# Tag Info

### Why would you teach Calculus before teaching Real Analysis?

You may as well ask: Why teach elementary school children how to perform whole-number arithmetic without teaching them the Peano axioms first? Why teach high-school Algebra without starting with the ...
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The student changed something which was indeterminate ($\infty-\infty$) into something which was not ($\infty\cdot \infty$). How does that not merit a perfect score? Changing indeterminate expressions ...
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### How rigorous should high school calculus be?

Not very rigorously at all, but that doesn't (and shouldn't) mean just memorizing calculations. (I should add that I'm basing this on my experience teaching calculus to non-major college students, ...

### What do you do in order to drag out lectures?

What you are describing is so far outside of my, and I suspect most educators, experience that it appears to be literally incredible. The very strongest Universities in the country, with some of the ...
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### Is there a canonical name for a polynomial-like expression allowing for negative powers?

There is also the term "Laurent series", where we allow an infinite number of terms... $$\dots +5x^{-3} + 2x^{-2} + 3x^{-1} + 2 + 4x - 7x^2+\dots$$ So perhaps yours is a "finite ...

### Do I really need to cover solids of revolution in my Calculus I class?

An operation is born when we recognize the regularity in repeated reasoning. Take multiplication for example. If we are living lives which involve even a modest amount of arithmetic thinking, we will ...

### Why are calculators not allowed in post-secondary exams?

As a professor/teacher I have some insight. You just answered your own question: "my level of math right now is not basic. So, like many, I tend to forget the basics of math, like adding fractions, ...

### Why is differential calculus often presented before integral calculus?

While there are no theoretical difficulties with developing integration first ("from scratch" measure theory books demonstrate this), it presents some pedagogical challenges. Most problems ...

### What is the best way to intuitively explain the relationship between the derivative and the integral?

You start by noticing that the Riemann sums (multiplication followed by addition) and the difference quotients (subtraction followed by division) undo each other. Their limits -- the integral and the ...
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### How to properly define volume for beginner calculus students?

It depends somewhat on the style of the course, but the majority of calculus students do not need a formal definition of volume or area, in my experience. They have studied geometry and (usually) ...

### Why don’t we teach a topological view of continuity instead of epsilon-delta?

Yes. You are crazy to be spending time on another effort to bring real analysis style rigor into a calc 1 course. In this case, even off brand real analysis. Instead of a "different" way ...

### Why do we teach that every line is a linear function?

The usage you object to is, in fact, the original meaning of "linear". "Linear" means "having to do with lines". The notion of "linear" in the sense of "linear transformation" is a more modern, ...

### Why do no students know to change the limits of integration when doing substitutions?

Some students are instead taught to change the substitution variable back into the original variable before evaluating the antiderivative at the bounds. \int\limits_0^2 2x\cos(x^2) \;\mathrm{d}x =...
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### An introductory example for Taylor series (12th grade)

One practical reason for choosing a Taylor Series approximation of a function over the function itself is if you are able to compute using only the four arithmetic operations. For example, if you are ...