Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
1
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0answers
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
votes
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 ...
8
votes
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 ...
10
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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? ...
25
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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
votes
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 ...
7
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0answers
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 ...
11
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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) \;?...
12
votes
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 ...
2
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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
votes
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 ...
1
vote
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
vote
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 ...
1
vote
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
votes
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
votes
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).
$$
The approximation is good when $x$ is close to $x_0$. This approximation is very useful because the ...
-2
votes
2answers
84 views
What are the uses of calculus in every day life? [closed]
Can anybody say me what is differentiation and integration and what is the use of these ?
5
votes
1answer
94 views
Tables of primitives with indication of solution method
I am looking for an extensive source (often called "table of integrals") listing primitives of various classes of functions including the "elementary" ones (rational functions, functions involving ...
5
votes
3answers
303 views
Justifying the multi-variable chain rule to students
Suppose that $f(x,y,z) = x + 2xy^2 - yz$, and that $\gamma(u,v) = \langle uv, u\sin(v), u\cos(v)\rangle$. Use the chain rule to calculate $\partial(f \circ \gamma)/\partial u$.
This is an exercise ...
2
votes
1answer
139 views
Can $y^{(n)}$ be used as a way of representing higher order derivatives?
I have never seen this notation, but I think that it follows in a similar vein for function notation. So if $y=f(x)$, then $y''=f''(x)$.
Then by that, can we say that
$$f^{(n)}(x)=y^{(n)}$$
3
votes
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
votes
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, ...
5
votes
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 ...
7
votes
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
votes
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
votes
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
votes
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 ...
16
votes
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
votes
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. ...
19
votes
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. ...
4
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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.
9
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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
votes
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:...
1
vote
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:...
-1
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
4
votes
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 ...
9
votes
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 ...
10
votes
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 $\...
6
votes
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 ...
2
votes
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 ...
