SUNY Geneseo Department of Mathematics

Tangent Planes

Friday, October 14

Math 223 01
Fall 2022
Prof. Doug Baldwin

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Anything You Want to Talk About?

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Colloquium

Another one! Next Tuesday!

Prof. Jenna Zomback, Williams College and Geneseo class of 2017.

“Ergodicity: From Local Statistics to Global Analysis and Back”

Tuesday, October 28, 4:30 - 5:20 PM.

Newton 204.

Today’s colloquium is 4:00 - 5:00

Tangent Planes and Related Ideas

Based on section 3.4 in the textbook.

Key Ideas

The equation for a tangent plane at point (x0, y0, f(x0,y0) ):

Equation of tangent plane is a reference point plus rates of change times amounts of change

Linear estimation uses the tangent plane to approximate the actual function, so the equation for linear estimation is exactly the equation for the tangent plane.

Differentiability is defined as having a tangent plane, but there are easier ways to test for it than to use the definition directly.

Tangent Planes

Consider the paraboloid z = 4x2 + y2.

Find the equation for the tangent plane at point (1, 1, 5).

Start by finding the partial derivatives of z, then evaluate them at x = 1, y = 1 and plug the results into the tangent plane equation:

Find tangent plane by finding and evaluating derivatives and plugging them into tangent plane equation

Linear Estimation.

Using the same paraboloid as above (i.e., z = 4x2 + y2), what do you estimate z to be when x = 1.01 and y = 0.98?

One approach is to plug x - 1 and y - 1 (i.e., the changes in x and y from the reference point) into the tangent plane equation from above:

5 plus 8 times X minus 1 plus 2 times Y minus 1 is 5.04 when X minus 1 is 0.01 and Y minus 1 is negative 0.02

Another approach is to simplify the equation of the tangent plane, and then plug x and y (1.01 and 0.98) into the simplified equation:

Tangent plane simplifies to 8 X plus 2 Y minus 5 which is 5.04 when X is 1.01 and Y is 0.98

Both approaches produce the same estimate.

What’s the actual value (you’re welcome to use a calculator)?

5.0408, according to Mathematica used as a calculator.

The ideas of tangents and linear estimation extend to more dimensions by just adding more “derivative times change in variable” terms.

Differentiability

Is that paraboloid differentiable at (x,y) = (1,1)?

Yes, but the easy way to realize it is to use the theorem that says that if a function and its first derivatives are all continuous at (x0,y0), then the function is differentiable there. This is much simpler if you need to check differentiability than going to the definition in terms of an error expression and limits.

Next

Generalizing the chain rule to multivariable functions.

Please read “Chain Rules for One or Two Independent Variables” and “The Generalized Chain Rule” in section 3.5 of the textbook.

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