Questions tagged [calculus]

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

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7
votes
1answer
611 views

How to (or should one) distinguish between lowercase and uppercase letters orally when lecturing?

I sometimes teach calculus in English whereas it's not my native language. For example, during a course about antiderivatives, how do you (orally) pronounce $f$ vs $F$? Which are the best? "the ...
9
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2answers
168 views

An intuitive explanation of l'Hôpital's rule for ∞/∞

L'Hôpital's rule for the indeterminate form $\frac00$ at finite points can be given a nice intuitive explanation in terms of local linear approximations. See for instance this textbook or this one. ...
8
votes
6answers
392 views

Should Euler's formula $e^{ix}=\cos x+i\sin x$ be seen as a definition rather than something to prove?

There are a lot of "proofs" of the identity $e^{ix}=\cos x+i\sin x$ in textbooks, using either differential equations or power series. However, I find those proofs often misleading, because it appears ...
2
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2answers
137 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 ...
7
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5answers
2k views

Teaching asymptotic notations at the beginning of calculus [duplicate]

I'm thinking about teaching calculus by firstly introducing the asymptotic notations (big-Oh, little-oh, and $\sim$), secondly explaining their "arithmetic" (things like how to sum little-oh's and ...
0
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4answers
214 views

Is “Volume of Solids of Revolution” a part of Cal I or Cac II

I took Calc I in preparation for the CLEP. I did not learn about solids of revolution. Is this something that is normally part of Calc I and that most Calc II classes will expect that I can do already?...
4
votes
5answers
127 views

How to teach sketching a parametric curve?

I feel very confused when teaching students how to sketch a parametric curve like $$x(t)=e^{-t}t; y(t)=t^{2}+t.$$ Here students are supposed to know derivatives, increasing-decreasing,... In ...
6
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3answers
152 views

(Riemann integrability) How do you explain this to a high school student?

The following question was in a high school teacher's guide: Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ defined by $$f(x)=\begin{cases} x & x\in\mathbb{R}\setminus\mathbb{Q}\\ 2x & x\...
3
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1answer
122 views

Is there a rigrous and relatively complete book/lecture notes for selfstudying several variable calculus?

I would like to study university level mathematics on my own. I found rigorous books on single variable calculus and topology and learned those. But what would one recommend for several variable ...
15
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9answers
4k views

How important is knowledge of trig identities for use in Calculus

I have a question regarding tutoring a calculus student. They need to prove trig identities such as $$\frac{1}{1-\sin x}+\frac{1}{1+\sin x}=2\sec^2x.$$ Doing this kind of problem is very tedious and ...
-1
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1answer
125 views

Questions about the calculus AP [closed]

I'm not sure if this is the proper place to ask this question, but I figured I'll try anyway. If I have the option of taking a Calculus AP class, and am thinking of pursuing a career in either ...
11
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5answers
919 views

A different symbol for the indefinite integral/antiderivative?

Examples. An indefinite integral (or antiderivative) of $\cos$ is $\sin$: $$\int \cos = \sin.$$ Edit: There has been much unexpected confusion with the above statement. I define the above statement ...
1
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2answers
68 views

Retain problems and combat regression in learning

Regressive Learning It's a really stressful situation. I can achieve but not retain expertise in maths problems. History 6 months back, I studied integration in Calculus at college. I learnt it all ...
14
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7answers
426 views

How should students say in words the notation for a limit?

$$\lim_{x\rightarrow a} f(x)=L$$ Which way should students best get in the habit of? The limit of $f(x)$, as $x$ approaches $a$, equals $L$ The limit of $f(x)$ equals $L$, as $x$ approaches $a$ The ...
8
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1answer
89 views

Studies on more simple problems, or fewer difficult problems?

I'm a new adjunct faculty member. I haven't taught math yet (I'm teaching a related subject for now), but I've been thinking about my approach to teaching Calculus I should I be asked to teach it in ...
4
votes
3answers
134 views

Is there a pre-calculus introduction to the formal definition of a limit?

To give an example of what I mean, I'll answer a similarly worded question: “is there a pre-calculus introduction to the derivative?” I would say yes, since there already are the ideas of a slopes of ...
17
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5answers
8k views

What is the most difficult concept to grasp in Calculus 1?

I would say it is not the Fundamental Theorem of Calculus, but rather some notion connecting limits and continuity, perhaps the $(\epsilon,\delta)$-definitions of limits and continuity. But I would be ...
2
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3answers
133 views

Self-Study of Calculus Two [closed]

I just finished self study of Calculus One and I am looking to begin Calculus Two. Is there a free platform that can help me with this?
0
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6answers
424 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 ...
2
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1answer
99 views

Student-friendly / efficient approach to computing Taylor coefficients of infinite binomial series expansions?

I’m working on a section of a course covering Taylor expansions, and have found that, although there is great notation for simplifying the formula for the coefficients of a general infinite binomial ...
6
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4answers
322 views

Ideas for the introduction of the derivative?

I want to introduce to my class to the derivative, but I am still searching for a good, realistic context that isn't too hard to understand, without seeming to be contrived. Do you have an ideas for ...
-1
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1answer
144 views

Practical applications of integration by substitution where integrand is unknown

I posted this question on the Mathematics Stack Exchange a while ago, and got no responses, so I thought I would ask it here. I'm looking for any real-life applications of integration by substitution ...
4
votes
1answer
223 views

Resources for teaching calc III

I was very unsatisfied with how I taught Calc III a couple years ago, and this summer I have to do it again. Are there any general resources for teaching this course? It seems like there should be, ...
4
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2answers
153 views

Introducing derivative concept and definition

I need to give a short presentation on introducing a class of engineering students to the concept and definition of the derivative. I'm to assume that the students are currently at the appropriate ...
5
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2answers
249 views

Unconstrained/Constrained optimization real life example

I am in charge of some practice lesson for Calculus II. I have to show how to apply the theory for unconstrained optimization (mainly Hessian analysis) and constrained optimization (Lagrange ...
9
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3answers
154 views

Physical devices for exploring calculus or pre-calculus

I saw this partial derivative machine yesterday, and it got me excited about other physical devices for exploring calculus concepts in a "lab" setting (e.g. make a prediction, collect data, etc.) Do ...
11
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9answers
3k views

How to explain what's wrong with this application of the chain rule?

