Questions tagged [calculus]

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

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5
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2answers
469 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
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4answers
204 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
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4answers
329 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 ...
5
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2answers
191 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 ...
8
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1answer
321 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 ...
11
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4answers
836 views

Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ — why?

I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than ...
12
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3answers
258 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 ...
17
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8answers
7k views

“Real world” examples of implicit functions

When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences. So I'm looking for (relatively simple) ...
5
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3answers
455 views

Advice on Full Time Community College Math Instructor Position

I have to do a teaching demo on the following: Please treat the committee as students in your Calculus II class. Please take 12-15 minutes to introduce your lesson on the Taylor Series and its ...
6
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3answers
139 views

Is there a point at which it makes decidedly more sense to learn about a “linear approximation” to a function, rather than a “tangent”?

I'm tutoring a first-semester calculus student, and we were looking over the slides the teacher has used. After teaching (or rather, repeating, for those who completed AP high school math) basic ...
4
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4answers
256 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 ...
5
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2answers
167 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 ...
6
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4answers
527 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 ...
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0answers
56 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. ...
10
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2answers
376 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
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3answers
448 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 ...
12
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6answers
711 views

teach that $\frac10$ not defined properly

there're some students, who belive that $$\frac10 = \infty $$ I need to teach them that this is not true and $\frac10 $ is undefined, mathematically and give a good picture (for their minds) what is ...
2
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1answer
260 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
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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 ...
18
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4answers
540 views

“Function” vs “Function of …”: how much does it contribute to students difficulties?

Most textbooks I've seen (and teachers I've met, myself included) are rather careless about the distinction between variables and functions. For example, when we write $y=f(x)$ we all know that $f$ ...
10
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4answers
907 views

How to teach calculus (book recommendation)

I'm going to teach calculus for the first time to undergraduate students. I would like to know if there is some book about how to teach the concepts of calculus (e.g. limits, derivatives, etc.).
1
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1answer
259 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
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1answer
244 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
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3answers
292 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 ...
21
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3answers
804 views

Tutoring a recalcitrant/awkward/exasperating student---special needs?

As part of my duties at a GTA, I spend several hours per week in our department's drop-in tutoring center. The center is open to all students enrolled in 100- and 200-level math courses, with the ...
0
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0answers
96 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
455 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?
4
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4answers
1k views

How are the basic trigonometric functions introduced to students?

The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is ...
0
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1answer
173 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
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3answers
232 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
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4answers
301 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 ...
8
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9answers
6k views

Why are calculators not allowed in post-secondary exams?

Before you downvote this question, I actually want an answer to this. Is the calculator going to give me my derivative? No. Is it going to give me my integral? No. It can sure give me the answer to my ...
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
78 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 ...
4
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1answer
167 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
127 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 ...
22
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4answers
558 views

Keeping quicker students engaged and interested throughout a course

In a college math course one is bound to find a fairly broad range of students in terms of their quickness in understanding the material. This is due to many reasons, including differing mathematical ...
63
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14answers
2k views

Encouraging class participation

I teach calculus to college students, and find it very difficult to get them to speak up in class when I ask questions, or when I'm trying to get a pulse for how much they understand. I think ...
4
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2answers
149 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 ...
2
votes
1answer
169 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 $...
4
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5answers
761 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
182 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 ...
5
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2answers
495 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 ...
15
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1answer
360 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 ...
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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2answers
428 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 ...
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 ...
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 ...

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