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I've noticed that my students (primarily first-year students taking their first college math course) have a lot of trouble with inequalities. Not the math, the purpose: they seem to have a preconception that math is supposed to be about finding things exactly, and every time we talk about a bound, I get multiple people asking why anyone would ever care about such a thing.

I've been sort of waving my hands and saying "Well, if you build a bridge, you care about knowing the smallest weight which might cause it to break..." but I think it might help to be more concrete. I'm wondering if anyone can suggest good techniques or examples which illustrate the importance of bounds. (Given that the students aren't mathematicians, uses outside of math which make sense even in a world with computers that produce calculations accurate to 30 decimal places are particularly useful.)

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It seems to me that some of modern physics might be a good example; the smallest of the small or the biggest of the big are hard to get numbers for exactly. I'm not the right person to talk about this, though. –  Mike Miller Apr 15 at 18:14
    
In what context are you teaching these inequalities: the definitions of calculus? computing to a specified accuracy? constraints in optimization? I'd have different approaches for the different cases. –  Matt F. Apr 15 at 18:42
    
There are two big appearances in our calculus course: when we talk about using the tangent line to approximate a function, and when we use sums to approximate integrals. –  Henry Towsner Apr 15 at 20:10
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6 Answers 6

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Here's something I've done in 2nd semester calculus classes.

On the blackboard draw a big schematic graph of $y = \frac{1}{x}$ from $x=1$ to $x=n,$ with $1,\,2,\,\ldots,\,n-1,\,n$ labeled on the $x$-axis. Looking at the right and left hand Riemann sums with $\Delta x = 1$ gives

$$\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n-1} + \frac{1}{n} \;\; < \;\; \int_{1}^{n} \frac{1}{x}\,dx \;\; < \;\; 1 +\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n-1}$$

Letting $S(n)\; = \; 1 +\frac{1}{2} + \cdots + \frac{1}{n}$ be the sum of the first $n$ terms of the harmonic series, we get

$$S(n) - 1 \; < \; \ln{n} \; < \; S(n) - \frac{1}{n}$$

The left inequality gives $S(n) < \ln{n} + 1.$

The right inequality gives $\ln{n} + \frac{1}{n} < S(n).$

Hence, we get

$$\ln{n} + \frac{1}{n} \; < \; S(n) \; < \; \ln{n} + 1$$

Thus, we see that $0< S(n) - \ln{n} < 1.$ This allows us to obtain extremely accurate estimates (relatively speaking) to the sum of the first $10^{100}$ terms of the harmonic series or the first $10^{1000}$ terms of the harmonic series. It also provides a jumping off point to talking about whether this difference has a limit when $n \rightarrow \infty,$ and what that limit might be.

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I had the exact same problem as the students (I disliked analysis because the bounds seemed so arbitrary). I overcame this by seeing very precise results coming from imprecise bounds (one example is the Squeeze theorem). However, it might be easier to show them they use bounds all the time.

Examples In Real life

  1. Imagine you are going to McDonalds. You see a man with a briefcase full of hundred dollar bills. He pulls one out and pays for his sandwich with it. He says, "I didn't know which meal I would get, so I wanted to make sure I have enough money". This is a stupid example, right? But if your students start talking about why it's stupid, that could help.
  2. If you have a washer full of laundry, it seems that you can always fit in one more sock. Can you fit in infinitely many socks? Give me a number of socks that can't fit into a washer. Can you give me the smallest number of socks that can't fit into a washer?
  3. For what temperatures does ice stay frozen (at normal pressures, etc)? Can you express this without using inequalities or bounds?
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#1 is a deceptively good example. I love it. –  Mike Miller Apr 15 at 18:43
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You might consider appealing to computationally intractable problems that can be wrangled by mathematical bounding. This could double as an argument against those students who might say, "Why bother learning math? Can't I just make my computer solve things for me?"

I believe that students of a variety of ages can understand an argument that $R(3,3)=6$. Depending on whether you're talking to high-schoolers or undergrad math majors, you might phrase the setup of the problem differently, or go into different depths in the argument. But they can all follow along.

