Units

# Unit 2: Tangent Plane

## The Concept

For a 2D curve $y=f(x)$, there is at most one tangent line to a point $(x_0, y_0)$ on the curve.  The equation of tangent line to 2D curve $y=f(x)$ at point $(x_0, y_0)$ is

$y=y_0+f'(x_0)(x-x_0)$.

The tangent plane in 3D is an extension of the above tangent line in 2D. For a 3D surface $z=f(x,y)$, there are infinitely many tangent lines to a point $(x_0, y_0, z_0)$ on the surface; these tangent lines lie in the same plane and they form the tangent plane at that point.

Recall that two lines determine a plane in 3D space. Thus, one usually uses two special tangent lines to determine a tangent plane and these two tangent lines are related to the partial derivatives (i.e., $f_x$ and $f_y$) of the surface function $z = f(x,y)$. The equation of the tangent plane to surface $z = f(x,y)$ at point $(x_0, y_0, z_0)$ is

$z = z_0 + f_x(x_0,y_0)(x - x_0) + f_y(x_0,y_0)(y-y_0)$.

## The Plot

Now, you should engage with the 3D plot below to understand the tangent plane. Follow the steps below to apply changes to the plot and observe the effects:

1. Input a 3D surface function in the function box in the plot. The function can be a single variable function or a double variable function.
2. Adjust point $P$ using the sliders or by dragging the point on the graph below.
3. The tangent plane equation will be depicted on the plot.

## Self-Checking Questions

Check your understanding by solving the following questions:

1. Find the equation of the tangent plane to the surface defined by the function $x^2+10xyz+y^2+8z^2=0,P(−1,−1,−1)$
2. Find the equation of the tangent plane to the surface defined by the function $h(x,y) = ln(x^2) + y^2$ at Point $(x_0,y_0) = (3,4)$. 