Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
346
questions
11
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16answers
5k views
How does one tutor an A-level student past the derivative paradox?
Background: I am new to this site, but have 1500 reputation on the main Maths Stack. I am (age-wise) a secondary student of maths, but for a very long time have been informally learning at home and ...
14
votes
8answers
6k views
Why is differential calculus often presented before integral calculus?
Why is differential calculus often presented before integral calculus?
Note: I'm still learning calculus at the moment.
It seems that many elementary calculus texts describe differential calculus ...
5
votes
1answer
189 views
Making the leap from Pre-Calculus to Calculus
This question is targeted at teachers who taught both low and high level mathematics. I have a group of students that I'm currently teaching precalculus and they seem to be doing really well in all ...
13
votes
7answers
2k views
An introductory example for Taylor series (12th grade)
I (a student) am doing a presentation on Taylor series in my class (12th grade, in Germany if this is relevant). I am looking for a good example where you can see when Taylor series might be useful. ...
2
votes
2answers
168 views
How to explain continuity and differentiability to highschool students? [closed]
I would say continuity is the idea that points numerically close to the input point of a function agree with the value of the function at that point. Now, suppose I introduce the idea of a derivative, ...
6
votes
2answers
528 views
Calculus limits taught in the US vs Spain?
So, I realize this can be a broad question, so I'll narrow it down. I have lived in Spain and own several Math textbooks from that country (the equivalent of 8th grade and high school Math). Has ...
5
votes
4answers
231 views
Showing applications of calculus to intro students
So I'm wrapping up my second year teaching high school calculus, and my first as an AP course. In addition to review, I'd like to end the year by giving the students a taste of the kinds of real-...
2
votes
4answers
363 views
Is it necessary to teach the definition of a limit for engineering majors? [closed]
I have always wondered whether it is necessary or not. For me, it seems that it is enough to teach them the intuitive idea, that is, limit is just an approximation of a certain process.
what do you ...
8
votes
1answer
338 views
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 ...
12
votes
3answers
278 views
Are there any applications of $x^x$?
I'm teaching Calculus I. It's time for the derivative of $x^x$. In previous semesters, I've told
students we mainly do this just for closure, so that we know that we can find derivatives of every ...
5
votes
2answers
178 views
Tabular Fundamental Theorem of Calculus
This semester in first-semester Calculus I've been trying to focus on how to do Calculus calculations when given a table of data since this seems to be of importance to science majors. There are ...
1
vote
0answers
59 views
History of business calculus/linear algebra curriculum
I will be teaching a combination of business calculus and business linear algebra, two classes that have been around awhile at my school. I’m assuming people are familiar with these types of classes. ...
6
votes
4answers
551 views
Distance Between Curves in First-Semester Calculus
In the optimization section of Calculus 1 a common problem is to find the minimum distance between a curve and a point. I'd like to extend this idea and be able to compute the minimum distance between ...
10
votes
2answers
401 views
Why are so many online sources “wrong” about directional derivatives?
I noticed many seemingly reputable online sources have "incorrect" description of
directional derivatives for real-valued functions in several variables.
Here, by "incorrect" I ...
8
votes
3answers
498 views
Why teach the algebraic Calculus?
In the context of a standard undergraduate Calculus sequence, I've noticed there is a big emphasis on teaching the algebra part of Calculus. What I mean by this is that a student may feel more ...
5
votes
2answers
201 views
Term for candidates for inflection points
The critical points of a function $f(x)$ are candidates for local extrema, i.e., if a function changes from increasing to decreasing, or vice versa, it must happen at a critical point.
Is there an ...
2
votes
1answer
275 views
Are there any university programs that “supersize” calculus courses?
Most differential calculus courses begin with the theory (and analysis) of differentiation, followed by computations, and likewise integral calculus courses. That's a lot for a three credit course, ...
19
votes
11answers
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Why do we still teach the determinant formula for cross product? And is it as bad as I think it is?
The cross product is an important vector operation in in any serious multivariable calculus course. In most textbooks that I'm aware of, right after the definition, we always introduce the ...
1
vote
1answer
289 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?
4
votes
1answer
248 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. ...
11
votes
3answers
305 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 ...
0
votes
0answers
98 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-...
3
votes
3answers
726 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?
0
votes
1answer
200 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 ...
6
votes
3answers
240 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 ...
3
votes
4answers
311 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
3k 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
79 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 ...
4
votes
1answer
169 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
128 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
173 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
780 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
184 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
181 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
505 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
440 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
371 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
143 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 ...
19
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 ...
6
votes
1answer
209 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
199 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
137 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
514 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. ...
69
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
458 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
146 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 ...
