# Questions tagged [calculus]

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

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### 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 ...
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 ...
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 ...
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?
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).
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### 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 ...
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 ...
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 &...
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 ...
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.
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### 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 ...
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 ...
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 ...
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. ...
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, ...
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### 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 ...
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### 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-...
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 ...
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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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 ...
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 ...
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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### 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 ...
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?
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### 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. ...
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 ...
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-...
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?
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### 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 ...
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 ...
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### 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 ...
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### 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 ...
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 ...