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Questions tagged [calculus]

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

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8 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 ...
7
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3answers
840 views

Is is 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 ...
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3answers
234 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 ...
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3answers
211 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
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4answers
131 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? ...
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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
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2answers
126 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
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7answers
484 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 ...
0
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2answers
349 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 ...
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136 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 ...
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5answers
315 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) \;?...
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3answers
345 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 ...
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0answers
127 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
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2answers
244 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 ...
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0answers
47 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
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1answer
71 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 ...
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2answers
294 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
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4answers
192 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
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4answers
324 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
84 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
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1answer
94 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
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3answers
303 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 ...
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1answer
139 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 $$f^{(n)}(x)=y^{(n)}$$
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2answers
135 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) ...
29
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4answers
600 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, ...
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8answers
300 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 ...
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3answers
318 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
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2answers
166 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 ...
3
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0answers
90 views

How to explain concepts of limit and continuity to non-mathematical students

How to explain fundamental concepts of limits and continuity to a non-mathematical background student? I am a PhD student in Mathematics working in Differential Geometry. As a part of my teaching ...
7
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2answers
201 views

How to teach Leibniz and Newton's notation

There has been many posts here and in MSE about different notations of differentiation. See for example this, this and this. However those questions only deal with the common misunderstanding about ...
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7answers
2k views

“Real world” examples of implicit functions

When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences. So I'm looking for (relatively simple) ...
5
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1answer
330 views

How to teach calculus effectively to non-math (and noisy) students?

How can I teach calculus effectively to non-math students (chemical engineering students, for example)? Sorted by priority, the problems I'm experiencing are: Students cannot keep their voice down. ...
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8answers
6k views

Why do no students know to change the limits of integration when doing substitutions?

I've TAed and tutored calculus for years and of the hundreds of students I've interacted with, it is always a shock when I tell them to change the limits of integration when they do substitutions. ...
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3answers
211 views

Could you recommend a book for studying calculus 1?

I would like to practice a large quantity of exercises from limit calculation, derivatives, sequences and series and finaly integrals.
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5answers
436 views

teach that $\frac10$ not defined properly

there're some students, who belive that $$\frac10 = \infty $$ I need to teach them that this is not true and $\frac10 $ is undefined, mathematically and give a good picture (for there minds) what ...
26
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16answers
3k views

Grading a limit problem

In an exam we have Question (5points) find $\lim_{x\to\infty}(x-\sqrt{x})$. A student answered: $\lim_{x\to\infty}(x-\sqrt{x}) =\lim \sqrt{x}(\sqrt{x}-1)=\infty \cdot\infty=\infty$. My question is:...
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0answers
58 views

Treating infinity as numbers in exams, [duplicate]

In an exam we have Question (5points) find $\lim_{x\to\infty}(x-\sqrt{x})$. A student answered: $\lim_{x\to\infty}(x-\sqrt{x}) =\lim \sqrt{x}(\sqrt{x}-1)=\infty \cdot\infty=\infty$. My question is:...
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1answer
135 views

What textbook is this chapter from? [closed]

I was looking for a good explanation text on single-variable calculus, especially limits, introduction and examples. I found this pdf and I really like it and would like to read the whole book. I was ...
6
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3answers
168 views

What to teach in a class of 45 minutes about the mean value theorem

I would like some advises which contents to teach in a 45 minutes class to first year undergraduates about the mean value theorem. I'm thinking to do the following: Motivation The theorem and the ...
6
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4answers
210 views

Why do we teach estimation in Statistics and Mathematics?

Please forgive me if this is naive. I am only an aspiring educator after all. Why do we still teach estimation when there are easily accessible/teachable exact techniques? Why do statisticians teach $...
3
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1answer
109 views

Deriving Jerk Equations without using Calculus

I am thinking about the links between SUVAT equations (constant acceleration), and equations for motion when higher-order measurements are constant (for example, when jerk is constant, or snap is ...
7
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2answers
155 views

Simple initial value problems - pros and cons of different methods

Consider the problem: Find $f(x)$ if $f’(x)=4x$ and $f(3)=12$ I have always done this, and taught it, as a two-step problem: First, find the general anti-derivative, $f(x)=2x^2+C$, and then plug ...
6
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6answers
313 views

Ideas for a 2 weeks project focused in polynomial functions

Right now I’m teaching precalculus in high school and I want to propose a project to my students about polynomial functions. They already know enough about quadratic functions and we study variation ...
6
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2answers
126 views

Alternative ways of thinking about the one-variable Riemann integral for elementary calculus,

I think I've done a decent job with teaching my students limits and derivatives so far in elementary calculus -- they were particularly intrigued with how easy and how accurate a first-order, linear ...
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3answers
416 views

When analytic form of derivatives is preferred over numerical form?

Is there a specific example when the analytic form of a derivative $\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$ is preferred to the numerical form $\frac{f(x+h)-f(x)}{h}$, $h \ll 1$? Are there cases when the ...
10
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4answers
590 views

Is Calculus Necessary?

That title is a quote from Fred Roberts: Fred Roberts. "Is Calculus Necessary?" Proceedings of the Fourth International Congress on Mathematical Education. 1980. p.52ff. "Calculus is not ...
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5answers
325 views

Activities for biology undergraduates taking integral calculus

After searching for applications of calculus for biology students, I've found that many of the results are all either contrived exercises, or are way over the heads of students that are seeing ...
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4answers
372 views

Surrounding a subject and strangling it to death versus concentrating on the main point

Standard calculus textbooks begin by introducing limits, including limits of a fraction as the numerator and denominator approach $0,$ limits of a fraction as the numerator and denominator approach $\...
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3answers
287 views

What is the ULTIMATE Calculus syllabus

After such amazing answers I got here for a related question (link at the end if someone still wants to share with me their views)... Here is the concept: If you were to create the ULTIMATE Calculus ...
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0answers
77 views

what is the standard subdivision or classification of calculus related rates problems?

I am working on a project where I have to group/classify calculus problems. Now with most the calculus topics, it's usually obvious how it's divided in various textbooks, but when it comes to related ...