Questions tagged [calculus]

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

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2
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1answer
180 views

Honors Precalculus: what topics to cut?

We’re precalculus honors teachers. In this year of Covid and reduced instructional time, what topics can we cut (Demana textbook) that would not hurt our kids in either calc AB or BC?
5
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1answer
230 views

Looking for a calculus books with very specific requirements

I plan to record lectures for a MOOC on Calculus sometime next year. The MOOC is targeted at an undergraduate audience that comprises engineers, math majors as well as majors in the sciences, etc. ...
12
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3answers
268 views

Usefulness of $u$-substitution in and beyond early Calculus?

My students, when presented with an integral (source) like $$\int (2x+2)e^{x^2+2x+3} \ dx$$ are apt to recognize derivative patterns like $u' e^{u}$ and reverse-engineer anti-derivatives rather than ...
1
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0answers
87 views

What notation do they use for mathematical expressions in Polish schools?

I thought of something like Polish notation all by myself and asked the question https://cs.stackexchange.com/questions/111067/could-we-define-the-decimal-notation-of-a-natural-number-as-a-series-of-...
2
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3answers
295 views

Are differential equations considered calculus and included in a calculus class or is it its own class?

Are differential equations considered calculus and included in a calculus class or is it its own class? Also, if it is its own class then what calculus classes does it come after?
1
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1answer
138 views

Teaching Hours for AP Calculus AB [closed]

What are the estimated hours for teaching AP Calculus AB for students aiming for a 5? Similar question, how many hours of practice will the student need to put it. Some Clarifications based on ...
7
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3answers
220 views

Am I responsible to help a student who does not understand/know some prerequisites of a course?

I am teaching Calculus III this semester and a student signed up for this course after completed Calculus I and II in a different institution. I quickly realised that this student does not understand ...
4
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4answers
285 views

Teaching calculus in AP without the limit definition

Years ago as a college freshman I was taking my first calculus course. Another freshman skipped it because he had calculus in Advanced Placement in high school. I mentioned we were learning the limit ...
5
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4answers
2k views

Examples of real-life vector fields for vector calculus

My two main ones are Electrostatic force field $\mathbf{E}\left(\mathbf{r}\right)=\frac{Q}{4\pi\epsilon_0 \left|\left|\mathbf{r}\right|\right|^3}\mathbf{r}$ and Gravitational force field, $\mathbf{F}\...
2
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1answer
72 views

Appropriate context for teaching derivative (undergraduate/graduate)

(Repost from MO, where the question will eventually be closed.) This question is related to lectures I have to make concerning differential calculus in one variable, but the students are quite ...
9
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4answers
3k views

Can we skip Newton's Method?

I am teaching an introductory calculus course for high school juniors and seniors. It is not formally described as an AP Calculus course, but it is supposed to map roughly onto Calculus AB. The ...
5
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1answer
164 views

Low-tech ways of visualizing multivariable and vector calculus

One way, which is the most obvious, is do sketches of 3d shapes that tend to be the ones that we can all draw (like rectangle, cone, cylinder, sphere, etc.) Another way is by analogy so even if we can'...
2
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2answers
125 views

Analogy for cylindrical shells

The analogy for cross-sections is easy since we can think of how slices of bread can make up a loaf. But what would be the analogy for cylindrical shells? Regarding shapes, apparently there's ...
16
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6answers
2k views

Are there direct practical applications of differentiating natural logarithms?

The textbook I am using to teach Calculus I includes in the exercises of most chapters a number of interesting real-world applications of the concepts from that chapter. However, the chapter on the ...
4
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2answers
113 views

What is a good way to teach Taylor expansion of multi-variable calculus?

I found teaching Taylor expansion for multivariable functions rather challenging. It is a bit complicated to prove and to to compute. So what happened to me last year was that my students simply ...
4
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5answers
734 views

What strategy for picking convergence tests for series do you teach?

Without getting bogged down in details, I'll list the names only. It seems that the strategy I generally use is this: Divergence test first Is it a recognizable form? p-series or geometric? a) No ...
4
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1answer
181 views

When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, should we consider only those $(x, y)$ in the domain of $f$?

When evaluating the limit of $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$, we should or should not consider only those $(x, y)$ in the domain of $f(x, y)$ ? I am confused by different practices of ...
2
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1answer
154 views

In single variable calculus, do you distinguish between critical and singular points?

