Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
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0answers
23 views
Student-friendly / efficient approach to computing Taylor coefficients of infinite binomial series expansions?
I’m working on a section of a course covering Taylor expansions, and have found that, although there is great notation for simplifying the formula for the coefficients of a general infinite binomial ...
2
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2answers
194 views
Ideas for the introduction of the derivative?
I want to introduce to my class to the derivative, but I am still searching for a good, realistic context that isn't too hard to understand, without seeming to be contrived. Do you have an ideas for ...
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1answer
123 views
Practical applications of integration by substitution where integrand is unknown
I posted this question on the Mathematics Stack Exchange a while ago, and got no responses, so I thought I would ask it here. I'm looking for any real-life applications of integration by substitution ...
3
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1answer
192 views
Resources for teaching calc III
I was very unsatisfied with how I taught Calc III a couple years ago, and this summer I have to do it again. Are there any general resources for teaching this course? It seems like there should be, ...
3
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2answers
144 views
Introducing derivative concept and definition
I need to give a short presentation on introducing a class of engineering students to the concept and definition of the derivative. I'm to assume that the students are currently at the appropriate ...
2
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0answers
100 views
Unconstrained/Constrained optimization real life example
I am in charge of some practice lesson for Calculus II.
I have to show how to apply the theory for unconstrained optimization (mainly Hessian analysis) and constrained optimization (Lagrange ...
7
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3answers
142 views
Physical devices for exploring calculus or pre-calculus
I saw this partial derivative machine yesterday, and it got me excited about other physical devices for exploring calculus concepts in a "lab" setting (e.g. make a prediction, collect data, etc.) Do ...
11
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9answers
3k 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 ...
8
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1answer
270 views
Real World use of the Function $(\sin{x})^x$
Today in my calculus class we were going over L'Hopital's Rule and were dealing with limits of the following form
$$h(x)=f(x)^{g(x)}$$
Three examples we considered are as follows:
$(1)\; \...
1
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2answers
92 views
Are questions on overlapping solids of revolutions without prior definitions and instructions fair given that there are divided interpretations?
If words of command are not clear and distinct, if orders are not thoroughly understood, the general is to blame. But if his orders are clear, and the soldiers nevertheless disobey, then it is the ...
3
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1answer
151 views
When teaching someone how to prove a function is uniformly continuous, using epsilon/delta, which example would be among the simplest?
I've taught how to use $\epsilon, \delta$ to prove that a function is continuous at a point, and I'm about to teach how to prove that a function is uniformly continuous over an open interval.
Usually,...
10
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8answers
414 views
How does knowing more about mathematics help one's teaching of lower level course, such as calculus?
A question has been asked about why great mathematicians are not necessarily great teachers. On the other hand, I am wondering if knowing more mathematics actually helps with one's teaching of lower ...
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2answers
100 views
Activities for calc based physics
I was sort of thrown into teaching calculus based physics to a bunch of non-physics majors, who have taken one semester of calculus, and are poor with that material. It is only a 50 minute per week ...
4
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4answers
212 views
Would teaching nonstandard calculus in an introduction calculus course make it easier to learn?
Nonstandard calculus is a reformulation of calculus that is based on infinitesimals instead of epsilon-delta definitions. Of course, people had tried to use infinitesimals in calculus before; in fact, ...
5
votes
3answers
192 views
Calculus workbook suggestions
Context: I am an assistant professor of mathematics at a small institution in the US.
Our department uses Stewart's Essential Calculus for our calculus sequence, but I find that my students and I are ...
2
votes
1answer
64 views
Is there a point at which it makes decidedly more sense to learn about a “linear approximation” to a function, rather than a “tangent”?
I'm tutoring a first-semester calculus student, and we were looking over the slides the teacher has used.
After teaching (or rather, repeating, for those who completed AP high school math) basic ...
1
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6answers
251 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 ...
4
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5answers
195 views
How to prepare for lecturing in a non-fluent foreign language?
