Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
0
votes
1answer
73 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 ...
6
votes
0answers
105 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 ...
-3
votes
0answers
74 views
Find a bounded function $f$ such that $sin(x)<f'(x)<sin(x)+1$ [closed]
This question came up in my diffyqs discussion today. I was curious as to what you fine people had to say.
My intuition says it can't happen because the derivative is mostly positive... but my ...
9
votes
5answers
231 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) \;?...
10
votes
3answers
327 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
votes
0answers
121 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
229 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
70 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
216 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
184 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, ...
5
votes
3answers
238 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).
$$
The approximation is good when $x$ is close to $x_0$. This approximation is very useful because the ...
-2
votes
2answers
78 views
What are the uses of calculus in every day life? [closed]
Can anybody say me what is differentiation and integration and what is the use of these ?
5
votes
1answer
89 views
Tables of primitives with indication of solution method
I am looking for an extensive source (often called "table of integrals") listing primitives of various classes of functions including the "elementary" ones (rational functions, functions involving ...
5
votes
3answers
299 views
Justifying the multi-variable chain rule to students
Suppose that $f(x,y,z) = x + 2xy^2 - yz$, and that $\gamma(u,v) = \langle uv, u\sin(v), u\cos(v)\rangle$. Use the chain rule to calculate $\partial(f \circ \gamma)/\partial u$.
This is an exercise ...
2
votes
1answer
137 views
Can $y^{(n)}$ be used as a way of representing higher order derivatives?
I have never seen this notation, but I think that it follows in a similar vein for function notation. So if $y=f(x)$, then $y''=f''(x)$.
Then by that, can we say that
$$f^{(n)}(x)=y^{(n)}$$
3
votes
2answers
134 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) ...
28
votes
4answers
557 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
294 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
308 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
165 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 ...
2
votes
0answers
88 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
197 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
287 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
votes
3answers
204 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.
5
votes
3answers
296 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
133 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
167 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
205 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
103 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
152 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
281 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
124 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
367 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
votes
4answers
580 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
280 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 ...
9
votes
4answers
356 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
276 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
74 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 ...
7
votes
0answers
115 views
How can I deal with the time pressure of teaching a short course?
I am an undergraduate applied math student. In about a month, I will be teaching two nine-hour math courses (one precalculus, one calculus) to a small group of motivated high school students. My broad ...
8
votes
4answers
251 views
What's the best way to explain multivariable limit problems to students who are not familiar with $\epsilon-\delta$ proofs?
For example,
$\displaystyle \lim_{(x,y,z)\rightarrow(0,0,0)} \frac{x^2y^2z^2}{x^2+y^2+z^2}$
This question is from 8th edition of Stewart Calculus textbook. My fellow graduate student TAs and ...
6
votes
0answers
123 views
Which calculus textbook is aligned the most with the CollegeBoard course description?
The CollegeBoard website lists many AP calculus BC references.
But it also mentions that "The materials on this List range in alignment from 59% to 100%."
So, which of them is aligned the most with ...
16
votes
6answers
529 views
Is it a good idea to have one or two or three classes on basic logic before teaching $\varepsilon$-$\delta$ in Calculus?
I am teaching Calculus I and will be teaching it again. To me, the $\varepsilon$-$\delta$ definition of limit is one of the key ideas of Calculus; learning calculus without learning $\varepsilon$-$\...
5
votes
3answers
121 views
Notebook software for exploring integral approximation with finite sums?
Calc 2 (integration) courses often begin by introducing the idea of approximating the area under curves by rectangles, drawing pictures like this one of $y=\sqrt{x}$
Here we can approximate the area ...
3
votes
1answer
165 views
calculus without analytic geometry
How much of introductory calculus can be learned without using analytic geometry or for that matter any algebraic notations but simple euclidean geometry? Are there any resources(new ones not the old ...
17
votes
2answers
463 views
Tutoring a recalcitrant/awkward/exasperating student---special needs?
As part of my duties at a GTA, I spend several hours per week in our department's drop-in tutoring center. The center is open to all students enrolled in 100- and 200-level math courses, with the ...
