0
votes
0answers
295 views

How to present calculus? Asking for some excellent intuitive referance.

$\text{Dear}$ mathematicians, amateurs, learners, students et al; I learned calculus when I was 13 years old, I was at the time able to evaluate some easy derivatives, integrals, some tricky limits ...
2
votes
1answer
35 views

Revenue Function - Silly Definition

I'm teaching the section 4.7 on optimization in Stewart Calculus. It has a subsection on "Applications to Business and Economics." There the author defines the price function $p(x)$ to be the price ...
1
vote
4answers
181 views

Applications of inflection points

Recently, I was teaching maxima, minima and inflection points to first year engineering students. I motived extrema by giving practical examples of optimization problems, but when a colleague asked me ...
4
votes
0answers
64 views

Transitioning to Higher Level Mathematics

I am just finishing grade 12 pre-calculus at my school and have strong interest in math. The problem is, it seems some important elements of higher level math are not in my schools curriculum that are ...
27
votes
6answers
2k views

Why is there antagonism towards extended real numbers?

In my backstory, I was introduced to the geometric concept of infinity rather young, through reading about the inversive plane. In the course of learning calculus, I'm pretty sure I formed a concept ...
5
votes
2answers
172 views

How to explain lagrange multipliers to a lay audience?

So I will be giving a seminar to a scientifically mature lay audience (think bio/social science undergrad level). I have been told that I should count on less than half the audience to have experience ...
0
votes
0answers
12 views

Request for an excellent Hnote on mean value theorems in calculus I for teaching engineering students

I was wondering if anybody could give me a link to a note of the mean value theorems for science and engineering students. This'll be for the teaching purpose, and I'm looking for a suggestion of ...
1
vote
1answer
29 views

Variational characterization of gradient?

Let $f: \mathbb{R}^n \to \mathbb{R}$ be a differentiable function. One way to define the gradient of $f$ is as the vector whose inner product with any other vector gives the directional derivative in ...
0
votes
0answers
38 views

Line Integral Problem best or easier solved using geometry?

Does anyone have any recommendation on a line integral problem involving vector fields (aka work) such that evaluating the resulting line integral using parameterization would be significantly ...
1
vote
2answers
86 views

Convention verses memory: The quotient rule v product rule for derivatives

I have long wondered why the product rule is taught the way it is. ${ d(UV)=Udv+Vdu}$ Don't get me wrong, I am not a complete NOB when it comes to calc, but the quotient rule states $${d(\frac ...
4
votes
1answer
53 views

Natural discontinuities

As I stare at a cube-shaped building whose side has length $100$ meters, while walking westward parallel to its north wall at a location $100$ meters north of the building, the distance to farthest ...
4
votes
2answers
351 views

Examples of open ended calculus “class project” ideas

I have instructed calculus I an II, each once, at the college level and would like to emphasize that math is not just about memorizing formulas and concepts for a test and that applied math is not a ...
1
vote
2answers
70 views

Why is Cauchy condition for convergence not formulated in a simpler way?

The standard definition of a Cauchy sequence (e.g. it's given in Wikipedia and most textbooks I remember; admittedly those are mostly older ones) is: for every positive real $ε > 0$ there is a ...
11
votes
3answers
140 views

Applications of functions of the form $f(x)^{g(x)}$

Early on in my calculus education, I learned how to take the derivative of $x^x$ by re-writing it in the form $e^{x\ln x}$. More generally, this technique is helpful in finding the derivative of ...
2
votes
0answers
57 views

(Actual) applications of basic differential and integral methods

If this isn't the place, I apologize: At the end of my calculus class, we asked the students (among other things) what some applications of calculus methods are. Disappointingly, many focused on the ...
1
vote
0answers
92 views

Question about limits

I am quite new on SE. I see a lot of question about integrals, series, limits. I am wondering if there is a limit to teachers (or textbooks) imagination in these areas.
1
vote
1answer
187 views

Parabola & Area Proving (Integral)

This is not a homework question. I am a new teacher (just graduated) and a student asked me this question. The points A(3,9) and B(-2,4) lie on the parabola y=x^2. The line y=x+6 joins A and B. The ...
4
votes
3answers
202 views

What is the best way to solve this high school exercise?