I am giving some lectures on a calculus course in Norwegian. My Norwegian (or, rather, Scandinavic) is good enough to do so mostly without resorting to English, but I would, of course, like to improve....
4
votes
2answers
128 views
How to explain linear approximation to an equation to calculus students?
I am, at the moment, teaching calculus to students whose majors are, for example, biology, biochemistry, chemistry and geology. The course book is Claudia Neuhauser's "Calculus for biology and ...
4
votes
4answers
271 views
Recommended list of things calculus students should be required to memorise?
I am seeking a list of topics that students taking calculus should memorise. Some topics from Calculus I might include:
$\varepsilon-\delta$ definition of limit;
Definition of the derivative of a ...
17
votes
6answers
1k views
Is there a place to buy physical models to demonstrate the Calculus shell, disk, and washer methods?
I know a math teacher who is going to teach a calculus class that will include the shell, disk, and washer methods for calculating volumes. My question is, is there some 3D kit she could use to ...
1
vote
1answer
124 views
When self teaching, should I learn set theory before continuing ap calculus?
I am studying ap calculus now, before I move onto differential equations etc., but the thing I am unsure of is, should I learn set theory before continuing on my ap calculus sections?
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3answers
299 views
The royal road to calculus
In the early 1900s Felix Klein lay out his vision for secondary mathematics curriculum. He wanted schools to teach calculus, so that universities would not be burdened by it. And at the core of the ...
12
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3answers
1k views
Is it a bad idea to use an old textbook such as Differential and integral calculus, with examples and applications for calculus course?
I am wondering if it is a bad idea to use an old textbook, such as
Differential and integral calculus, with examples and applications by George A. Osborne. This book was published in 1906 and there ...
11
votes
4answers
379 views
Multiple students writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ — why?
I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than ...
10
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3answers
233 views
Formats for Calculus instruction at different colleges and universities
In the comments under another question, a couple of people expressed interest in how Calculus is taught at the University of Michigan. I'm not convinced a question that narrow is appropriate for this ...
7
votes
4answers
155 views
Making physical 3D models
I was thinking to make classroom illustrations of some 3D mathematical objects, such as graphs of 2 variable functions, minimal surfaces, etc.
My question is, what would be a good way to go about it? ...
26
votes
3answers
4k views
Difference between high school and college calculus courses
I am curious why students who take calculus in high school often do so poorly in college calculus. I am an instructor at an engineering college and I've noticed a decent number of students who have ...
2
votes
2answers
165 views
Line Integral Motivation
Is there a case to be made that the topic of line integrals should only involve vector fields?
My colleagues and our textbook take the position that line integrals should only be taught from a vector ...
7
votes
7answers
547 views
List of realistic extremum problems
I am a student who would like to become a teacher, so I am currently following courses in education. One of the things I learned, is that students like authentic, realistic problems. However, much of ...
1
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2answers
366 views
Integrating derivatives over functions problem
I had a question from a student which I'm unable to answer.
We were practicing the rule $\int \frac{f'(x)}{f(x)} \, dx=\ln(f(x))$.
A student noticed that if applied naively it gives the following ...
7
votes
0answers
148 views
Are there standard questions for testing how an instructor grades calculus?
My institution is now in the process of "standardizing" our calculus classes. One issue we have is the variation among instructors in grading problems. I am interested if there are ways to objectively ...
11
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5answers
332 views
Iconic image to explain the fundamental theorem of calculus?
Is there some single, persuading visualization that can be used to convince students of the intuitive truth of the fundamental
theorem of calculus, in the form
$$
\int_a^b f(t) \, dt = F(b) - F(a) \;?...
12
votes
3answers
368 views
Examples (for beginners) of real functions which are not given by elementary formulae
Question: What are good examples of functions $f: \Bbb R \rightarrow \Bbb R$ (or $f: D \rightarrow \Bbb R$ with $D \subseteq \Bbb R$) which are not just given by "a formula" (or finitely many formulae ...
