SUNY Geneseo Department of Mathematics

Tangent Planes

Tuesday, March 21

Math 223
Spring 2023
Prof. Doug Baldwin

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Misc

PRISM Leadership Recruiting

PRISM is recruiting people now who want to be PRISM officers next year, so that there’s time for the current officers to train them a bit.

Look for flyers around the math department with more information about applying.

PRISM’s math apparel sale is starting too. See email for information about pre-ordering, pricing, etc.

Second-Year Advising

The math department requires group advising for second-year math majors, to alert you to considerations in planning upper-level courses and directions through the major(s), etc.

Everyone has to attend one of these sessions sometime in your second year.

Two sessions have been scheduled for this semester: Thursday, March 30, 3:00 pm and Wednesday, April 5, 3:30 pm; both in Fraser 108.

Tangent Planes

Based on “Tangent Planes” and “Linear Approximations” in section 3.4.

Key Ideas

The equation for the plane tangent to function f(x,y) at point (x0,y0):

Z equals F of X 0 and Y 0 plus F sub X of X 0 and Y 0 times X minus X 0, plus F sub Y of X 0 and Y 0 times Y minus Y 0

Use tangent planes to estimate values. This is basically the definition of linear approximation for 2-variable functions.

Questions

The cross product and tangent planes? The textbook uses a cross product to derive the general equation for a tangent plane, but I wouldn’t recommend using that approach to find the equations for specific planes. Use the general equation instead.

Example

Suppose f(x,y) = x2 + y2. Find a tangent plane at f(1, -1), and then use it to estimate the value of f(1.01, -0.98).

To find the tangent plane, treat (1, -1) as (x0, y0), and then plug numbers into the general equation:

Finding the equation for a plane tangent to a function at a given point, using the function's derivatives

Geometrically, the tangent plane equation says that to get from a known point on the plane (point f(x0, y0)), you move up or down by an amount given by the slope of the plane in the x direction times the change in x value, and also up or down by an amount given by the slope in the y direction times the change in y:

Curved surface with displacements in height depending on displacements of X and Y

To use this idea to approximate a value for the function, just plug the x and y values at which you want the approximation (should close to x0 and y0 for best results) into the tangent plane equation:

F of 1.01 and negative 0.98 is about 2 plus 2 times 1.01 minus 1 minus 2 times negative 0.98 plus 1

Differentiability

Based on “Differentiability” in section 3.4.

We’ll look at this again tomorrow, but we did develop some key ideas and questions to concentrate on.

Key Ideas

The definition of differentiability.

Differentiability implies continuity, and continuity of a function and its first derivatives implies differentiability.

Smoothness.

Questions

Need an example of differentiability.

What does the error term in the definition of differentiability mean?

Next

Finish differentiability.

Time permitting, start looking at the chain rule for multivariable functions.

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