Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
327
questions
2
votes
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
votes
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
votes
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
vote
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
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?
...
8
votes
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
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 ...
2
votes
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
$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{t^2} dt = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}...
2
votes
1answer
136 views
Making epsilon-delta proofs not just precalculus
In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it ...
8
votes
1answer
248 views
Terminology for parts of limit notation
When we talk about: $$\lim_{x\to{c}}f(x)=L$$ Is there a formal name for the number "$c$"?
I know of course that it means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It just would be nice to be ...
1
vote
1answer
107 views
How to use Calculus to give an explanation for a video by Lewin [closed]
In the video by the well-known physicist Water Lewin Link Here,
in which he demonstated a series of experiments of rolling objects. Many are quite surprising. At the end of the video, the attended ...
7
votes
7answers
2k views
Teaching Calculus I to engineers
I am in a research project where one of our jobs is improving the first year university experience for our students. One of the topics we are looking into is changing the way we teach our introductory ...
9
votes
8answers
3k views
Are there any proofs of Euler's Formula that do not rely on calculus?
The most common way I have seen Euler's formula
$$
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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.$$
Here students are supposed to know derivatives, increasing-decreasing,...
In ...
6
votes
3answers
162 views
(Riemann integrability) How do you explain this to a high school student?
The following question was in a high school teacher's guide:
Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ defined by
$$f(x)=\begin{cases} x & x\in\mathbb{R}\setminus\mathbb{Q}\\
2x & x\...
5
votes
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
votes
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
votes
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
votes
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 ...
