Questions tagged [calculus]

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

Filter by
Sorted by
Tagged with
3
votes
1answer
259 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 ...
6
votes
2answers
214 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
946 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
158 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
3k 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 ...
13
votes
4answers
941 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 ...
10
votes
5answers
2k 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
445 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
408 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
94 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 ...
8
votes
1answer
180 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
345 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 ...
7
votes
0answers
157 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 ...
15
votes
6answers
608 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
146 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
303 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 ...
22
votes
4answers
988 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 ...
7
votes
2answers
240 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 ...
8
votes
3answers
340 views

I'm worried that my struggles with calc 2 mean I won't be able to become a professor later

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. ...
8
votes
9answers
7k 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 ...
27
votes
4answers
955 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
254 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
votes
5answers
352 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
votes
1answer
176 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
votes
9answers
1k 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
416 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, ...
16
votes
12answers
961 views

Geometric intuition for $D(e^x) = e^x$

I'm teaching a preparatory course on mathematics at a university. The content is mostly calculus, manipulating expressions and solving equations and inequalities. I show a couple of simple derivations/...
0
votes
4answers
384 views

Double Integral: Area or Volume? [closed]

When we study double integral many Calculus textbooks state that for a region $R$ in the plane $$\iint_R1\ dA= \text{area bounded by }R $$ But double integral actually give the volume of a solid. ...
18
votes
4answers
566 views

"Function" vs "Function of ...": how much does it contribute to students difficulties?

Most textbooks I've seen (and teachers I've met, myself included) are rather careless about the distinction between variables and functions. For example, when we write $y=f(x)$ we all know that $f$ ...
6
votes
2answers
182 views

Calculus I known material, getting students engaged

I'm currently teaching Calculus I problem solving sessions for college freshmen. The problem is that the majority of students have studied the material in high school (what's a function, even/odd, ...
8
votes
3answers
514 views

Teaching Asymptotes

Yes, I've read a number of definitions In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. ...
6
votes
1answer
1k views

Most Commonly-Adopted Calculus Textbooks

I am looking for an article (or other source) that addresses the calculus textbook industry, specifically which calculus textbooks are most commonly adopted. I'm primarily interested in that of the U....
11
votes
9answers
9k views

Why do we study ordinary differential equations?

What is a good answer to the question: Why should one study ordinary differential equations? I would give the answer: ODEs are used in many models to determine how the state of this model is changing ...
2
votes
1answer
126 views

Proving convergence or divergence of series: tips and recommendations

This is a follow up of my question on MSE. Which tips and recommendations would you give students who want to investigate series about convergence or divergence? So far we have collected: It is ...
11
votes
1answer
600 views

A very tricky pseudo-proof of $0=-1$ through series and integrals

Dealing with a recent question I spotted a very nice exercise for Calc-2 students, i.e. to find the mistake in the following lines. Lemma 1. For any $n\in\mathbb{N}$, we have: $$ \int_{0}^{1} x^n\...
15
votes
6answers
894 views

What is a better way to explain these claims about limit are not true in general?

As a TA who led calculus* 1 and 2 discussion section and holds office hour** in the previous year, I heard the following (wrong) arguments several times. $\displaystyle \lim_{x\to \infty} \sqrt{x+1}-\...
5
votes
0answers
193 views

SMSG Calculus Usage?

Did anyone out there have experience with SMSG's Calculus text? Our school system (Amherst, MA) used SMSG texts from my 6th grade class onward. But for HS Calculus (1969) we didn't use the two-part ...
4
votes
2answers
560 views

Why "plug in numbers" when solving inequality?

Let me use this example, Solve $x^3-4x>0$ After factorization, we have $$(x+2)x(x-2)>0$$, in order to have product of several numbers positive, even(0,2,4,...) of them have to be negative ...
7
votes
1answer
258 views

Calculus Text that Uses Sequences to Define Limits

The last time I taught the first semester of calculus, I decided to go the route of teaching them limits of functions via characterization by sequences. I found that many students were able to grasp ...
14
votes
4answers
560 views

Is there research for or against such an approach in teaching calculus?

Copying from Calculus Made Easy by Silvanus Thompson (2nd ed., 1914): CHAPTER I:TO DELIVER YOU FROM THE PRELIMINARY TERRORS The preliminary terror, which chokes off most fifth-form boys from ...
5
votes
3answers
377 views

Telescoping sums to introduce integral and derivative relationship

In order to introduce the fundamental theorem of Calculus to a 18 year-old class I was thinking about starting with simple telescoping sums (finite sums). It would be important to have examples that ...
9
votes
3answers
236 views

Applications of MVT for Integrals, suitable for calculus 1

I'm about to give a first-semester calculus lecture covering the mean value theorem for integrals: If $f$ is continuous on $[a,b]$, then there is some $c\in(a,b)$ such that $(b-a)f(c)=\int_a^b f(x)\,...
3
votes
1answer
781 views

Why is the convergence of infinite series covered in Calculus II?

I'm teaching AP Calculus BC for the first time this year. The AP curriculum is an attempt to cover most of the material in a two-semester university freshman calculus series, and so I am reminded of ...
-1
votes
1answer
111 views

books of mathematics [closed]

a)I am trying to get a book that would give describe all the coordinate systems and their transformations(e.g. cartesian,polar,spherical,homogeneous,curvilinear,generalized,etc). b) And I need to ...
5
votes
1answer
351 views

Functions can be divided into odd and even components - name of theorem?

I'm explaining to a student that all functions can be divided into odd and even (symmetric and anti-symmetric) components. It is easy to prove (basic algebra or Taylor series) but is not referenced in ...
9
votes
1answer
399 views

Alternative limit for e

I have recently worked with some students motivating the development of $e^t$ and $e^{t i}$ as summing change over time, basically informally solving differential equations. My motivation for this is ...
4
votes
4answers
2k views

How are the basic trigonometric functions introduced to students?

The fundamental trigonometric functions $\sin(x)$ and $\cos(x)$ are used throughout the sciences, but I believe students are often introduced to a very limited initial understanding where it is ...
17
votes
5answers
3k views

Source of conceptual, multiple choice calculus questions

I'd like to give my Calculus 1 class periodic multiple choice questions that really test conceptual understanding. Ideally, I'd like these questions to require very little computation. I know that a ...
25
votes
10answers
6k views

Why would you teach Calculus before teaching Real Analysis?

Let's assume our students are actual aspiring mathematicians. Why would we introduce our students to Calculus rather than Real Analysis? After all, "Calculus is a subset of Real Analysis". He will ...
9
votes
2answers
384 views

How can we neatly explain chain rule of differentiation

My students are often getting confused while using chain rule for complicated functions. For example $$f(x)=\tan^3\left(\sqrt{x^2+x+1}\right)$$ Most of the students wrote $f'(x)$ wrongly as $$f'(x)=...

1 2 3
4
5
8