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This example shows how to approximate gradients of a function by finite differences. It then shows how to plot a tangent plane to a point on the surface by using these approximated gradients.

Create the function $$f(x,y)={x}^{2}+{y}^{2}$$ using a function handle.

f = @(x,y) x.^2 + y.^2;

Approximate the partial derivatives of $$f(x,y)$$ with respect to $$x$$ and $$y$$ by using the `gradient`

function. Choose a finite difference length that is the same as the mesh size.

[xx,yy] = meshgrid(-5:0.25:5); [fx,fy] = gradient(f(xx,yy),0.25);

The tangent plane to a point on the surface, $$P=({x}_{0},{y}_{0},f({x}_{0},{y}_{0}))$$, is given by

$$z=f({x}_{0},{y}_{0})+\frac{\partial f({x}_{0},{y}_{0})}{\partial x}(x-{x}_{0})+\frac{\partial f({x}_{0},{y}_{0})}{\partial y}(y-{y}_{0}).$$

The `fx`

and `fy`

matrices are approximations to the partial derivatives $$\frac{\partial f}{\partial x}$$ and $$\frac{\partial f}{\partial y}$$. The point of interest in this example, where the tangent plane meets the functional surface, is `(x0,y0) = (1,2)`

. The function value at this point of interest is `f(1,2) = 5`

.

To approximate the tangent plane `z`

you need to find the value of the derivatives at the point of interest. Obtain the index of that point, and find the approximate derivatives there.

x0 = 1; y0 = 2; t = (xx == x0) & (yy == y0); indt = find(t); fx0 = fx(indt); fy0 = fy(indt);

Create a function handle with the equation of the tangent plane `z`

.

z = @(x,y) f(x0,y0) + fx0*(x-x0) + fy0*(y-y0);

Plot the original function $$f(x,y)$$, the point `P`

, and a piece of plane `z`

that is tangent to the function at `P`

.

surf(xx,yy,f(xx,yy),'EdgeAlpha',0.7,'FaceAlpha',0.9) hold on surf(xx,yy,z(xx,yy)) plot3(1,2,f(1,2),'r*')

View a side profile.

view(-135,9)