Questions tagged [calculus]
For questions applying to calculus courses. Topics include derivatives, integrals, limits, continuity, series, application questions, etc.
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Why are volumes of revolution typically taught in Calculus 2 and not Calculus 3?
Solids of revolution are typically taught in Calculus II for most undergraduate students or in AP Calculus BC for most high school students. However, it seems to me that this topic is far more ...
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Emphasizing the decreasing condition in the Integral Test or in the AST (in Calculus II): is it worth the time?
The title is basically the question. But I guess I should expand a little. For the background, I'm teaching at a large public university in the US. Out student body is mixed in terms of their ...
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Does this explanation of integration and the Fundamental Theorems of Calculus make any sense?
First, Sue Pemberton (Pure Mathematics 1 Coursebook, 2018, Cambridge University Press) introduces
integration as the reverse process of differentiation ...
$\int x^3 \mathrm d x$ is called the ...
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Earliest real-world uses of Calculus and Linear Algebra
I want to illustrate in class that real-world applications of mathematics might take time to come to fruit. In this context, I want to find what the earliest real-world applications of Calculus and ...
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A visualization for the quotient rule
Context: first year didactics of mathematics course for middle school teacher students (in Norway).
I have a reasonable visualization for the product rule of derivatives: Consider a rectangle with ...
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Advice and Remedial Algebra Resources for Students Committed to Calculus
I've got a student in my introductory calculus course. They're failing because they lack algebra skills. They understand the concepts just fine, and can articulate their understanding fine, but get ...
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What theorems from single-variable calculus break down in the multi-variable context?"
In teaching multi-variable calculus, it's insightful to discuss with students not only how certain concepts from single-variable calculus extend to multiple variables but also where these extensions ...
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Motivating a definition of "gap" in a line just barely more advanced than the one used in the typical first-year calculus course
Imagine a course barely getting into some topics more theoretical than what is done in the typical very staid first-year calculus course, and the kind of students for whom such a course is appropriate....
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What is the terminology for "self-referral" integrals in calculus?
In the topic of integration and anti-derivatives in Calculus we come across cases where the attempt at integration by parts brings us back to the original integral, the most basic example being $\int ...
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Define logarithmic function by functional relation [closed]
My son was working the other day with exercises such as: Find all the mappings $f:\mathbb{N}\rightarrow\mathbb{Z}$ verifying
$$\forall m,n \in \mathbb{N}, f(m+n)=f(n)+f(m).$$
As another example: Find ...
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Any examples of calculus sequence that deemphasizes calculation tricks?
I'm considering creating a series of classes that explore deeper ideas in calculus without overemphasizing the various computational tricks used in integration and differentiation.
My vision is ...
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How can we explain intuitively the convergence and divergence of these two series?
It is known that $\displaystyle\sum_1^{\infty} \frac{1}{n^{1.000001}}$ converges while $\displaystyle\sum_{n\text{ is a prime number}}\frac{1}{n}$ diverges. Though we can logically prove these results,...
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Is this a viable Calculus 1 question?
A person is standing next to a hot air balloon. At the same time, the person starts moving away from the balloon at 5 ft/sec and the balloon rises straight into the air at a rate of 12 ft/sec. Is the ...
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Why not think of derivatives as fractions?
Back in high school—back in the 1900s, as my sons say—when our calculus teacher was introducing the chain rule...
$\frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx}$
...he made a special point of ...
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"Real life" examples of limits of functions at finite points
This is more specific than this similar question on math.SE, since I'm not satisfied with the answers there.
Question:
Can you provide an interesting, natural and simple example of some physical/...
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Students can't seem to grasp the intent of tangent lines and getting general trends of derivatives from graphs
Background
I'm informally helping a few students with college Calc 1. This isn't the first time I've aided people with calculus, and so they've sought me for help, though I don't consider myself to ...
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What would be a good pacing for teaching this calculus 2 course?
Next semester I'm going to lecture calculus 2 in an institution I just joined. However, when I had calculus 2 back then the syllabus was very different, it mainly covered several variable calculus up ...
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Sourcing and verifying calculus applications
There are many questions on this site about specific (or not-so-specific) applications of calculus to the "real world". However, one issue I've noticed in using textbooks for this purpose ...
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Is there any university or college in any country where failure and dropout rates in Calculus are not so high?
Calculus is a foundational mathematics course that is often seen as a bottleneck for STEM majors. However, it is also a course that is notorious for its high dropout rates. In the United States, for ...
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How to properly define volume for beginner calculus students?
I'm interested in opinions based on experience about how to introduce volume for beginner calculus students. Below I present some observations and specific questions.
In Stewart's book, the volume of ...
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How to explain that integral calculate areas?
I teach internal combustion engines theory in a technical school. I have an elementary knowledge of calculus and my students lack even this.
I want to intuitively explain them what is the pdV integral,...
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Calculus at competitive level or Olympiad level
What are the topics of Calculus which can be useful for competitive level and the students are not exposed to it ?
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Topics covered in Calculus I and II (university level) that aren't covered in the AP Curriculum
I teach AP Calculus BC at my high school and we have AP Calculus AB as a pre-req for taking BC. So most of my students are coming in with a strong calculus foundation, and I can spend less time on the ...
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Why don’t we teach a topological view of continuity instead of epsilon-delta?
I would like a critique of this approach to teaching continuity to calculus 1 students.
Show them that for an increasing function on (a,b) we have that (a,b) is contained in the set of solutions to $...
user22312
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Process of finding limits for multivariable functions
I was tutoring a student today and they asked a question which made me curious.
