Questions tagged [calculus]

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

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28 votes
6 answers
2k views

Calculus problems arising from real research problems

I am visiting my in-laws for the holidays. My sister in law is a statistician. She asked me to take a stab at a calculus problem which was coming up in her research. The Lambert $W_0$ function is ...
3 votes
6 answers
961 views

Is this motivation for the concept of a limit a good one?

tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...
5 votes
8 answers
9k views

What do you do in order to drag out lectures?

I posted earlier about how I was surprised that a typical Calculus 1 course that meets 3-4 hours each week for 15 weeks only barely manages to reach the fundamental theorem by the end of the course. ...
3 votes
6 answers
529 views

Why is calculus important for pre-service Math basic school teachers?

Why should pre-service math basic school teachers take calculus courses? Should this course be different from calculus courses offered to engineering?
2 votes
1 answer
133 views

Calculus/analysis as basis for basic school topics

I'm looking for examples of facts/results in basic school that are supported by calculus/analysis. Here is an example: Consider the infinite decimal expansion (a usual topic in basic school): basic ...
6 votes
4 answers
2k views

What are some rationales to teach Computer Science students Sequences and Series?

I was asked to teach the following topics to undergraduate Computer Science (CS) students in a discrete math course: Sequences (definition, convergent sequences, find the limit of a convergent ...
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0 votes
1 answer
111 views

What are specific set of tools Partial Differential Equations provide in studying a system? [closed]

I know what are PDEs, but I am looking to identify the major strengths of PDEs. If I have to convince a pool of engineers to use PDEs for solving a problem, let's say stress distribution in a body. ...
-4 votes
2 answers
254 views

Is it nonsensical to try to 'prove' Euler's 'formula' in real numbers? What is Wikipedia/proofwiki even doing? [closed]

Edit re the close vote: I guess this 1 of those questions whose on-topic-ness depends on the answer. If the answer is no, then well maybe it's off-topic. But if the answer is yes, then I believe it's ...
  • 564
11 votes
1 answer
1k views

Why Massively Multivariate open online calculus class (M2O2C2) on Coursera was discontinued?

I know that MOOCs were generally unsuccessful. However, I felt that M2O2C2 in Coursera was a great (at least my favorite) course and it's a pity it was removed. Does anyone have any info - will it ...
  • 113
5 votes
4 answers
2k views

Skipping a calculus topic (squeeze theorem)

Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School. We are on ...
6 votes
5 answers
1k views

Explaining difference between improper integrals that converge and diverge

How would you explain the difference in the results given by integration of the two functions $y = \frac{1}{1+x}$ and $y = \frac{1}{1+x^2}$ ? The graphs of these two function look so similar on ...
4 votes
0 answers
237 views

What are your experiences with Buck’s Advanced Calculus?

I stumbled across the book when searching for rigorous alternatives to Rudin with some solutions. It’s an “old school” (1965) calculus text but, I think, covers similar material to Rudin in a more ...
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6 votes
1 answer
351 views

Multivariable Calculus Project Ideas

Next semester, I am going to teach a small section of advanced high school students a class of Multivariable Calculus (it's about 3-4 students that have completed AP Calculus BC). Multivariable ...
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1 vote
0 answers
86 views

References to learn modern functions applied to integration and numerical series problems and how to teach them to Calculus students

I think most of us have met integration problems concerning the trigonometric, polynomial, exponential, hyperbolic and power functions in the calculus courses. But many of the problems in this website ...
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5 votes
2 answers
939 views

Is "Annular Ring" redundant?

I've come across the term annular ring in parentheses following washer in my calculus textbook: "has the shape of a washer (an annular ring)". The definition of the word "annular" ...
  • 153
2 votes
0 answers
116 views

Generating exercises about extrema of $f(x,y)$

(I have asked this question on math.stackexchange.com and according to a suggestion from comments I am re-asking it here.) I can generate many examples of functions $f(x,y)$ for which finding local ...
-1 votes
5 answers
303 views

Is it considered a mistake to use different correct notation for writing intervals?

Standard definition of writing interval states that it should be written (a,b) where a<b Due to this being arbitrary and just a convention that we all use, would it be considered a mistake to write ...
4 votes
4 answers
451 views

Student forgets to remove dx after integrating

I am tutoring another US college student in a Calculus 1 class. Initially, she was having trouble with basic concepts, but after much prodding most of the conceptual difficulties seem to have been ...
  • 247
4 votes
1 answer
223 views

AP Calculus BC Guidance

Hello wonderful educators. I am hoping to get some help on a tough situation I am in. So first, a little bit of background facts: This is my first time teaching an AP class. I've adjuncted for Calc I ...
  • 1,992
3 votes
2 answers
168 views

Good Examples of Equations Derived from Elementary Calculus

I'm collecting additional enrichment content for my calculus students. I'm looking for examples of equations that are used in various fields, but which can be derived at least somewhat ...
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18 votes
6 answers
5k views

Do I really need to cover solids of revolution in my Calculus I class?

I will be teaching Calculus 1 soon, using Stewart's Calculus: Early Transcendentals as a reference. I can't help but recall my time in high school AP Calculus and my first semester undergraduate ...
8 votes
5 answers
376 views

Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?

I'd like to ask my students whether some real function is differentiable at a certain $x_0$. I prefer not telling them that they have to use the definition of the derivative, but to instead present a ...
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12 votes
3 answers
3k views

How do I sketch a good gaussian curve freehanded, or by using only common sketching tools?

I'm a lousy artist. If I want my Gaussian curves to be accurately drawn when I use a whiteboard, or work with pen & paper, what are my options? Is there a way to use a straight edge, or compass, ...
1 vote
1 answer
187 views

Introducing direct substitution in an intro calculus course

I'm revisiting the materials I've put together for students taking a non-proof-based intro to calculus, and my goal is for them to have a clear but rough sense of a limit as a bound (basically enough ...
  • 229
0 votes
3 answers
326 views

How can I visualize differential equations and Integration in real life?

How can we understand differential equations and Integration in real life so that we can understand calculus easily. All we do here, at university level is memorize calculus and get the answer. We ...
7 votes
8 answers
3k views

Nice examples of limits to infinity in real life

I have to teach limits to infinity of real functions of one variable. I would like to start my course with a beautiful example, not simply a basic function like $1/x$. For instance, I thought of using ...
6 votes
1 answer
220 views

More advanced (free) alternatives to Geogebra and Math3D?

I teach vector calculus. I love both Math3D and Geogebra. But I have reached a limit in terms of what these programs can do. Some examples of features that I wish Math3D had: Draw vector fields with ...
4 votes
4 answers
228 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 ...
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2 votes
2 answers
413 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
1 answer
294 views

Distance from the wall when reading a sign? A Calc I/II problem [closed]

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 ...
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0 votes
2 answers
350 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
2 answers
184 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
4 answers
338 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 ...
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7 votes
0 answers
167 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 ...
  • 71
8 votes
4 answers
678 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
2 answers
114 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
5 answers
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.
15 votes
17 answers
7k 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 ...
  • 269
17 votes
9 answers
7k 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 ...
user avatar
5 votes
1 answer
247 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
7 answers
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. ...
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2 votes
2 answers
215 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
2 answers
663 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 ...
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4 votes
4 answers
329 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-...
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2 votes
4 answers
551 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 ...
  • 121
7 votes
1 answer
486 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 ...
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13 votes
3 answers
376 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 ...
  • 18.5k
3 votes
2 answers
244 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 ...
  • 6,685
1 vote
0 answers
66 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. ...
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7 votes
6 answers
788 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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