Partial derivatives and the tangent plane

Recall that if $f$ is a function of two variables, its partial derivatives are the functions $f_x$ and $f_y$ defined by

\begin{eqnarray*} \frac{\partial f}{\partial x}&=&f_x(x,y)= \lim_{h\rightarrow 0}\frac{f(x+h,y)-f(x,y)}{h}\\ \frac{\partial f}{\partial y}&=&f_y(x,y)= \lim_{h\rightarrow 0}\frac{f(x,y+h)-f(x,y)}{h} \end{eqnarray*}

In the other hand, the equation of the tangent plane to a given surface $z=f(x,y)$, at $\big(a,b,f(a,b)\big)$, is

\begin{eqnarray*} z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b), \end{eqnarray*}

or equivalent

\begin{eqnarray*} (x,y,z)=\big(a,b,f(a,b)\big)+\lambda \big(1,0,f_x(a,b)\big)+\mu \big(0,1,f_y(a,b)\big),\qquad \lambda,\,\mu\in \mathbb R. \end{eqnarray*}

Simulation

This simulation shows the geometric interpretation of the partial derivatives of $f(x,y)$ at point $\text{A}\in \mathbb R^2$. It also shows the tangent plane at that point.

Things to try:

Change the function f(x,y).
Drag the point A or type specific values on the boxes.
Select the object you want to show: Tangent plane, f_x or f_y.
Use right click and drag the mouse to rotate the 3D view or click on View button.

Sorry, the simulation is not supported for small screens.

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Worksheet exemplar

The following file contains activities and problems associated with the simulation.