left(2\&comma;3\right)\cdot \left(y-3\right)\right)$

$27x-32y\&plus;25$

Obtain $z\&equals;f\left(2\&plus;\mathrm{dx}\&comma;3\&plus;\mathrm{dy}\right)\doteq f\left(2\&comma;3\right)\&plus;\mathrm{df}$ 

$z\doteq f\left(2\&comma;3\right)\&plus;\mathrm{f\_\_x}\left(2\&comma;3\right)\cdot \left(x-2\right)\&plus;\mathrm{f\_\_y}\left(2\&comma;3\right)\cdot \left(y-3\right)$

$z\doteq \left(27x-32y\&plus;25\right)$

$f≔\left(x\&comma;y\right)\to 3 x^2\&plus;5 x y-7 y^2\&plus;4\&colon;$$$

Obtain the tangent plane at $\left(2\&comma;3\right)$ 

$\mathrm{Student}:-\mathrm{VectorCalculus}:-\mathrm{TangentPlane}\left(f\left(x\&comma;y\right)\&comma;x\&equals;2\&comma;y\&equals;3\right)$

Obtain the first-degree Taylor polynomial approximation

$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{TaylorApproximation}\left(f\left(x\&comma;y\right)\&comma;\left[x\&comma;y\right]\&equals;\left[2\&comma;3\right]\&comma;1\right)$

$27x-32y\&plus;25$

Obtain $z\&equals;f\left(2\&plus;\mathrm{dx}\&comma;3\&plus;\mathrm{dy}\right)\doteq f\left(2\&comma;3\right)\&plus;\mathrm{df}$ 

$z\doteq f\left(2\&comma;3\right)\&plus;\left({\mathrm{D}}_1\left(f\right)\left(2\&comma;3\right)\cdot \left(x-2\right)\&plus;{\mathrm{D}}_2\left(f\right)\left(2\&comma;3\right)\cdot \left(y-3\right)\right)$

$z\doteq \left(27x-32y\&plus;25\right)$