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Questions tagged [calculus]

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

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61 votes
4 answers
6k 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 ...
abnry's user avatar
  • 852
33 votes
14 answers
2k views

Revisiting topics from previous courses [closed]

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.) ...
Henry Towsner's user avatar
74 votes
17 answers
10k 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 ...
Stephen Herschkorn's user avatar
49 votes
14 answers
6k views

Should we avoid indefinite integrals?

I am very uncomfortable with indefinite integrals, as I have difficulty giving them a precise sense that matches how they are written and the usual meaning of other symbols. For example, when one ...
Benoît Kloeckner's user avatar
12 votes
7 answers
3k views

How can you be perfect at maths (highschool)?

I'm in my last year of highschool. And I'm aiming for a perfect grade in maths. The problem is that this year is the hardest year of maths I have ever faced in my entire life. Especially derivation ...
Vincent's user avatar
  • 137
65 votes
14 answers
3k 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 ...
Jared's user avatar
  • 2,223
59 votes
24 answers
73k 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 ...
Chris Cunningham's user avatar
22 votes
4 answers
1k views

How can I choose a free calculus textbook?

As I have been recently informed, it is a good idea to consider free calculus textbooks for college and university courses. However, this feels risky to me, because: I don't know anyone who is using ...
Chris Cunningham's user avatar
46 votes
16 answers
33k 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, ...
Jim Belk's user avatar
  • 8,279
39 votes
11 answers
3k 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 ...
Anschewski's user avatar
  • 4,811
32 votes
10 answers
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 ...
Brian Rushton's user avatar
30 votes
4 answers
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 ...
Chris Cunningham's user avatar
25 votes
6 answers
2k 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 ...
user avatar
39 votes
14 answers
7k 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 ...
Frank Thorne's user avatar
  • 2,249
37 votes
7 answers
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) \, ...
Mike Shulman's user avatar
  • 6,580
33 votes
11 answers
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 ...
Brian Rushton's user avatar
33 votes
12 answers
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 ...
Brian Rushton's user avatar
31 votes
6 answers
3k 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 ...
user5249's user avatar
  • 311
28 votes
8 answers
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}{...
Frank Thorne's user avatar
  • 2,249
25 votes
10 answers
7k 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 ...
ClassicEndingMusic's user avatar
23 votes
8 answers
4k 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 ...
Chris Cunningham's user avatar
23 votes
9 answers
3k views

The definition of natural log and e

I'm asking this question from the point of view of an introductory non-rigorous calculus instructor. Calculus textbooks have different approaches about how to define $e$ and $\ln$. For example, my ...
Chris Cunningham's user avatar
21 votes
6 answers
2k views

Physical applications of higher terms of Taylor series

Depressingly many of the physical "applications" of Taylor series that I can find in textbooks and online are actually just applications of linear approximation, since they only take the constant and ...
Mike Shulman's user avatar
  • 6,580
19 votes
2 answers
866 views

Emphasizing the discrete in early undergraduate education?

From time to time, I have come across course ideas emphasizing the discrete over the continuous, such as Peter Saveliev's Fantasy Math curriculum (update: see also his material on discrete calculus) ...
J W's user avatar
  • 4,753
35 votes
11 answers
2k 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 ...
Gamma Function's user avatar
29 votes
4 answers
4k 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 ...
Brendan W. Sullivan's user avatar
27 votes
16 answers
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:...
Muath Karaki's user avatar
26 votes
7 answers
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....
Chris Cunningham's user avatar
25 votes
5 answers
829 views

"Function" vs "Function of ...": how much does it contribute to students difficulties?

Most textbooks I've seen (and teachers I've met, myself included) are rather careless about the distinction between variables and functions. For example, when we write $y=f(x)$ we all know that $f$ ...
Michael Bächtold's user avatar
23 votes
2 answers
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 ...
Akiva Weinberger's user avatar
23 votes
7 answers
7k views

Is the reciprocal function continuous?