Follow this up with asking about $R(4,4)$. If they can't find the exact number "by hand", could they program a computer to do it? Work with them to realize how many possible 2-colorings of $K_n$ there are, and estimate how long it would take to check one graph for a monochromatic $K_4$. (You can be hand-wavey here, depending on the audience.)

Continue for $R(5,5)$ and $R(6,6)$. Try to explain how the computational time required explodes. Explain how we have no hope of simply letting a computer run to figure out an answer. By proving better and better bounds, we can narrow the search space, but without doing that, we're essentially knowledge-less.

Then, hit 'em with this quote (idea attributed to Erdős, words by Joel Spencer):

Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of $R(5,5)$ or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for $R(6,6)$. In that case, he believes, we should attempt to destroy the aliens.

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A more "practical" example to this argument is maybe the travelling salesman problem. Optimal solutions for large inputs are very hard to find, but finding solutions being at most $x\%$ away from the optimal solutions are far easier. –  Toscho Apr 16 at 18:35
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Another interesting example appears when evaluating infinite series. I have always personally found it fascinating that, for example, with an alternating series, we can use the terms of the series to accurately estimate the final sum; more importantly, we can declare how accurate our estimation is without even knowing the final sum.

Many calculus textbooks have this idea expressed as an inequality: "For $\sum (-1)^n a_n$, where $a_n\geq 0$, the remainder of the $n$-th partial sum obeys $|R_n|\leq a_n$."

But I think far too few stress this idea: we can say something like, "Adding up the first 100 terms yields 10 decimal places of accuracy, even though we don't know what number we're approximating so accurately!"

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An example they might already know from secondary education are definite integrals. The integral is bounded in both directions by partial sums. For every precision $h$, the partial sums aren't exact. But for infinite precision, the limits are exact (for Riemann-integrable functions).

If the students don't know this definition of definite integrals but are fit in modelling, you can introduce it as modelling the area by rectangle/trapezoids.

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Ah, but this is precisely the problem. We're covering definite integrals (which they've seen before, but are relearning), and their response is: "why should we care about bounding the area when we can just find it exactly?" –  Henry Towsner Apr 15 at 18:50
    
@HenryTowsner, give them a fiendish integral, where they can (easily) give bounds for the integrand, and thus give bounds on the integral. –  vonbrand Apr 15 at 19:07
    
@vonbrand: Ah, I misspoke. Even if they can't find it exactly, they don't see why having bounds would be interesting. This was the context---they met the easy upper and lower bounds on an integral by bounding it in boxes, and asked: "When would we use this? Will it help us calculate integrals?" –  Henry Towsner Apr 15 at 19:26
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@HenryTowsner Ah, ok. Another example: $\lim\limits_{x\to 0}\frac{\sin x}{x}$ without L'Hopital. Proof is here: youtube.com/watch?v=GAkdisUIMPQ –  Toscho Apr 15 at 19:54
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@HenryTowsner It sounds like the students are thinking exclusively in terms of known techniques to get answers (i.e. algorithmically) rather than in terms of problem solving (i.e. mathematically). Understanding the relationship between partial sums and integrals is crucial to really understanding the nature of integrals, but it won't necessarily help them compute integrals if the only integrals they're seeing are ones for which they already "know" some algorithm. But algorithms must be invented and their accuracy must be proven somehow.... –  Kyle Strand Apr 15 at 20:00
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You mentioned two examples, and I'm enthusiastic about one of them.

For using the tangent line to approximate a function, I sympathize with the students. "Use the derivative of $y=\sqrt{x}$ at $x=25$ to approximate $\sqrt{26}$ by hand" is not impressive. So I wouldn't focus much on this topic in a calculus class. If students continue mathematically, they'll see natural situations without closed-form expressions, and that will be a better time to see the power of linearization.

But for using sums to approximate integrals, I would give them hard problems. What is $\int_0^1 (e^{1-x^2}-1)\ dx$ or $\int_1^e\negthinspace \sqrt{1-\ln y}\ dy$ or $\int_0^1 x \sin(1/x)\ dx$, or the area between the axes and $x^2+x^3+y^4+y^5=1$? Bounds are easier to appreciate for problems that can only be solved numerically.

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