Can you share with me how would you best solve this exersise to a high school student? Show that $f(x)=x^2-6x+2$ , $x\in(-\infty,3]$ is $1-1$ and find its inverse.
2
votes
3answers
106 views

Derivative in interesting way

I am supposed to give a 15-20 minutes math lecture, where I am expecting around 20-30 people. The lecture is about derivative. Since this would be my first "class", I would appreciate any suggestions ...
1
vote
2answers
71 views

An Integral With an Odd Function That Isn't Contrived

When ever I teach calculus, single or multivariable, there is always the point in the text when the author covers odd functions and then gives an example of an integral to evaluates to $0$ because the ...
3
votes
2answers
121 views

Trying to teach supremum and infimum.

I'm helping out my former calculus teacher as a volunteer calculus advisor, and I have under my supervision 5 students. They've already had an exam and... well, they failed. I read their exams and I ...
7
votes
4answers
210 views

A Handwaving Proof of a Specific Existence and Uniqueness Theorem

My problem is as follows: Given the second order homogeneous linear differential equation with constant coefficients $$a\frac{d^2y}{dx^2}+b\frac{dy}{dx}+c\,y(x)=0,$$ is there a good heuristic ...
1
vote
1answer
101 views

A way to teach Archimedean property

A student asked me how to understand the Archimedean property, I tried to re-read with him what he has already done in class (well, actually copy from the blackboard in class). However I think I'm not ...
12
votes
7answers
710 views

Should a high school introductory calculus class teach $\varepsilon$-$\delta$ proofs?

It seems to me that most high school students are comfortable with the intuitive notion of a limit ("as $x$ gets arbitrarily close to $c$, $f(x)$ gets arbitrarily close to $L$") and gain little ...
31
votes
12answers
2k views

explaining the derivative of $x^x$

You set the following exercise to your calculus class: Q1. Differentiate $y(x) = x^x$. A student submits the following solution: Let $g(a)=a^x$ and $f(x)=x$. Then $y(x) = g(f(x))$, so by ...
17
votes
6answers
779 views

Why is the definition of “limit” difficult to understand at first?

Tomorrow I teach my students about limits of sequences. I have heard that the definition of limit is often difficult for students to understand, and I want to make it easier. But first I need to ...
5
votes
0answers
43 views

Optimal partition for a riemann integral

I am a statistician tasked with teaching an elementary calculus course. I am about to teach Riemann sums. The breakpoints for the rectangles (the partition) that make up the Riemann sum need not be ...
5
votes
1answer
380 views

The Constant Function Theorem first of all $\,$?

I quote Thomas W.Tucker $\,$ "... By the way, I view the Constant Function Theorem as even more basic than the IFT. It would be nice to use it as our theoretical cornerstone, but I know of no way to ...
2
votes
0answers
278 views

These unknown uniformly differentiable functions

Let $f$ be defined on $[a,b]$ and there uniformly differentiable ($\,$the $\delta$ in the definition of derivative is independent of the point). Given $\epsilon>0$, choose a partition $P \, : \, ...
5
votes
1answer
307 views

How does one visualize a function with a discontinuous second derivative?

Let us assume that all functions are continuous. I was teaching my calculus students the other day. We were talking about what points of non-differentiability look like. Two ways a function can fail ...
16
votes
4answers
1k views

Common student mistakes/misconceptions in a first year calculus course

What are the common mistakes and misconceptions students make in a first year calculus course? More importantly: What can I do to prevent/rectify them? Context: Soon I will be doing ...
8
votes
2answers
356 views

Do students understand infinite series before they're informally introduced?