I'm curious the views of those who teach calculus. As you know the continuity of a function at a point is defined in terms of the limit in the typical course. I'd like to ask a pair of questions: ...
James S. Cook's user avatar
22 votes
3 answers
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 ...
Mikhail Katz's user avatar
  • 2,238
20 votes
5 answers
4k views

Source of conceptual, multiple choice calculus questions

I'd like to give my Calculus 1 class periodic multiple choice questions that really test conceptual understanding. Ideally, I'd like these questions to require very little computation. I know that a ...
Jared's user avatar
  • 2,223
19 votes
2 answers
700 views

Comparison of different concepts of integral

As the following math stack exchange question (and answers) show: https://math.stackexchange.com/questions/703212/is-dxdy-really-a-multiplication-of-dx-and-dy There are a lot of different ways to ...
kjetil b halvorsen's user avatar
16 votes
1 answer
1k views

Standards-based grading in calculus

A friend of mine recently tried a standards-based grading (SBG) approach for her Calculus II course. (You can read about Kate's experience on her blog.) I find this approach to evaluation very ...
François G. Dorais's user avatar
15 votes
6 answers
3k views

Is there any difference between teaching calculus for math and engineering students?

In our university both math and engineering students attend in the same calculus classes. There are arguments in our department about the possible influences of this approach on students. It seems ...
user avatar
14 votes
6 answers
4k views

Explaining the symbols in definite and indefinite integrals

I teach the definite integral before the indefinite for a few reasons, one of which is that I want students to recognize that the definite integral means area (not anti-derivative). If we do ...
Sue VanHattum's user avatar
  • 21k
14 votes
2 answers
1k views

What are some activities/projects I can assign to calculus students from bio/chem/physics majors to specifically motivate their interest?

(This question was proposed during the area51 phase.) It's common for chemistry/biology/physics majors to be required to take certain calculus courses. At my school, chem/bio students must take up ...
Brendan W. Sullivan's user avatar
11 votes
6 answers
1k views

How can I convince students that Fourier series are useful?

Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ...
matqkks's user avatar
  • 1,243
11 votes
3 answers
807 views

How to evaluate a proof that seems to be using induction informally?

In an Advanced Calculus course, students were asked to prove $$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+ \ldots +|a_n|$$ for $n$ real numbers $a_1,a_2,\ldots,a_n$ I am teaching assistant for this course, ...
JAEMTO's user avatar
  • 229
7 votes
8 answers
2k views

List of realistic extremum problems

I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of ...
Student's user avatar
  • 179
7 votes
5 answers
2k views

Constructing and sketching parabolas, conic sections and other curves

Whenever teaching or discussing parabolas, conic sections and other curves with my students, I always feel dissatisfied with the standard "find vertex, pick points, connect the dots" method to draw a ...
celeriko's user avatar
  • 5,070
6 votes
3 answers
947 views

Teaching Calculus Less Formally

I'm wondering if anyone knows of calculus books or other work towards teaching calculus in a less mathematically rigorous way. I'm thinking mostly of American-style college level calculus courses ...
Henry Towsner's user avatar
5 votes
2 answers
670 views

Which textbooks on College Algebra, Trigonometry, Pre-calculus, Calculus, Linear Algebra, ODE are written by world-class mathematicians?

For example, Trigonometry was written by Wolf-Prize winner Israel Gelfand, one of the top mathematicians in the 20th century. I am wondering if other world-class mathematicians have written textbooks ...
Zuriel's user avatar
  • 4,275
2 votes
6 answers
10k views

Is there a more telling name for "Calculus 2"?

I see a lot of places where "Calculus 1" is referred to as "Introduction to Calculus", or "Single-variable Calculus." "Calculus 3" is referred to as "Multiple-variable calculus." Is there an ...
Burt's user avatar
  • 694
-1 votes
2 answers
895 views

Is AP Calculus AB really necessary?

High school student here... This coming school year I'm scheduled to start AP Calculus AB and then my school is looking into taking Multivariable Calculus at a local university. The school's calculus ...
CaptainAmerica16's user avatar
33 votes
10 answers
11k 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 ...
Steven Gubkin's user avatar
33 votes
3 answers
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 ...
user avatar
29 votes
10 answers
2k 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 ...
Markus Klein's user avatar
  • 9,448
28 votes
10 answers
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 ...
Mike Shulman's user avatar
  • 6,580