Questions tagged [calculus]

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

360 questions
Filter by
Sorted by
Tagged with
384 views

2k views

Near-universal student mistake on $\lim_{x\rightarrow\infty}e^{x+1}/e^x$

On a recent first-semester calculus exam, I gave a bunch of limits. The student was supposed to use L'Hospital's rule if possible, or if not, explain why it didn't work and evaluate it by some other ...
969 views

As a TA, how to reduce imprecise notations/statements in students' exams

I'm not a course instructor, just a TA of the first quarter calculus course who lead discussion sections and grade exams. When grading the midterm, I found large number of students showed some ...
1k views

Why do we teach that every line is a linear function?

Teaching my precalculus class today, I noticed something very simple that I hadn't taken into account previously. The definition in our textbook read: "A linear function is a function defined by ...
682 views

Natural, rich, calculus questions

We have the good fortune of having "lab sections" here at my college. I'm interested in conducting some activities in the spirit of this talk. However, even in my stash of inquiry-based learning ...
504 views

Using $dx$ for $h$ in the definition of derivative

Is it mathematically correct to write $$f'(x)=\lim_{dx\to0}\frac{f(x+dx)-f(x)}{dx},$$ rather than $$f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}?$$ If not, what is the difference? If so, why isn't this ...
741 views

How to use a CAS in teaching calculus

I want to introduce calculus students to computer algebra systems (CAS) like Sage, Geogebra, and Wolfram Alpha in college Calculus 1 and 2. While I believe in the value of learning to do calculus by ...
2k views

Should we teach trigonometric substitution?

This is the question that was not asked here. Also related is this question, but both presuppose that it will be taught and ask about how best to do it. My question here is, suppose we are designing ...
239 views

Colleges that showcase their calculus material online

Since I will be creating a calculus course, I am hoping to find calculus material used in the best colleges across the globe as a reference. Lecture notes and exercise sheets are highly appreciated. ...
854 views

Writing a Calculus textbook for a course I am creating

Using LaTeX, I am attempting to write a single-variable calculus textbook that gives the reader an understanding of calculus and its applications without a lot of the fluff I have seen in many other ...
988 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.).
323 views

Brief book on calculus to read before studying the analysis

I am going to start studying the analysis texts (Rudin-PMA, Apostol-MA, Pugh-RMA) on the first week of August. I have a good proof skills through working on Artin's Algebra and Hoffman/Kunze's Linear ...
3k views

Good examples of Lagrange multiplier problems

I've noticed that most Lagrange multiplier problems I've seen can be solved with other methods. Often the method of Lagrange multipliers takes longer than the other available methods. I don't like ...
990 views

Intergration by differentiating will get you $0$ marks - but how to explain why?

When integrating and differentiating, sometimes one direction is easy and the other is harder. A nice example is $\frac{d}{dx}\tan x=\sec^2x$, where differentiating is easy but integration (without ...
365 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, ...
1k views

Is Knuth's suggestion on teaching calculus a good idea?

Note: I myself am not a math educator, though I plan to be one someday. In this letter, Donald Knuth suggests an alternate way of teaching calculus, based on big-O (introduced via a related big-A ...
1k views

Should we tell students to never replace parts of an expression by their limits when taking a limit?

Let me explain. Suppose we want to calculate $\lim\limits_{n\to\infty} n^2-n$. Since this limit is indeterminate, one way to do it is to write it as $\lim\limits_{n\to\infty} n^2(1-1/n)$. Since $n^2$ ...
462 views

I'm a math grad student, and next semester I start TAing a calculus class for the first time. We all know about the standard recitations: instructor gives short lecture on some more difficult topic ...
2k views

What are non-math majors supposed to get out of an undergraduate calculus class?

When I teach a course for math majors (an analysis course out of Rudin, say), I have a more or less clear idea of what the students should take away from the course, having been in their shoes some 15 ...
228 views

Examples where roots are necessary for the solution

I currently write an article where I want to introduce roots. Thus I need to motivate them. Here I said, they can be used to find solutions of equations like $x^n=a$. Now I want to make some examples, ...
257 views

Counterexamples to "stable digit" theory of error estimates

When covering issues related to error estimates in a calculus course, students find the technique of making estimates (definition of limit, Newton's method, numerical integration, remainder formula ...
777 views

The purpose of mathematics in a liberal education when it is not a prerequisite to other subjects

Suppose a calculus classroom is full of students majoring in Classical Greek or music or literature or sociology or pre-medical studies or any of many subjects that do not require the course as a ...
475 views

Good lessons/activities for one-day subs

In my school district, and I'm sure most others, every teacher needs to have a set of "emergency lesson plans", in case they are sick or need to be out for a day, so that the substitute can have ...
249 views

621 views

The 'epsilon-delta' method for teaching limits

Weierstrass' method for handling limits with the epsilon and delta symbols is very useful for rigorous analysis of math but it is terrible in terms of any intuitive approach to limits. There are are ...
2k views

How can students self-check derivatives?

It is a good thing for students to self-check their work. The results of some calculations can be checked easily. For example, the solutions to an equation can be substituted back into the original ...
589 views

What are the major obstacles to crowdsourcing a competitive, free calculus text?

It is well known that Allen Hatcher has created a free textbook for algebraic topology that is high enough quality to be used in a large number of graduate courses in the united states, saving ...
759 views

One of the best algebra-teaching games I've seen is the "Four 4's" game, where students have to take 4 fours and construct every number from 1-100 using only those fours and algebraic operations: 44/...
5k views

What basic algebra skills and techniques are most important for calculus students to know?

In my experience, algebra is one of the biggest stumbling blocks to calculus students. For instance, sign errors are common, and exponent laws (and log laws!) cause a lot of headaches. Many courses ...
Is there any good reason that in educational materials, I consistently see the formula for calculating geometric series in canonical form as: $$\sum_{k=0}^{n-1} ar^k = a \frac{1-r^n}{1-r}$$ While an ...