SUNY Geneseo Department of Mathematics

Differentiability

Wednesday, March 22

Math 223
Spring 2023
Prof. Doug Baldwin

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Anything You Want to Talk About?

Errors in Problem Set 6’s Mathematica problem involving “Recursion limit”? These can happen when you accidentally try to define a variable in terms of itself. In this particular question, if you use “t” as the variable in r(t) but also try to use it as a name for the tangent vector.

What plotting command to use for the plot(s) in Problem Set 7? The Plot3D command is probably best.

Download this notebook to see examples connected to both of the above answers.

Mid-Semester Feedback

There’s a survey in Canvas with which you can give me feedback on how this course is going for you. It’s completely optional and anonymous (even though Canvas calls it a “quiz”).

I’ll report back on any suggestions at the beginning of next week, and look for any more input periodically after that.

Differentiability

Based on “Differentiability” in section 3.4.

Definition

Based on how fast the surface of a function converges to its tangent plane around point (x0,y0). If the difference between the surface and the tangent shrinks much faster than the distance from (x0,y0) to point (x,y) does, then the function is differentiable at (x0,y0). Note that talking about the tangent plane implies that the partial derivatives of the surface exist, so this is in some ways an extension of the idea of differentiability of single-variable functions (where differentiability just requires a derivative to exist — now the derivatives have to exist and something else has to be true too).

Tangent plane separated from function F of X and Y by decreasing error E of X and Y

The “error term,” E(x,y) is defined as what you have to add to the tangent plane to get the actual function value:

F of X and Y equals F of X 0 and Y 0 plus F sub X of X minus X 0 plus F sub Y of Y minus Y 0 plus E of X and Y

Example

The book’s exercise 3.4.3, explained carefully. Show that f(x,y) = 3x - 4y2 is differentiable at (-1,2).

Testing differentiability involves multiple steps, something like this:

  1. Figure out x0, y0
  2. Claculate f(x0,y0), fx(x0,y0), etc., and build the equation for the tangent plane.
  3. Find E(x,y)
  4. Find the limit from the definition and see if it’s 0.

To use these steps on the given example, start by identifying x0 and y0 from the problem:

For F of X and Y equals 3 X minus 4 Y squared at point negative 1 and 2, X 0 is negative 1 and Y 0 is 2

Then find the equation for the tangent plane, just as we did yesterday.

Tangent plane equation is Z equals 3 X minus 16 Y plus 16

To find E(x,y), notice from its definition that you can find it by subtracting the equation for the tangent plane from the equation for the function itself.

E of X and Y equals negative 4 times the quantity Y minus 2 squared

Start finding the limit by writing out exactly what it is, and simplifying a bit:

Factor negative 4 out of limit of error term over distance from point X 0 Y 0 to point X Y

Now the “Squeeze Theorem” (which says, informally, that if some function, f, has a value in between two others, g and h, and g and h have the same limit at some point, then f must also have that limit) is very handy for finding the remaining limit:

Using Squeeze Theorem to show that limit of error term over distance from X 0 and Y 0 to X and Y is 0

Next

The chain rule for 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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