Finding and Plotting Equations of Tangent Planes to Surfaces in R³

   

Course: Any class with multivariable calculus components.

 

Uses the Merlot website ‘Surface Plotter’ by Yanto Suryono.

 

Background: To find the equation of a tangent plane to a surface z = f(x,y), consider a function of three variables g(x,y,z) = f(x,y) - z = 0. Since we know that a normal vector to the surface at (x₀,y₀,z₀) is normal to the level surface of g, g(x,y,z) = 0, it will be normal to the surface z = f(x,y). Then the equation of the tangent plane to the surface z = f(x,y) at the point (x₀,y₀,z₀) is given by the formula:

 

g_x(x₀,y₀,z₀)(x – x₀) + g_y(x₀,y₀,z₀)(y – y₀) + g_z(x₀,y₀,z₀)(z – z₀) = 0

 

The symbols g_x, g_y, g_z stand for the partial derivatives of the function g with respect to x, y, and z. Note that g_z will always = -1 in this situation. Hence the plane’s equation can be easily solved for z in terms of x and y. 

 

For more information about this, visit the site http://www.ac.cc.md.us/~donr/CalcIII/unit3/lesson7/u3l7.html if you need a better understanding.

 

First problem:

1. Given the surface z = 4 – x² – y², go to the site http://www.fedu.uec.ac.jp/~yanto/java/surface/index.html and plot the surface. You might want to expand the z axis a bit to get a better view. Also note that x² must be typed as x^2.

 

 

 

 

2. Find the equation of the tangent plane to the surface at the point (2,0,0) as described in the above lesson.

 

 

 

 

3. Solve your tangent plane equation for z.

 

 

 

 

 

4. Plot the graph of the tangent plane on the same axes as the surface, and rotate it various ways until you are convinced that the plane is tangent to the surface.

 

 

Second problem:

1. Solve the equation x² + 4y² + z² = 36 for z. (Two solutions!)

 

 

2. Plot the ellipsoidal surface. Note: use sqrt(   ) for square root, and you must type 4*y not just 4y. (Note it will take two surfaces!)

 

 

 

3. Remove one of your plots (the upper one!) and replace it with the equation of the tangent plane at the point (2,-2,-4) and inspect the graph for accuracy, checking your answer.

 

 

 

4. Remove the other plot, restore the upper surface, and replace the tangent plane equation with the tangent plane at the point (2.-2,4).