Questions tagged [calculus]

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

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2 votes
7 answers
1k views

Earliest real-world uses of Calculus and Linear Algebra

I want to illustrate in class that real-world applications of mathematics might take time to come to fruit. In this context, I want to find what the earliest real-world applications of Calculus and ...
11 votes
5 answers
3k views

A visualization for the quotient rule

Context: first year didactics of mathematics course for middle school teacher students (in Norway). I have a reasonable visualization for the product rule of derivatives: Consider a rectangle with ...
9 votes
2 answers
439 views

Symmetric version of product and quotient differentiation rules

The usual way of writing the product rule and the quotient rule in differentiation is $$(fg)'=f'g+fg'$$ $$\left(\frac{f}{g}\right)'=\frac{f'g-fg'}{g^2}\quad\text{where}\quad g\ne 0$$ A few years ago, ...
30 votes
7 answers
28k views

Early vs. late transcendentals

There seem to be two approaches to calculus education: Early transcendentals: introduce polynomials, rational functions, exponentials, logarithms, and trigonometric functions at the beginning of the ...
25 votes
11 answers
6k 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 ...
21 votes
6 answers
2k views

Physical applications of higher terms of Taylor series

Depressingly many of the physical "applications" of Taylor series that I can find in textbooks and online are actually just applications of linear approximation, since they only take the constant and ...
39 votes
14 answers
7k views

How to make Calculus II seem motivated, interesting, and useful?

I am due to teach Calculus II in the fall at an American state university. Our calculus sequence is somewhat slow, due to the fact that many of our students come with limited backgrounds. Most of our ...
39 votes
11 answers
3k views

Reasons for (not) distinguishing $f$ from $f(x)$

Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. In teaching at school and university, notation is quite often mixed up, e.g. the function is ...
22 votes
6 answers
4k views

What theorems from single-variable calculus break down in the multi-variable context?"

In teaching multi-variable calculus, it's insightful to discuss with students not only how certain concepts from single-variable calculus extend to multiple variables but also where these extensions ...
42 votes
5 answers
3k views

How can I help a student who has a "wrong" kind of enthusiasm?

Alice (not real name) is a student in one of my Math 100 (calculus) classes. It's a course offered by my college as a dual credit course at a high school, so the whole class is about 17/18 years old, ...
7 votes
2 answers
105 views

Advice and Remedial Algebra Resources for Students Committed to Calculus

I've got a student in my introductory calculus course. They're failing because they lack algebra skills. They understand the concepts just fine, and can articulate their understanding fine, but get ...
11 votes
2 answers
915 views

What is the terminology for "self-referral" integrals in calculus?

In the topic of integration and anti-derivatives in Calculus we come across cases where the attempt at integration by parts brings us back to the original integral, the most basic example being $\int ...
-1 votes
4 answers
546 views

Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course

Imagine a course barely getting into some topics more theoretical than what is done in the typical very staid first-year calculus course, and the kind of students for whom such a course is appropriate....
7 votes
10 answers
9k 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 ...
13 votes
11 answers
4k views

How to explain what's wrong with this application of the chain rule?

Yesterday a student in my calculus class attempted something like this: Problem statement: Find the derivative of $3^{(5x+1)}$ with respect to $x$. Proposed solution: Let the inner function be ...
2 votes
1 answer
189 views

Define logarithmic function by functional relation [closed]

My son was working the other day with exercises such as: Find all the mappings $f:\mathbb{N}\rightarrow\mathbb{Z}$ verifying $$\forall m,n \in \mathbb{N}, f(m+n)=f(n)+f(m).$$ As another example: Find ...
8 votes
4 answers
3k views

Any examples of calculus sequence that deemphasizes calculation tricks?

I'm considering creating a series of classes that explore deeper ideas in calculus without overemphasizing the various computational tricks used in integration and differentiation. My vision is ...
6 votes
4 answers
2k views

How can we explain intuitively the convergence and divergence of these two series?

It is known that $\displaystyle\sum_1^{\infty} \frac{1}{n^{1.000001}}$ converges while $\displaystyle\sum_{n\text{ is a prime number}}\frac{1}{n}$ diverges. Though we can logically prove these results,...
49 votes
14 answers
6k views

Should we avoid indefinite integrals?

I am very uncomfortable with indefinite integrals, as I have a hard time giving them a precise sense that matches the way they are written and the usual meaning of other symbols. For example, when ...
27 votes
7 answers
17k views

Why not think of derivatives as fractions?

Back in high school—back in the 1900s, as my sons say—when our calculus teacher was introducing the chain rule... $\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$ ...he made a special point of ...
2 votes
3 answers
291 views

Is this a viable Calculus 1 question?

A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon rises straight into the air at a rate of 12 ft/sec. Is the ...
13 votes
6 answers
5k views

"Real life" examples of limits of functions at finite points

This is more specific than this similar question on math.SE, since I'm not satisfied with the answers there. Question: Can you provide an interesting, natural and simple example of some physical/...
15 votes
15 answers
7k views

Students can't seem to grasp the intent of tangent lines and getting general trends of derivatives from graphs

Background I'm informally helping a few students with college Calc 1. This isn't the first time I've aided people with calculus, and so they've sought me for help, though I don't consider myself to ...
12 votes
4 answers
748 views

Good ways of explaining the idea of epsilon-delta limits to bio & chem majors?

I am cross-posting from MSE. The students in my calc classes tend to be primarily bio/chem majors, and not very much math / physics / engineering. I feel like there are pretty good ways to talk ...
1 vote
1 answer
229 views

What would be a good pacing for teaching this calculus 2 course?

