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

1
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2answers
90 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
166 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
223 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
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2answers
72 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
86 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
292 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
133 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
120 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
518 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
288 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
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3answers
299 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
164 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
81 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
193 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
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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
votes
1answer
254 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
191 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
290 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 ...
25
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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:...
1
vote
0answers
55 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
132 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
165 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
198 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
99 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
151 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
254 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
121 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
281 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
535 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
244 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
353 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
273 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
73 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
113 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
239 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
118 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
521 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
119 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
162 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
452 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 ...
6
votes
2answers
189 views

How to catch students from different subjects' interest to math?

I have just started to teach Calculus to freshmans and sophomores who study non-mathematical subjects, e.g., international relations, psychology. They have to take few mathematics classes -including ...
7
votes
3answers
155 views

Younger student worried about upcoming mathematics

I have just turned 18 and am in calculus BC (calc 1 & calc 2). I most certainly grasp and understand the concepts of calc 1 however every once in a while a I seem to struggle with the calc 2 work. ...
5
votes
7answers
1k views

Why are calculators not allowed in post-secondary exams?

Before you downvote this question, I actually want an answer to this. Is the calculator going to give me my derivative? No. Is it going to give me my integral? No. It can sure give me the answer to my ...
26
votes
4answers
615 views

The Undergraduate Responsibility Gradient

We tell undergraduate students that they should study two to three hours for every hour they spend in class. We know that many students don't follow through with this nearly to the degree that they ...
3
votes
1answer
227 views

How should I deal well-known versus the obvious rubric?

I happen to be a student in America taking AP Calculus BC, or Calculus II, and recently, I had the following problem: Determine whether the following integral converges and evaluate it if it does: ...
16
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5answers
275 views

Frequent calculus error: replacing interior part of an expression with its limit

For example $$\lim\limits_{n\to\infty}\left(1+\frac{1}{2n+1}\right)^{n} =\lim\limits_{n\to\infty}{1}^{n}=1\,.$$ Here the student has replaced the sub-part $\frac{1}{2n+1}$ with its limit $0$, but he ...
7
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1answer
154 views

Lipschitz continuity before standard definition of continuity

In Practical Analysis in One Variable, Donald Estep introduces Lipschitz continuity early on, delaying the standard definition of continuity, along with uniform continuity, until the beginning of his ...
16
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9answers
664 views

Evaluating integrals geometrically, without using the fundamental theorem of calculus

I'm designing a lesson for an Introduction to Integral Calculus class, and I want to encourage students to evaluate integrals without just going straight for the antiderivative and using the ...
8
votes
9answers
270 views

Good metaphor to explain the difference between pointwise and uniform convergence

What could be a good "layman" metaphor for illustrating the difference between uniform and pointwise convergence of function series? I am teaching calculus to engineering undergrads; for many of them, ...