Questions tagged [calculus]

For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.

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70
votes
17answers
9k views

How shall we teach math online?

Many universities, including mine, are now requiring we teach our courses online because corona. How shall we do this? Let’s brainstorm here. Some challenges: My school provides limited online ...
63
votes
14answers
2k views

Encouraging class participation

I teach calculus to college students, and find it very difficult to get them to speak up in class when I ask questions, or when I'm trying to get a pulse for how much they understand. I think ...
59
votes
4answers
5k views

Is it worth grading calculus homework?

I am a young math educator. I've TAed four semesters of calculus for various instructors. Some instructors have required me to grade selected problems in homework sets. Another required me simply to ...
54
votes
24answers
60k views

Optimization problems that today's students might actually encounter?

Our students are not fencing in farm fields, cutting wires and folding them, or designing windows, so they are often uninspired by the optimization problems we give them. They seem like something that ...
46
votes
10answers
4k views

Should we avoid indefinite integrals?

I am very uncomfortable with indefinite integrals, as I have a hard time giving them a precise sense that matches the way they are written and the usual meaning of other symbols. For example, when ...
43
votes
16answers
29k views

How is calculus helpful for biology majors?

It's common for majors in biology to take calculus courses, and many calculus textbooks (and calculus professors) try to cater to these students by including applications to biology. My question is, ...
39
votes
10answers
2k views

Reasons for (not) distinguishing $f$ from $f(x)$

Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. In teaching at school and university, notation is quite often mixed up, e.g. the function is ...
36
votes
13answers
6k views

How to make Calculus II seem motivated, interesting, and useful?

I am due to teach Calculus II in the fall at an American state university. Our calculus sequence is somewhat slow, due to the fact that many of our students come with limited backgrounds. Most of our ...
35
votes
4answers
2k views

How can I help a student who has a "wrong" kind of enthusiasm?

Alice (not real name) is a student in one of my Math 100 (calculus) classes. It's a course offered by my college as a dual credit course at a high school, so the whole class is about 17/18 years old, ...
33
votes
3answers
2k views

A calculus book that uses differentials?

All introductory calculus books that I have seen spend most of their chapters on differential calculus talking about derivatives, with at most a short section defining differentials as $dy = f'(x) \, ...
32
votes
11answers
8k views

How can I teach my students the difference between a sequence and a series?

Sequences and series are related concepts but differ extremely from one another. I feel that students in integral calculus frequently mix them up. Part of the problem is that: Sequences are usually ...
32
votes
11answers
1k views

Epsilons and deltas in a first calculus course

In a freshman calculus course for non-majors; Is it to the benefit of the students to include discussion of epsilons and deltas? To what extent, if any, should they be used? For example, just to ...
31
votes
10answers
9k views

Simple "real world" l'Hôpital's rule problem?

I am on a team which is writing a set of lecture notes for differential calculus. I am using a format of "Break ground" which poses a problem, "Dig in" which develops the tools to solve the ...
31
votes
10answers
2k views

Should students be asked to use more than one notation for the derivative in an introductory calculus class?

There are many, many ways of writing the derivative of a function $y=f(x)$: $$\frac{d}{dx}y, \frac{dy}{dx},\frac{d}{dx}f(x), \frac{df}{dx}, \dot y, D_x f,f',y',f'(x),f_x$$ and so on. Students often ...
30
votes
6answers
1k views

What holds your students back in Calculus?

I teach Precalculus to high school kids, and I know a lot of you all teach Calculus. What are some issues that your students have in Calculus classes that you wish had been addressed in a ...
30
votes
3answers
2k views

Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$

On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other ...
29
votes
11answers
5k views

For calculus students, what should be the intuition or motivation behind series?

I've noticed that series are one of the most difficult portions of calculus for new students to learn. I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as ...
29
votes
14answers
2k views

Revisiting topics from previous courses

I teach calculus to students who have almost all taken calculus before. (Primarily first-year college students who took calculus in high school but didn't perform well enough to skip the course.) ...
29
votes
3answers
14k views

Difference between high school and college calculus courses

I am curious why students who take calculus in high school often do so poorly in college calculus. I am an instructor at an engineering college and I've noticed a decent number of students who have ...
28
votes
10answers
1k views

What are argument one can give to students on the definition $0^0$?

From high school to introduction courses in university, the expression $0^0$ is some (psychological) problems. High school students just apply it to their calculator and either the result is $1$ or ...
28
votes
6answers
2k views

What are non-math majors supposed to get out of an undergraduate calculus class?

When I teach a course for math majors (an analysis course out of Rudin, say), I have a more or less clear idea of what the students should take away from the course, having been in their shoes some 15 ...
28
votes
4answers
4k views

Open-Source Math Textbooks

It seems to me that an open-source model could work quite well for textbooks, with issues being raised by the users of the book and different forks of the project being created for different ...
28
votes
6answers
22k views

Early vs. late transcendentals

There seem to be two approaches to calculus education: Early transcendentals: introduce polynomials, rational functions, exponentials, logarithms, and trigonometric functions at the beginning of the ...
27
votes
4answers
949 views

The Undergraduate Responsibility Gradient

We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they ...
27
votes
4answers
3k views

Students use WolframAlpha. Can we change calculus instruction to exploit it while discouraging 'cheating'?

