As known from functions with one variable, the tangent line at a point *x*_{0} is used, as a first approximation (*y*_{1}) of the function *y* = *y*(*x*) , around this point:

$${y}_{1}={y}_{0}+{\left(\frac{\text{d}y}{\text{d}x}\right)}_{0}\left(x-{x}_{0}\right)$$ |
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where the index 0, denotes at point *x*_{0}.

**The first approximation of a function of n variables**

$$y=y\left({x}_{1},\mathrm{....},{x}_{n}\right)$$ |
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around the point

$${\left({x}_{1},\mathrm{....},{x}_{n}\right)}_{0}=\left({x}_{1,0},\mathrm{....},{x}_{n,0}\right)$$ | (1) |
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**has the form of**:

$${y}_{1}={y}_{0}+{\displaystyle \sum _{k=1}^{n}{\left(\frac{\partial y}{\partial {x}_{k}}\right)}_{0}\left({x}_{k}-{x}_{k,0}\right)}$$ | (2) |
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since for the point (1)

$$\begin{array}{l}{y}_{1}={y}_{0}\text{\hspace{1em}}\text{and}\\ \frac{\partial {y}_{1}}{\partial {x}_{k}}={\left(\frac{\partial y}{\partial {x}_{k}}\right)}_{0}\text{\hspace{1em}}k=1,\text{\hspace{0.17em}}\mathrm{...},\text{\hspace{0.17em}}n\end{array}\}$$ | (3) |
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As seen from (2), the first approximation itself is in general a function of *n* variables. **Care should be taken that the function and the derivatives are continuous at the point (1)**.

**In the case of a function of two variables,**

$$z=z\left(x,y\right)$$ | (4) |
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**the first approximation is the tangent plane**:

$${z}_{1}={z}_{0}+{\left(\frac{\partial z}{\partial x}\right)}_{0}\left(x-{x}_{0}\right)+{\left(\frac{\partial z}{\partial y}\right)}_{0}\left(y-{y}_{0}\right)$$ | (5) |
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It is called "tangent", since its derivatives are equal to these of the function at the point (*x*, *y*, *z*)_{0}, but is it a plane?

**A plane in a three dimensional space is defined as any linear combination of the Cartesian coordinates:**

$$Ax+By+Cz=D$$ | (6) |
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**where at least one of the constant factors of the coordinates ( A,B,C) does not vanish.** Indeed any intersection of (6) with the planes of constant

**A point of a function of many variables with vanishing and continuous derivatives is called a stationary point**. As in the case of a function of a single variable, a stationary point indicates that for an infinitesimal deviation from this point, the function remains constant. In the case of two variables (5), **the tangent plane at a stationary point** is

$$z={z}_{0}\text{\hspace{1em}}\text{where}\text{\hspace{0.28em}}{z}_{0}\text{\hspace{0.28em}}\text{is}\text{\hspace{0.28em}}z\text{\hspace{0.28em}}\text{at that point}$$ | (9) |
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The following ** example** will be used to illustrate the tangent plane.

$$z=3\left[1-{\left(\frac{x}{4}\right)}^{2}-{\left(\frac{y}{3}\right)}^{2}\right]$$ | (10) |
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The section of this function with a constant *z* plane is

$${\left(\frac{x}{4}\right)}^{2}+{\left(\frac{y}{3}\right)}^{2}=1-\frac{{z}_{c}}{3}\text{\hspace{1em}}\text{with}\text{\hspace{1em}}{z}_{c}=\text{constant}$$ | (11) |
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which is an ellipse for *z _{c}*<3, a point for

$$\begin{array}{l}x=y=0\\ z=3\end{array}\}\text{\hspace{1em}}\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{point}\text{\hspace{0.17em}}\text{of}\text{\hspace{0.17em}}\text{maximum}$$ | (12) |
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The sections with constant *x* or *y* values are parabolas:

$$\begin{array}{l}z=-\frac{{y}^{2}}{3}+3\left[1-{\left(\frac{{x}_{c}}{4}\right)}^{2}\right]\text{\hspace{1em}}{x}_{c}=\text{constant}\\ z=-\frac{3}{16}{x}^{2}+\left[3-\frac{{y}_{c}^{2}}{3}\right]\text{\hspace{1em}}\text{\hspace{0.28em}}{y}_{c}=\text{constant}\end{array}\}$$ | (13) |
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The shape is similar to a paraboloid of revolution, except that instead of circles there are ellipses, and therefore it is **called an elliptic paraboloid**.

**The tangent plane** at point (*x*,*y*,*z*)_{0}, according to (5) is:

$$z={z}_{0}-\frac{3{x}_{0}}{8}\left(x-{x}_{0}\right)-\frac{2{y}_{0}}{3}\left(y-{y}_{0}\right)$$ | (14) |
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For the maximum (12), the tangent plane becomes

$$z=3$$ | (15) |
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which corresponds to a stationary point (9).

This example (9) and some display of the tangent planes are shown in **Fig. Elliptic paraboloid**.