Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
356
questions
3
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2answers
115 views
Average Cost to Velocity Analogy
In my Business Calculus class (U.S. college-level), we discuss three aspects of cost: Total Cost $C(q)$, Marginal Cost $MC(q)$, and Average Cost $A(q)$ where $q$ is quantity produced. The defining ...
2
votes
0answers
118 views
Is Stewart Calculus a good book for AP calculus exam prep?
I have some background from completing Silvanus Thompson's book, but I didn't fully grasp the later chapters in it. I'm going to use khan academy and maybe other resources as a supplement.
How useful ...
0
votes
1answer
217 views
Distance from the wall when reading a sign? A Calc I/II problem
Here is a question on a test I asked recently in a Calculus I class I teach on Saturdays. These are students who are planning to major in Math, and they are already taking AP Calculus BC in their high ...
0
votes
2answers
165 views
How to explain the difference between a constant function and a linear function?
Both function types contain a straight line on a plane, so if a student asks for the difference between them what would be a standard or recommended way to explain it?
-4
votes
2answers
169 views
What is the most fundamental continuous function in calculus after a constant (totally straight) line?
What would be to teach in most countries and education systems?
(Taught before "high education frames", i.e. before doing bachelor of arts in mathematics).
0
votes
4answers
291 views
implicit differentiation, formula of a tangent line
I need to understand "implicit differentiation" and after that I need to be able to explain it to a student.
Here is an example:
Find the formula of a tangent line to the following curve at ...
7
votes
0answers
149 views
How to recognize possible dyscalculia in a student?
I am looking for input/advice regarding whether a student I just began tutoring may have dyscalculia - and, if so, how to go about broaching the subject / assisting them as best as possible.
I'd ...
4
votes
1answer
292 views
Exponential & logarithm in a high school calculus class
So recently I was teaching high school calculus to a high school class and I was wondering about the pedagogically best way to make students actually understand why the derivatives of the exponential &...
1
vote
2answers
104 views
How can I effectively learn and master Math and Statistics for Data Science?
I completed a BSc in Computer Science recently and am going on to do an MSc in Data Science. However, the only focussed math module I had during CS was in the first year and I didn't do too well. I ...
5
votes
5answers
2k views
Is Calculus Made Easy by Silvanus P. Thompson a good book for first-time calculus learners?
Specifically the one updated by Martin Gardner. I'm not studying as part of a high school or college course (I, in the near future, will though) just as a personal project.
12
votes
16answers
6k 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
205 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 ...
15
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
181 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
553 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 ...
4
votes
4answers
238 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
385 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
349 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
298 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 ...
4
votes
2answers
185 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
561 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
428 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
537 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
213 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
292 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, ...
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votes
11answers
5k views
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
310 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
255 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
321 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
101 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
2k 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
233 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
250 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
343 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
81 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
176 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
205 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
830 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
189 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
245 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
539 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
464 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
385 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 ...