In some texts, a critical point is when the derivative exists and is zero, and a singular point is when the derivative does not exist. So I suppose, at $x=0$, $|x|$ would have a singular point while $...
5
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2answers
472 views

Intuition or geometry for Partial Fractions

When teaching partial fractions, there's probably no way to escape the heavy algebra necessary for partial fractions, but I'm wondering how to introduce the idea in a way that is intuitive or ...
2
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2answers
400 views

Do you mention the continuity and the differentiability of the empty function

My main question is directly related to the title: "Do you mention that (in its domain) the empty function is everywhere continuous and everywhere discontinuous?" (and a similar question ...
15
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1answer
346 views

Analogies for grad, div, curl, and Laplacian?

I want to try making some calculation-less questions about vector calculus identities that are solely based upon picture diagrams of vector fields, or fields that could be sketched out by hand. The ...
17
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4answers
3k views

What are some of the open problems that can be suitably introduced in a calculus course?

I feel it may be a good idea to introduce some related open problems in a calculus course. Surely I am not expecting my students to solve any one of them, though I cannot say it is absolutely ...
4
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2answers
141 views

Graphing program for conceptualizing calculus

I'm taking integral calculus at the moment. I was understanding everything quite well until we started learning about finding volume of a solid of revolution. I understand the concept, but practicing ...
18
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6answers
5k 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 ...
5
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1answer
109 views

Ideas and/or references for Projects for a Business Calculus course

I have undertaken the teaching a Business Calculus course for this semester (spring II). The various assesments for the students, include quizzes/hw/midterms/final exams, adjusted with suitable ...
7
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2answers
193 views

Do you teach different proofs or calculations of same question?

Recently I asked a question on math.stackechange about the most ways to differentiate the same function and it didn't seem to generate any interest - rather, the reason why I'd ask such a question was ...
5
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1answer
129 views

Resources for improving computational skills at the high school/university transition

Teaching first year undergraduates, I've noticed that what gives them the most trouble is simple computations like factoring, expanding, handling fractions, powers, especially when variables and other ...
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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? ...
8
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1answer
377 views

How to conduct online testing for Calculus?

Due to COVID-19, I have been planning to transition to online teaching (which, of course, includes online testing as well). The LMS that we use is Blackboard which is integrated with Proctorio. ...
67
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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 ...
2
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4answers
243 views

Proof that convergent Taylor Series converge to actual value of function

Taylor series (or Maclaurin Series) are the only way to get values for some functions, such as $$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{t^2} dt = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}...
2
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1answer
136 views

Making epsilon-delta proofs not just precalculus

In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it ...
8
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1answer
248 views

Terminology for parts of limit notation

When we talk about: $$\lim_{x\to{c}}f(x)=L$$ Is there a formal name for the number "$c$"? I know of course that it means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It just would be nice to be ...
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1answer
107 views

How to use Calculus to give an explanation for a video by Lewin [closed]

In the video by the well-known physicist Water Lewin Link Here, in which he demonstated a series of experiments of rolling objects. Many are quite surprising. At the end of the video, the attended ...
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7answers
2k views

Teaching Calculus I to engineers

I am in a research project where one of our jobs is improving the first year university experience for our students. One of the topics we are looking into is changing the way we teach our introductory ...
9
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8answers
3k views

Are there any proofs of Euler's Formula that do not rely on calculus?

The most common way I have seen Euler's formula $$ re^{i\theta} = r(\cos\theta+i\sin\theta) $$ introduced in a classroom environment is to substitute $i\theta$ into the series expansion of the ...
2
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3answers
131 views

How to teach integrals motivated by the work done in moving an object?

I am now teaching Calculus of several variables this semester. In apllications of integrals, the problem of finding the work done in moving an object under a force $F$ is one of the most common ...
14
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8answers
2k views

How should I introduce the Chain Rule

I'm halfway through my first year of teaching AP Calculus to high school seniors. It's been going generally well, but I'm feeling like I really could have done better getting them into the Chain Rule....
10
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1answer
715 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 ...
11
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4answers
332 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
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6answers
499 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 ...
3
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2answers
167 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
233 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
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5answers
145 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
162 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\...
5
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2answers
227 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 ...
16
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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
138 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
1k 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 ...

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