Next semester I'm going to lecture calculus 2 in an institution I just joined. However, when I had calculus 2 back then the syllabus was very different, it mainly covered several variable calculus up ...
0 votes
2 answers
137 views

Sourcing and verifying calculus applications

There are many questions on this site about specific (or not-so-specific) applications of calculus to the "real world". However, one issue I've noticed in using textbooks for this purpose ...
1 vote
1 answer
382 views

Is there any university or college in any country where failure and dropout rates in Calculus are not so high?

Calculus is a foundational mathematics course that is often seen as a bottleneck for STEM majors. However, it is also a course that is notorious for its high dropout rates. In the United States, for ...
20 votes
5 answers
4k views

Source of conceptual, multiple choice calculus questions

I'd like to give my Calculus 1 class periodic multiple choice questions that really test conceptual understanding. Ideally, I'd like these questions to require very little computation. I know that a ...
7 votes
5 answers
3k views

How to properly define volume for beginner calculus students?

I'm interested in opinions based on experience about how to introduce volume for beginner calculus students. Below I present some observations and specific questions. In Stewart's book, the volume of ...
33 votes
12 answers
5k views

For calculus students, what should be the intuition or motivation behind series?

I've noticed that series are one of the most difficult portions of calculus for new students to learn. I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as ...
18 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 ...
35 votes
11 answers
2k views

Epsilons and deltas in a first calculus course

In a freshman calculus course for non-majors; Is it to the benefit of the students to include discussion of epsilons and deltas? To what extent, if any, should they be used? For example, just to ...
11 votes
3 answers
629 views

Terminology for parts of limit notation

When we talk about: $$\lim_{x\to{c}}f(x)=L.$$ Is there a formal name for the number "$c$"? I know that the notation means "$L$ is the limit of $f(x)$ as $x$ approaches $c$". It ...
6 votes
7 answers
2k views

How to explain that integral calculate areas?

I teach internal combustion engines theory in a technical school. I have an elementary knowledge of calculus and my students lack even this. I want to intuitively explain them what is the pdV integral,...
1 vote
1 answer
221 views

Calculus at competitive level or Olympiad level

What are the topics of Calculus which can be useful for competitive level and the students are not exposed to it ?
25 votes
5 answers
819 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 $f$ ...
17 votes
7 answers
5k views

Why don’t we teach a topological view of continuity instead of epsilon-delta?

I would like a critique of this approach to teaching continuity to calculus 1 students. Show them that for an increasing function on (a,b) we have that (a,b) is contained in the set of solutions to $...
6 votes
2 answers
1k views

Topics covered in Calculus I and II (university level) that aren't covered in the AP Curriculum

I teach AP Calculus BC at my high school and we have AP Calculus AB as a pre-req for taking BC. So most of my students are coming in with a strong calculus foundation, and I can spend less time on the ...
19 votes
20 answers
8k views

How does one tutor an A-level student past the derivative paradox?

EDIT (two years later): I was saddened to realise that no-one seems to care at the school level. Everything I thought might be a problem ended up as a non-issue because no-one challenged anything. The ...
23 votes
7 answers
7k views

Is the reciprocal function continuous?

I'm curious the views of those who teach calculus. As you know the continuity of a function at a point is defined in terms of the limit in the typical course. I'd like to ask a pair of questions: ...
3 votes
5 answers
384 views

Average Rate of Change isn't/is Statistics

I have the common misconception in my business calculus classes that the Average Rate of Change, say from $x=1$ to $x=5$, is the statistical average of the rates on the four unit intervals $1$ to $2$, ...
4 votes
5 answers
356 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 ...
4 votes
6 answers
413 views

Which examples should we mention when teaching the concept of derivatives?

I am teaching Calculus for non-maths major students. As far as I know, when we teach about derivatives, we should mention "the rate of change". There are some practical examples to motivate this ...
1 vote
2 answers
979 views

Is Morris Kline's 'Calculus: An Intuitive and Physical Approach' a Good Book to Learn Calculus From ?'

Would I have to read a standard textbook in addition -- i.e., Stewart, etc. -- or would this be sufficient ? My interest is in applications (dynamical systems theory and physics in general).The book ...
2 votes
2 answers
785 views

Process of finding limits for multivariable functions

I was tutoring a student today and they asked a question which made me curious. We were working on the following question together. After explaining that we must look at the limit along the x axis, I ...
3 votes
3 answers
351 views

Interpreting the derivative as instantaneous rate of change in real phenomena

When interpreting the meaning of the derivative in real phenomena, it may seem that the interpretation is in conflict with the definition of the derivative itself. The confusion is caused by the units ...
5 votes
1 answer
343 views

How does the average level of expected mathematical sophistication at high school level increase?

I remember reading an old calculus book (years 1920-1930) and in the preface it was portrayed as revolutionary because it was for high school students. Nowadays, that is not revolutionary, because ...
4 votes
1 answer
160 views

How do you describe your experience using OER textbooks for calculus?

If you have used commercial as well as OPENSTAX OER textbooks for calculus I would like to know about your experience. How would you compare the two? Were there any disadvantages to using OpenStax?
5 votes
1 answer
481 views

Homework in a Flipped Classroom

I'm in the middle of teaching first-semester Calculus where, for the first time, I'm trying to implement a flipped classroom. (Background: Small university in U.S.; Calc 1 for STEM majors, 50 minute ...
17 votes
4 answers
2k views

Surprising examples of Cavalieri's principle

I showed my calculus students two circular cones with the same base and height, but one of them "right" and the other slanted, and asked which had a greater volume. They all answered correctly that ...

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