We were working on the following question together.
After explaining that we must look at the limit along the x axis, I ...
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Is Morris Kline's 'Calculus: An Intuitive and Physical Approach' a Good Book to Learn Calculus From?
Would I have to read a standard textbook in addition — i.e. Stewart, etc. — or would Kline's Calculus: an Intuitive and Physical Approach be sufficient? My interest is in applications: dynamical ...
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Interpreting the derivative as instantaneous rate of change in real phenomena
When interpreting the meaning of the derivative in real phenomena, it may seem that the interpretation is in conflict with the definition of the derivative itself. The confusion is caused by the units ...
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How do you describe your experience using OER textbooks for calculus?
If you have used commercial as well as OPENSTAX OER textbooks for calculus I would like to know about your experience. How would you compare the two? Were there any disadvantages to using OpenStax?
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Homework in a Flipped Classroom
I'm in the middle of teaching first-semester Calculus where, for the first time, I'm trying to implement a flipped classroom. (Background: Small university in U.S.; Calc 1 for STEM majors, 50 minute ...
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Is there ADA-compliance certification for mathematics text books?
What factors are there to consider when adopting a text as far as ADA (Americans with Disabilities Act) is concerned? Is there a certification? What do you look for in the digital version of the text? ...
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Definite integrals with equal limits
As a property of definite integrals, we teach that definite integrals are zero if the lower and upper limits are the same (Wolfram mathworld says this too). Is this valid in general?
In the case of ...
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Is there a canonical name for a polynomial-like expression allowing for negative powers?
When introducing the techniques of differentiation, polynomials come up all the time as great examples to familiarize students with the "power rule" and the linearity of differentiation.
A ...
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Calculus problems arising from real research problems
I am visiting my in-laws for the holidays. My sister in law is a statistician. She asked me to take a stab at a calculus problem which was coming up in her research. The Lambert $W_0$ function is ...
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Is this motivation for the concept of a limit a good one?
tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...
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What do you do in order to drag out lectures?
I posted earlier about how I was surprised that a typical Calculus 1 course that meets 3-4 hours each week for 15 weeks only barely manages to reach the fundamental theorem by the end of the course. ...
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Why is calculus important for pre-service Math basic school teachers?
Why should pre-service math basic school teachers take calculus courses?
Should this course be different from calculus courses offered to engineering?
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Calculus/analysis as basis for basic school topics
I'm looking for examples of facts/results in basic school that are supported by calculus/analysis. Here is an example:
Consider the infinite decimal expansion (a usual topic in basic school): basic ...
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What are some rationales to teach Computer Science students Sequences and Series?
I was asked to teach the following topics to undergraduate Computer Science (CS) students in a discrete math course:
Sequences (definition, convergent sequences, find the limit of a convergent ...
user19945
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What are specific set of tools Partial Differential Equations provide in studying a system? [closed]
I know what are PDEs, but I am looking to identify the major strengths of PDEs. If I have to convince a pool of engineers to use PDEs for solving a problem, let's say stress distribution in a body. ...
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Is it nonsensical to try to 'prove' Euler's 'formula' in real numbers? What is Wikipedia/proofwiki even doing? [closed]
Edit re the close vote: I guess this 1 of those questions whose on-topic-ness depends on the answer. If the answer is no, then well maybe it's off-topic. But if the answer is yes, then I believe it's ...
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Why Massively Multivariate open online calculus class (M2O2C2) on Coursera was discontinued?
I know that MOOCs were generally unsuccessful. However, I felt that M2O2C2 in Coursera was a great (at least my favorite) course and it's a pity it was removed. Does anyone have any info - will it ...
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Skipping a calculus topic (squeeze theorem)
Background - I am tutoring a second year college sophomore for a class titled Single Variable Calculus, and whose curriculum looks to be similar to the AB calculus I tutor in my High School.
We are on ...
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Explaining difference between improper integrals that converge and diverge
How would you explain the difference in the results given by integration of the two functions $y = \frac{1}{1+x}$ and $y = \frac{1}{1+x^2}$ ?
The graphs of these two function look so similar on ...
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What are your experiences with Buck’s Advanced Calculus?
I stumbled across the book when searching for rigorous alternatives to Rudin with some solutions. It’s an “old school” (1965) calculus text but, I think, covers similar material to Rudin in a more ...
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Multivariable Calculus Project Ideas
Next semester, I am going to teach a small section of advanced high school students a class of Multivariable Calculus (it's about 3-4 students that have completed AP Calculus BC). Multivariable ...
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How does the average level of expected mathematical sophistication at high school level increase?
I remember reading an old calculus book (years 1920-1930) and in the preface it was portrayed as revolutionary because it was for high school students. Nowadays, that is not revolutionary, because ...
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Is "Annular Ring" redundant?
I've come across the term annular ring in parentheses following washer in my calculus textbook: "has the shape of a washer (an annular ring)". The definition of the word "annular" ...
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Generating exercises about extrema of $f(x,y)$
(I have asked this question on math.stackexchange.com and according to a suggestion from comments I am re-asking it here.)
I can generate many examples of functions $f(x,y)$
for which finding local ...
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Is it considered a mistake to use different correct notation for writing intervals?
Standard definition of writing interval states that it should be written (a,b) where a<b
Due to this being arbitrary and just a convention that we all use, would it be considered a mistake to write ...
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Student forgets to remove dx after integrating
I am tutoring another US college student in a Calculus 1 class. Initially, she was having trouble with basic concepts, but after much prodding most of the conceptual difficulties seem to have been ...
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