Differentials and Linear Approximation

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Anything You Want to Talk About?

How do the “beyond mastery” questions on problem sets figure in grades? They boost grades from the B+ range to the A- or A range. In other words, if you never answer one of those questions correctly but achieve the expected mastery on all course outcomes you’re likely to get a B+ for the course. On the other hand, to get an A or A- you should answer some of the “beyond mastery” questions, but certainly don’t need to answer all of them. (And notice that some questions are “beyond mastery” not because they’re harder than usual, but simply because they lead to interesting connections between what we’re doing and other areas in or outside of math.)

SI

The next SI session is on Wednesday (to keep rejuvenation day free), at 5:00.

Linear Approximation and (Mostly) Differentials

From “Linear Approximation of a Function at a Point” and “Differentials” in section 4.2 of the textbook, and this discussion of differentials.

Basic Ideas

How would you use linear approximation to estimate what 10.1² is?

“Linear approximation” is the idea of approximating a curved graph, which can be hard to calculate values on, by a straight line, which is easier to calculate values on. The line that is tangent to the curve at some known point is typically close enough to the curve to be a good approximation at other points near the known one.

In this case, the line tangent to the curve y = x² at known point (10,100) has slope dy/dx, so to estimate 10.1² you can multiply that slope by the change in x between 10 and 10.1 and add the result (which is the change in y) to the known y value at 10:

The general equation for linear approximation just generalizes this idea to an arbitrary function f, and arbitrary value a at which you know the function’s value and x near a at which you want to know it:

You can also capture this idea in terms of “differentials,” i.e., the “dy” and “dx” from derivatives. Conceptually, these are infinitesimally small changes in y or x, and as such can be manipulated independently. For example, the derivative of x² can be described in two ways:

We’ve also seen this as a way to write and understand the chain rule, namely something that breaks the ratio of 2 differentials into a product of two ratios, each involving a common intermediate differential:

In terms of differentials, linear approximation is multiplying the change-in-y to change-in-x ratio by a change in x in order to find a change in y (and then adding that change to a reference y value):

What Differentials Are and Aren’t

Differentials are “conceptually” infinitesimally small changes in x or y, but ultimately that’s a metaphor for how real numbers work — in reality you can never find a change that’s infinitesimally small enough to give exactly the right answer for, e.g., a derivative calculated literally as a non-zero change in y divided by a non-zero change in x.

This is why derivatives are defined as limits. The limit definition of a derivative is in fact based on a ratio of change in y to change in x, but plugging small but non-zero values of h into that definition generally won’t give you exactly the value the limit gives.

So differentials are metaphorical rather than real, but they are a good enough metaphor that doing algebra and arithmetic on differentials often works, and you’ll see it again, particularly with integrals.

Next

Another application of derivatives: finding minimum and maximum (collectively, “extreme”) values of functions.

Please read “Absolute Extrema” and “Local Extrema and Critical Points” in section 4.3 of the textbook by class time on Wednesday.

Also contribute to this discussion of extreme values by class time Wednesday.

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