Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
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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 ...
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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.)
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
user507
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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 ...
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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) \, ...
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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 ...
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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 ...
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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 ...
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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}{...
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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 ...
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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 ...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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:...
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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....
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"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 $f$ ...
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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 ...
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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:
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
user230
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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 ...
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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 ...
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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 ...
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Grade on proving |$a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$
In an Advanced Calculus course, students were asked to prove $$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+... +|a_n|$$
for $n$ real numbers $a_1,a_2,...a_n$
I am teaching assistant for this course, and one of ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
user507
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10
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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 ...
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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 ...
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28
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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 ...
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