Questions tagged [calculus]

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

Filter by
Sorted by
Tagged with
9
votes
2answers
384 views

How can we neatly explain chain rule of differentiation

My students are often getting confused while using chain rule for complicated functions. For example $$f(x)=\tan^3\left(\sqrt{x^2+x+1}\right)$$ Most of the students wrote $f'(x)$ wrongly as $$f'(x)=...
15
votes
7answers
3k views

Students strictly follow the steps and notations in sample problems without understanding them

It's just an observation, but I'll be highly appreciated if anyone with experience in teaching (or TA-ing) lower-division calculus can explain this phenomenon in detail. I'm one of a TAs who's ...
7
votes
1answer
235 views

Population of students taking freshman calculus

Is there any data available on the majors of students taking freshman calculus, including information on changes over time? In the past I think engineering students were probably a hefty majority, but ...
14
votes
4answers
3k views

Why do we teach calculus in high school rather than a different math course?

In most high schools (in America), I think it is safe to say that the highest math subject offered is calculus. But why is it calculus rather than number theory or some other branch of mathematics? ...
12
votes
3answers
660 views

Example of function with *all* the features of differential calculus at first-year level

I'm teaching a first-year calculus course, that is mid-way between a first intro to university-level calculus, and intro to real analysis (I'm based in Australia, for reference). We assume the ...
9
votes
3answers
459 views

Defining vertical tangent lines

In looking at the definition of vertical tangent lines in some popular calculus texts, I noticed that there are a few different definitions for this term, including the following: A function $f$ ...
4
votes
2answers
464 views

Demonstrating that integrals of some unbounded functions exist, and others do not

This is my first year teaching calculus. On a recent quiz, I asked my students to give an argument that $\int^0_1(1/x)dx$ does not exist. I was looking for arguments that appealed to Riemann sum ...
4
votes
1answer
297 views

How can I explain $\lim_{x \to \infty} \frac{e^x+e^{-x}}{e^x-e^{-x}}$ using L'Hôpital's Rule?

I have given a problem in limits to my students: $$\lim_{x \to \infty} \frac{e^x+e^{-x}}{e^x-e^{-x}}$$ Most of the students used direct substitution and identified that it is an indeterminate form $...
13
votes
3answers
471 views

What's the most effective way to introduce/motivate the anti-derivative of $\sec x$?

When teaching calculus, I often present the indefinite integral $$ \int \sec x \, dx = \ln | \sec x + \tan x | + C. $$ I am looking for ways to motivate this anti-derivative rather than just tell the ...
6
votes
3answers
7k views

Looking for realistic applications of the average and instantaneous rate of change

I am looking for realistic applications of the average AND instantaneous rate of change, that can serve as an entry point to calculus for students. The main-idea is to show them a (simplified) problem ...
10
votes
8answers
571 views

What topics should be included in a course matching these specifications?

I posted this question on m.s.e., where I upvoted the two answers, both of which said rather little by comparison to what the question asks. Hence this present posting. Say you have a calculus ...
10
votes
2answers
173 views

Could you suggest books, papers or problems that could be used as good "general" motivating examples of calculus application?

I would like to stress the kind of reference I am looking for: In statistics there are lots of motivating (and sometimes unexpected) examples that are interesting for everyone such as Birthday Problem,...
18
votes
6answers
2k views

What good is the phrase "Taylor series"?

I've taught integral calculus a few times, and in every course the students are confused about the distinction between Taylor series and power series. It's something I remember being confused by as a ...
9
votes
2answers
322 views

Good examples of non-convex optimization

I am looking for a function of on an interval with several local optima that appears in some mathematical model and which you can at least imagine that you want to optimize. I am teaching a calculus ...
5
votes
1answer
128 views

interactive web page for differential calculus

I apologize if this is a silly question. I look for free online web page for a differential calculus and theirs rules for higher school. By google shearch, I don't have an intersting results. For ...
21
votes
7answers
2k views

Good examples of functions defined as definite integrals of elementary functions?

I am writing some Calculus content, and I would like a "big list" of useful functions which are defined by definite integrals, but are not elementary functions. Two examples of such functions are $$ ...
31
votes
3answers
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 ...
14
votes
4answers
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 ...
16
votes
7answers
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 ...
22
votes
5answers
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 ...
13
votes
4answers
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 ...
16
votes
3answers
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 ...
22
votes
8answers
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 ...
3
votes
3answers
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. ...
5
votes
5answers
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 ...
9
votes
4answers
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.).
4
votes
1answer
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 ...
18
votes
5answers
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 ...
11
votes
8answers
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 ...
8
votes
2answers
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, ...
22
votes
2answers
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 ...
20
votes
3answers
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 $n^2$ ...
16
votes
2answers
462 views

Nontraditional calculus recitations

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 ...
28
votes
6answers
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 ...
5
votes
2answers
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, ...
13
votes
3answers
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 ...
21
votes
6answers
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 ...
16
votes
3answers
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 ...
8
votes
2answers
249 views

Is there a good notation for "ratio" comparable to the use of $\Delta$ for "difference"?

It is standard to use the symbol $\Delta$ to indicate a change in a quantity between two points on a curve, two rows on a table, and so forth. For linear functions, we write slope = $\Delta y / \...
2
votes
1answer
336 views

Exactly what do I want these Calculus I students to learn? [closed]

Anyone tasked with moving Calculus I from a textbook to students sooner or later asks her/himself that question and I am curious as to what people are thinking of when they do. But to make it more ...
7
votes
5answers
2k views

Constructing and sketching parabolas, conic sections and other curves

Whenever teaching or discussing parabolas, conic sections and other curves with my students, I always feel dissatisfied with the standard "find vertex, pick points, connect the dots" method to draw a ...
6
votes
4answers
312 views

Are convergent and divergent sequences a prerequesite for calculus or are they a part of it?

Basically the textbooks in my country are awful, so I searched on the web for a precalculus book and found this one: http://www.stitz-zeager.com/szprecalculus07042013.pdf However, it does not cover ...
17
votes
5answers
2k views

How to convince students of the integral identity $\int_0^af(x)dx=\int_0^af(a-x)dx$?

A common identity in integration is $\int_0^af(x)dx=\int_0^af(a-x)dx$. The steps to prove it (algebraically, ignoring the geometric method) are as follows. Let $u=a-x$ so $dx=-du$. $\int_0^af(a-x)...
9
votes
2answers
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 ...
16
votes
6answers
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 ...
18
votes
4answers
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 ...
9
votes
1answer
759 views

Open-ended tasks for teaching students about integration techniques

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/...
19
votes
2answers
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 ...
8
votes
5answers
896 views

Geometric Series Formula and Calculus

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 ...
11
votes
6answers
1k views

How can I convince students that Fourier series are useful?

Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ...

1
4
5
6 7 8