2
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0answers
246 views
a theorem to simplify continuity in Stewart's calculus: early transcendentals
I'm covering section 2.5 of Stewart (on continuity) and stewarts treatment seems needlessly complicated. It seems like the following theorem would streamline a lot of it:
If $f(x)$ and $g(x)$ are ...
5
votes
2answers
264 views
Is Calculus AB/BC a 'bad course?'
preface: I took AB and not BC but I feel both are similar in nature.
By bad course, I mean that I feel as though Calculus AB/BC sets students up for 'failure,'
for starters, how successful are BC ...
1
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0answers
59 views
“Small” real numbers [duplicate]
At least for me, my intuition for what numbers are large or small comes entirely from positive numbers.
I find it challenging to use the word "small" correctly when talking about negative numbers. ...
1
vote
1answer
79 views
How can I make “complex” graphs that combine multiple functions with a software?
Til today I've been using geogebra to sketch functions for my students quizzes or homework. Sometimes I use the ones that I found searching in google, but this takes a lot of time specially because I ...
1
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2answers
314 views
Is AP Calculus AB really necessary?
High school student here...
This coming school year I'm scheduled to start AP Calculus AB and then my school is looking into taking Multivariable Calculus at a local university. The school's calculus ...
5
votes
4answers
198 views
What is the value in creating distinguishing terminology between the $x$, $y$, and $(x, y)$ values of a possible point of extremum?
I've been out of a math program for about four years now. My wife is starting a CS degree, and finished her first calculus course last semester.
I tutored calculus throughout my entire undergrad, ...
6
votes
4answers
340 views
Should I describe the function $x \mapsto f(x_0) + f'(x_0)(x - x_0)$ as “linear” in a freshman calculus class?
One of the most important ideas of calculus is
$$
f(x) \approx f(x_0) + f'(x_0)(x - x_0).
$$
The approximation is good when $x$ is close to $x_0$. This approximation is very useful because the ...
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2answers
90 views
What are the uses of calculus in every day life? [closed]
Can anybody say me what is differentiation and integration and what is the use of these ?
5
votes
1answer
103 views
Tables of primitives with indication of solution method
I am looking for an extensive source (often called "table of integrals") listing primitives of various classes of functions including the "elementary" ones (rational functions, functions involving ...
5
votes
3answers
316 views
Justifying the multi-variable chain rule to students
Suppose that $f(x,y,z) = x + 2xy^2 - yz$, and that $\gamma(u,v) = \langle uv, u\sin(v), u\cos(v)\rangle$. Use the chain rule to calculate $\partial(f \circ \gamma)/\partial u$.
This is an exercise ...
1
vote
1answer
150 views
Can $y^{(n)}$ be used as a way of representing higher order derivatives?
I have never seen this notation, but I think that it follows in a similar vein for function notation. So if $y=f(x)$, then $y''=f''(x)$.
Then by that, can we say that
$$f^{(n)}(x)=y^{(n)}$$
3
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2answers
152 views
Should Euler's method be taught in calculus 1 courses?
It seems to me that Euler's method for solving ODEs is a topic that nicely illustrates the main strategy of calculus, which is to approximate a nonlinear function locally by a linear (or affine) ...
30
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4answers
726 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, ...
5
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8answers
340 views
Comparison Tests in Calculus
How should I teach Comparison Tests in Calculus II, and why?
Note that I will cover comparison tests in some way, and students will be expected to justify their answers to questions about series ...
6
votes
3answers
349 views
Teaching Calculus Less Formally
I'm wondering if anyone knows of calculus books or other work towards teaching calculus in a less mathematically rigorous way.
I'm thinking mostly of American-style college level calculus courses ...
4
votes
2answers
185 views
What is the ideal teaching style for Calculus exercises only?
A Calculus class for 1st year students may have two subclasses (with two different lecturers) :
The main class (which covers the theories and concept),
and the 'response' class (which provides and ...
