# Questions tagged [calculus]

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

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### 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 ...
6k views

### Should we avoid indefinite integrals?

I am very uncomfortable with indefinite integrals, as I have difficulty giving them a precise sense that matches how they are written and the usual meaning of other symbols. For example, when one ...
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### 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 ...
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### 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 Kline's Calculus: an Intuitive and Physical Approach be sufficient? My interest is in applications: dynamical ...
6k 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 ...
807 views

### How to evaluate a proof that seems to be using induction informally?

In an Advanced Calculus course, students were asked to prove $$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+ \ldots +|a_n|$$ for $n$ real numbers $a_1,a_2,\ldots,a_n$ I am teaching assistant for this course, ...
451 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, ...
29k 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 ...
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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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: ...
394 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 $2$, ...
358 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 ...
414 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 ...
873 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 ...
356 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 ...