(This question developed from a comment in the thread "Revisiting the chain rule".) Students know that WolframAlpha and other software/computational resources exist and will make use of them as they ...
26
votes
16answers
4k views

Grading a limit problem

In an exam we have Question (5points) find $\lim_{x\to\infty}(x-\sqrt{x})$. A student answered: $\lim_{x\to\infty}(x-\sqrt{x}) =\lim \sqrt{x}(\sqrt{x}-1)=\infty \cdot\infty=\infty$. My question is:...
26
votes
8answers
2k views

Unusual applications of integration

I am trying to teach my calculus students to apply integration by thinking about what they are integrating rather than just applying formulas. Calculus books are full of formulas like "to find the ...
26
votes
6answers
697 views

Would taking 5 minutes to explain the history behind a mathematical idea help stimulate learning the idea?

I read a paper in my "Research Issues in Mathematical Education" class that I have applied to the Undergraduate Calculus I and Calculus II class that I teach. I take five minutes to explain the ...
26
votes
3answers
913 views

Counterexamples in first year calculus

Many believe (I think rightly so) that the presentation of counterexamples should play an important role in the teaching upper level mathematics courses such as real analysis and topology. ...
25
votes
10answers
6k views

Why would you teach Calculus before teaching Real Analysis?

Let's assume our students are actual aspiring mathematicians. Why would we introduce our students to Calculus rather than Real Analysis? After all, "Calculus is a subset of Real Analysis". He will ...
24
votes
9answers
3k views

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

This is my first post so bear with me, but something I've been thinking about lately is: Why didn't I ever question the relationship between the derivative and the integral when I was taking calculus? ...
24
votes
8answers
3k views

Is there a simple explanation for calculus classes of why partial fractions work?

I'd be happy even with an explanation in the simplest case: an explanation of why expressions of the form $\frac{ax + b}{(x - c)(x - d)}$ with $c \neq d$ can always be rewritten in the form $\frac{A}{...
24
votes
4answers
6k views

How is teaching calculus in high school different from teaching calculus in college?

I've taught calculus in college for five years, and it's always interesting to see students coming in who already had calculus in high school. Many of them do very well, and don't even seem like they ...
23
votes
5answers
4k views

What is the proper verb for "doing" an integral?

It's time to write exams, and when writing in committee we often discover differences in usage between various instructors. Here's an example I noticed today. What is the proper verb to use in a ...
23
votes
7answers
4k views

Why are we so careful in saying that dy/dx is not a fraction?

Calculus instructors are mostly very careful to explain that $\frac{\mathrm{d}y}{\mathrm{d}x}$ is not a fraction, and multiplying both sides of an equation by $\mathrm{d}x$ is nonsense, wrong, or evil....
23
votes
6answers
747 views

How can you explain to students that they should not use the same variable in an integrand and in the limits of integration simultaneously?

When teaching Calculus, one thing that many teachers emphasize is that the variable of integration is a `dummy variable' that is unimportant. Around the same time, we introduce integrals with ...
22
votes
13answers
1k views

Historical tidbits to liven up calculus classes

What are some examples of math history that can be mentioned in calculus classes, either to liven things up or to provide additional perspective / insight on the material being learned? For example, ...
22
votes
8answers
2k views

Should we teach trigonometric substitution?

This is the question that was not asked here. Also related is this question, but both presuppose that it will be taught and ask about how best to do it. My question here is, suppose we are designing ...
22
votes
4answers
577 views

Keeping quicker students engaged and interested throughout a course

In a college math course one is bound to find a fairly broad range of students in terms of their quickness in understanding the material. This is due to many reasons, including differing mathematical ...
22
votes
6answers
1k views

Practical experience with teaching differentials in freshman calc?

There is a well known essay by Dray and Manogue which argues that differentials should be brought back into freshman calculus, and that we shouldn't worry too much about choosing a specific way of ...
22
votes
5answers
674 views

Natural, rich, calculus questions

We have the good fortune of having "lab sections" here at my college. I'm interested in conducting some activities in the spirit of this talk. However, even in my stash of inquiry-based learning ...
22
votes
2answers
1k views

Is Knuth's suggestion on teaching calculus a good idea?

Note: I myself am not a math educator, though I plan to be one someday. In this letter, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A ...
21
votes
11answers
8k views

What interesting properties of the Fibonacci sequence can I share when introducing sequences?

The Fibonacci numbers are one of the first sequences given as examples of sequences in many calculus textbooks as they have a definition that does not obviously have a closed form and they have many ...
21
votes
8answers
3k views

What is a good reason to change calculus texts?

Our college is switching to an Early Transcendentals calculus text, and this seems like a good time to consider which text we are using in general. Larson, Stewart, Thomas, Briggs/Cochran, etc are all ...
21
votes
7answers
2k views

Good examples of functions defined as definite integrals of elementary functions?

I am writing some Calculus content, and I would like a "big list" of useful functions which are defined by definite integrals, but are not elementary functions. Two examples of such functions are $$ ...
21
votes
3answers
851 views

Tutoring a recalcitrant/awkward/exasperating student---special needs?

As part of my duties at a GTA, I spend several hours per week in our department's drop-in tutoring center. The center is open to all students enrolled in 100- and 200-level math courses, with the ...
21
votes
8answers
815 views

Non-Rigorous Use of Differentials

Consider the following example of working "directly" with differentials. One way to approach the problem of determining the arc length of the graph of a single-variable function is to imagine the arc ...
21
votes
1answer
449 views

To what extent is math research in the United States subsidized by calculus teaching?

I read an interesting article earlier this year about the future of mathematics. It claimed that math research is possible in the United States because of service courses (especially calculus). ...
21
votes
3answers
1k views

Which universities teach true infinitesimal calculus?

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...
20
votes
6answers
6k views

How rigorous should high school calculus be?

In the UK, calculus taught in secondary school focuses mainly on computation of derivatives and integrals and solving simple differential equations. There is a small amount of discussion about limits ...

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