We introduce infinite sequences and series very thoroughly in calculus classes. We first define infinite sequences, then series, carefully discussing notions of convergence, etc., and discuss all ...
12
votes
7answers
697 views

Defining the derivative without limits

These days, the standard way to present differential calculus is by introducing the Cauchy-Weierstrass definition of the limit. One then defines the derivative as a limit, proves results like the ...
6
votes
1answer
106 views

Integral using multiple methods

I'm teaching a Calculus II class and we are covering integration techniques. We've covered $u$-substitution, integration by parts, trig integrals, trigonometric substitutions, partial fractions and ...
3
votes
2answers
137 views

Undergraduate approach to a problem about convex functions

I am preparing some sheets of exercises that I'll assign to my undergraduate students in biology (sophomore class, or first academic year in italian universities). This is the problem: Exercise. Let ...
1
vote
6answers
616 views

Is the Mean Value Theorem interesting to engineers, scientists, and others?

I am trying to decide whether to include whether to include the Mean Value Theorem in a calculus course I will be teaching. I am sort of leaning away from it, in light of the interesting discussion ...
1
vote
0answers
87 views

References on the equivalence of different definitions of integrability

While writing a chapter of a book about mathematical analysis, I decided to compare some definitions of integrability that are usually taught to sophomore students, in Italy. I briefly collect four ...
2
votes
0answers
198 views

Motivating questions for some topics in undergraduate calculus

Being a grad student I'm going to teach a whole class for the first time the coming summer and I'm looking for some motivational problems which I could use to introduce different topics. In other ...
7
votes
1answer
343 views

Implicit use of the Implicit Function Theorem when finding tangent lines to polar curves.

Recently I found myself having to teach students how to find the slope of a tangent line to a curve in $\mathbb R^2$ given in polar coordinates by the equation $r = f(\theta)$. The students' calculus ...
2
votes
1answer
105 views

weakly locally one-to-one?

Is there any standard name for this concept that is weaker than local one-to-one-ness? In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$. Or, if you like: In ...
3
votes
2answers
124 views

Advice for Calculus Tutoring

I am tutoring a friend in calculus. Right now, she is working on finding relative maxima and minima as well as Rolle's theorem. While she gets how to find relative maxima and minima she does not get ...
5
votes
3answers
321 views

Good lecture optimization problem involving $\ln x$ or $e^x$

I am teaching a Calc 1 of sorts, like a slightly easier version of Calc 1 with no trig. I want a good optimization/practical problem to do in lecture that involves $\ln x$ or $e^x$, to combine review ...
7
votes
3answers
528 views

The Power of Taylor Series

I am teaching a Calculus class and we are finishing up power/Taylor series this week. The last section of the chapter is on applications, but the only ones listed there are approximating non-rational ...
5
votes
2answers
760 views

Simpson's Rule and other Newton-Cotes Formulas

I am curious about the value of Simpson's rule (also called the parabolic rule or the 3-point rule) for approximating integrals. The calculus text I am now teaching from uses this rule any time an ...
10
votes
4answers
1k views

What is an intermediate definition for a tangent to a curve?

Most students come to calculus with an intuitive sense of what a tangent line should be for a curve. It is easy enough to give a definition of a tangent to a circle that is both elementary and ...
11
votes
9answers
2k views

Motivating infinite series

What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course? My students in this course are generally from biochemistry, ...
6
votes
6answers
1k views

Why Doesn't This Series Converge?

I am teaching a Calc II course and came across the following series when finding the interval of convergence for the Taylor series of $f(x)=\sqrt{x}$ centered at $x=1$: $$ \sum_{n=2}^\infty ...
10
votes
4answers
3k views

Is “locally linear” an appropriate description of a differentiable function?

In this answer on meta, Pete L. Clark said: I think the question concerns the idea that a differentiable curve becomes more and more like a straight line segment the closer one zooms in on its ...
11
votes
15answers
7k views

What concepts were most difficult for you to understand in Calculus?

I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in ...
8
votes
7answers
943 views

Can this standard calculus result be explained “intuitively”

Recently I stumbled upon someone who said he wanted to understand why $\arctan x = \int\dfrac{dx}{1+x^2}$ At first I was confused. This is an easy result in any integral calculus course. But then he ...