Yesterday a student in my calculus class attempted something like this: Problem statement: Find the derivative of $3^{(5x+1)}$ with respect to $x$. Proposed solution: Let the inner function be ...
8
votes
1answer
304 views

Real World use of the Function $(\sin{x})^x$

Today in my calculus class we were going over L'Hopital's Rule and were dealing with limits of the following form $$h(x)=f(x)^{g(x)}$$ Three examples we considered are as follows: $(1)\; \...
2
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2answers
101 views

Are questions on overlapping solids of revolutions without prior definitions and instructions fair given that there are divided interpretations?

If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame. But if his orders are clear, and the soldiers nevertheless disobey, then it is the ...
3
votes
1answer
160 views

When teaching someone how to prove a function is uniformly continuous, using epsilon/delta, which example would be among the simplest?

I've taught how to use $\epsilon, \delta$ to prove that a function is continuous at a point, and I'm about to teach how to prove that a function is uniformly continuous over an open interval. Usually,...
11
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8answers
454 views

How does knowing more about mathematics help one's teaching of lower level course, such as calculus?

A question has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with one's teaching of lower ...
1
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2answers
110 views

Activities for calc based physics

I was sort of thrown into teaching calculus based physics to a bunch of non-physics majors, who have taken one semester of calculus, and are poor with that material. It is only a 50 minute per week ...
4
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4answers
239 views

Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?

Nonstandard calculus is a reformulation of calculus that is based on infinitesimals instead of epsilon-delta definitions. Of course, people had tried to use infinitesimals in calculus before; in fact, ...
5
votes
3answers
213 views

Calculus workbook suggestions

Context: I am an assistant professor of mathematics at a small institution in the US. Our department uses Stewart's Essential Calculus for our calculus sequence, but I find that my students and I are ...
3
votes
1answer
73 views

Is there a point at which it makes decidedly more sense to learn about a “linear approximation” to a function, rather than a “tangent”?

I'm tutoring a first-semester calculus student, and we were looking over the slides the teacher has used. After teaching (or rather, repeating, for those who completed AP high school math) basic ...
2
votes
6answers
266 views

Which examples should we mention when teaching the concept of derivatives?

I am teaching Calculus for non-maths major students. As far as I know, when we teach about derivatives, we should mention "the rate of change". There are some practical examples to motivate this ...
4
votes
5answers
204 views

How to prepare for lecturing in a non-fluent foreign language?

I am giving some lectures on a calculus course in Norwegian. My Norwegian (or, rather, Scandinavic) is good enough to do so mostly without resorting to English, but I would, of course, like to improve....
4
votes
2answers
132 views

How to explain linear approximation to an equation to calculus students?

I am, at the moment, teaching calculus to students whose majors are, for example, biology, biochemistry, chemistry and geology. The course book is Claudia Neuhauser's "Calculus for biology and ...
4
votes
4answers
298 views

Recommended list of things calculus students should be required to memorise?

I am seeking a list of topics that students taking calculus should memorise. Some topics from Calculus I might include: $\varepsilon-\delta$ definition of limit; Definition of the derivative of a ...
17
votes
6answers
1k views

Is there a place to buy physical models to demonstrate the Calculus shell, disk, and washer methods?

I know a math teacher who is going to teach a calculus class that will include the shell, disk, and washer methods for calculating volumes. My question is, is there some 3D kit she could use to ...
1
vote
1answer
128 views

When self teaching, should I learn set theory before continuing ap calculus?

I am studying ap calculus now, before I move onto differential equations etc., but the thing I am unsure of is, should I learn set theory before continuing on my ap calculus sections?
10
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3answers
316 views

The royal road to calculus

In the early 1900s Felix Klein lay out his vision for secondary mathematics curriculum. He wanted schools to teach calculus, so that universities would not be burdened by it. And at the core of the ...
12
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3answers
1k views

Is it a bad idea to use an old textbook such as Differential and integral calculus, with examples and applications for calculus course?

I am wondering if it is a bad idea to use an old textbook, such as Differential and integral calculus, with examples and applications by George A. Osborne. This book was published in 1906 and there ...
12
votes
4answers
528 views

Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ — why?

I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than ...
10
votes
3answers
239 views

Formats for Calculus instruction at different colleges and universities

In the comments under another question, a couple of people expressed interest in how Calculus is taught at the University of Michigan. I'm not convinced a question that narrow is appropriate for this ...
7
votes
4answers
188 views

Making physical 3D models

I was thinking to make classroom illustrations of some 3D mathematical objects, such as graphs of 2 variable functions, minimal surfaces, etc. My question is, what would be a good way to go about it? ...
28
votes
3answers
5k 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 ...
1
vote
2answers
190 views

Line Integral Motivation

Is there a case to be made that the topic of line integrals should only involve vector fields? My colleagues and our textbook take the position that line integrals should only be taught from a vector ...
7
votes
7answers
622 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 ...
2
votes
2answers
369 views

Integrating derivatives over functions problem

I had a question from a student which I'm unable to answer. We were practicing the rule $\int \frac{f'(x)}{f(x)} \, dx=\ln(f(x))$. A student noticed that if applied naively it gives the following ...