Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
310
questions
0
votes
1answer
63 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
428 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
365 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
309 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
76 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
186 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
119 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?
...
7
votes
1answer
232 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. ...
62
votes
17answers
8k 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? Letās brainstorm here.
Some challenges:
My school provides limited online ...
2
votes
4answers
200 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
$$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{t^2} dt = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}...
1
vote
1answer
119 views
Making epsilon-delta proofs not just precalculus
In trying to find lecture-length videos of epsilon-delta proofs, I've found that almost all of them just start with the definition and then work through the algebra to get the answer. In effect, it ...
8
votes
1answer
232 views
Terminology for parts of limit notation
When we talk about: $$\lim_{x\to{c}}f(x)=L$$ Is there a formal name for the number "$c$"?
I know of course that it means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It just would be nice to be ...
1
vote
1answer
105 views
How to use Calculus to give an explanation for a video by Lewin [closed]
In the video by the well-known physicist Water Lewin Link Here,
in which he demonstated a series of experiments of rolling objects. Many are quite surprising. At the end of the video, the attended ...
7
votes
7answers
2k views
Teaching Calculus I to engineers
I am in a research project where one of our jobs is improving the first year university experience for our students. One of the topics we are looking into is changing the way we teach our introductory ...
10
votes
8answers
3k views
Are there any proofs of Euler's Formula that do not rely on calculus?
The most common way I have seen Euler's formula
$$
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
129 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....
9
votes
1answer
704 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
301 views
An intuitive explanation of l'Hôpital's rule for ā/ā
L'HĆ“pital'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
473 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
158 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
225 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
141 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.$$
Here students are supposed to know derivatives, increasing-decreasing,...
In ...
6
votes
3answers
159 views
(Riemann integrability) How do you explain this to a high school student?
The following question was in a high school teacher's guide:
Let $f\colon\mathbb{R}\rightarrow\mathbb{R}$ defined by
$$f(x)=\begin{cases} x & x\in\mathbb{R}\setminus\mathbb{Q}\\
2x & x\...
5
votes
2answers
217 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
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
133 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 ...
1
vote
2answers
76 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
votes
7answers
478 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
votes
1answer
93 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
153 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: āis there a pre-calculus introduction to the derivative?ā I would say yes, since there already are the ideas of a slopes of ...
18
votes
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
votes
3answers
143 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
votes
6answers
459 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
votes
1answer
108 views
Student-friendly / efficient approach to computing Taylor coefficients of infinite binomial series expansions?
Iām 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
votes
4answers
346 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 ...
0
votes
2answers
215 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
263 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
votes
2answers
164 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 ...
6
votes
2answers
404 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
votes
3answers
170 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 ...
12
votes
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
342 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
votes
2answers
107 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 ...
