fuzzy approach to continuity

A “Fuzzy-Minded” Approach to Continuity

Timothy Biehler, Professor of Mathematics, FLCC

Sean Maley, Assistant Professor of Mathematics, FLCC

Objective

• Teach continuity and limits in an introductory Calculus course in a way that equips students with a clear sense of both the meaning and utility of these concepts.

Traditional Approach: The “Epsilon-Delta” Definition and Continuity

Modern, Intuitive Approach to Limit and Continuity:

• Limits:

• Continuity:

Warm up

Warm up

(7.5 is the right answer. 7.48 is wrong. 7.56 is wrong. Anything other than 7.5 is a wrong answer!)

Next problem

Tim has diabetes. The amount of insulin he needs to take before breakfast is a function of the amount of carbs his breakfast will contain (in grams).   He needs 2.5 units regardless of what he eats, and then one additional unit for each six grams of carbs.   How much insulin does he need to take if his breakfast will contain 30 grams of carbs?

Look familiar?

In the math classroom• We tend to speak in “exact”

terms

• In math class, f(30)=7.5, and 30 grams of carbs require 7.5 units of insulin.

• Is it “exactly” 7.5 units? Or “pretty much” 7.5 units?

• Here’s what an insulin syringe looks like:

• Think you can hit exactly 7.5 units?

Exact VS RealityGoogle says the time required to get here from Canandaigua is 4 hours and 10 minutes.

Of course no one takes that to mean the exact time it took us to get from exactly FLCC to exactly here is exactly 4hrs and 10 minutes!

What does this have to do with continuity?

• In reality, Tim didn’t eat exactly 30g of carbs and take exactly 7.5 units of insulin.

• We understand the dose for around 30g of carbs is close to 7.5 units.

• Limits and continuity are what makes this OK.

In math class, we say f(30)=7.5 as though the values must be exact.

But we never say that.

But we really understand this as “f(30-ish)=7.5 or so”

What about limits?

Because when we are around x=30, f(x) is around 7.5

Discontinuity

• On AT&T International Data plan, a package covering up to 120 MB of cellular data costs $30, and overages cost $0.25 per MB. A package covering 300 MB of data costs $60, and you are switched to that plan automatically once your usage reaches 300 MB.

• What is the cost of around 300 MB?

• 299MB?

• 301MB?

Discontinuity

• What is the cost of around 300 MB?

Advantages to the “Fuzzy Approach”• Appeals to students’ common sense.• Leads to more sound graphical understanding as

well.• Another way to say “small changes in input small

changes in output”

Approachable

• Easier to explain why continuity is needed for differentiability later on

Connecting Ideas

• Clearly articulates the nature of continuity in realistic applications. (Rarely do students draw this connection on their own.)

Relevant

Making Epsilon and Delta Meaningful

• For any error tolerance in the output (epsilon) we can find a degree of precision in the input (delta).

• If we can measure the dose within +/- units on the syringe, how close are we on the carbohydrate target?

More to come…

• Thanks for coming! SLIDES: tiny.cc/fuzzycontinuity

• Tim Biehler, Professor of Mathematics, FLCC

• Timothy.Biehler@flcc.edu

• Sean Maley, Assistant Professor of Mathematics, FLCC

• Sean.Maley@flcc.edu