# Calculus/analysis as basis for basic school topics

I'm looking for examples of facts/results in basic school that are supported by calculus/analysis. Here is an example:

Consider the infinite decimal expansion (a usual topic in basic school): basic school means primary;secondary school but mainly high school!

$$a = 0.12345678910\ 11\ 12\ 13\ 14\ldots$$ (the Champernowne constant)

What does this expression mean? The answer $$a$$ is the limit of the sequence:

$$a_1 = 0.1, a_2 = 0.12, a_3 = 0.123,$$ etc.

How do we know that this sequence has a limit? Here calculus helps: every bounded monotonic sequence has limit.

Another example: Why does there exist a number $$a$$ such that $$a \cdot a = 11$$?

Answer from calculus: $$f(x) = x^2$$ is continuous, $$f(0) = 0$$ and $$f(4) = 16$$ as $$0 \leq 11 \leq 16$$, by the Intermediate value theorem the existence of the number $$a$$ (square root of $$11$$) follows.

Other examples?

• I have had to tell a combative high school sophomore that the reason that pi is a number even though it cannot be written as a fraction and can't be computed in finitely many steps has to do with properties of the number line that are typically not discussed until college. If he was curious, he could read about real analysis and the squeeze theorem, but the short of it was that the number line doesn't have any gaps, and so the "outer" and "inner" ways of approximating pi (via Archimedes) had to be narrowing in on a number on the number line and we defined that result as the number pi. Commented Aug 27, 2022 at 18:34
• In your first example, that every bounded monotonic sequence has a limit in the real numbers is told to students in a calculus course, but it is hardly explained in a serious way in such a course (in contrast to an analysis course).
– KCd
Commented Aug 28, 2022 at 12:38
• I can't imagine your example showing up in high school math classes. Nor would it show up outside an advanced math course, even in college. I'm curious in what country it is a "usual topic". Opal's example makes more sense. I wonder if that should be an answer? Commented Aug 28, 2022 at 15:14

Calculus is the reason radian angle measure is important (formulas like $$(\sin x)’ = \cos x$$ are incorrect using degrees or any other angle measure besides radians) and the reason $$e$$ and $$\ln x$$ are important (look at formulas for $$(b^x)’$$ and $$(\log_b x)’$$). Before calculus there is no natural reason for those to seem like genuinely relevant topics.

The chain rule is arguably the reason that function composition needs to be emphasized earlier in high school. At least I remember that when I was learning calculus, once the chain rule appeared I thought “Aha, that’s why we spent all that time on function composition before.”

• Logarithms are extremely important and relevant because they transform products (which are hard) into sums (which are easy). You don't need to know how to integrate 1/x for logarithm to become relevant.
– Stef
Commented Sep 5, 2022 at 14:47
• @Stef reread the OP's question. It asks for results in more basic math that are supported by calculus, and that's what my answer addresses: the importance of logarithms to the base e is because of calculus, not for any reasons based on earlier math. The "relevant topics" I was writing about in my answer were not trigonometric functions and logarithm functions overall, but certain normalizations that are used with them: radians in trigonometric functions and the preference for base $e$ in exponential and logarithm functions. Algebraic rules don't justify those conventions, but calculus does.
– KCd
Commented Sep 5, 2022 at 15:56
• There is no need for this "reread the OP's question" bullshit. Either ignore my criticism or use it to improve your answer, but there's no need to be passive aggressive or overly defensive here. Have a good day.
– Stef
Commented Sep 5, 2022 at 16:00