All posts in category Calculus

Convexity and Orthogonality at Saddle Points

Today, the Texas Section of the Mathematical Association of America is holding its annual conference. Like many other professional conferences these days, this conference will be held virtually, and so my contribution to the conference is saved on YouTube and is available to the public.

Here’s the abstract of my talk: “At a saddle point (like the middle of a Pringles potato chip), the directions of maximum upward concavity and maximum downward concavity are perpendicular. The usual proof requires a fair amount of linear algebra: eigenvectors of different eigenvalues of a real symmetric matrix, like the Hessian, must be orthogonal. For this reason, the orthogonality of these two directions is not often stated in calculus textbooks, let alone proven, when the Second Partial Derivative Test for identifying local extrema and saddle points is discussed. In this talk, we present an elementary proof of the orthogonality of these two directions that requires only ideas from Calculus III and trigonometry. Not surprisingly, this proof can be connected to the usual proof from linear algebra.”

If you have 12 minutes to spare, here’s the talk.

Leave a comment

by John Quintanilla on April 9, 2021 • Permalink

Posted in Calculus, Linear Algebra

Tagged derivative

Posted by John Quintanilla on April 9, 2021

https://meangreenmath.com/2021/04/09/convexity-and-orthogonality-at-saddle-points/

A Clean Calculus Joke

The change of position over time is velocity.

The change of velocity over time is acceleration.

The change of acceleration over time is jerk.

And the change of jerk over time is an election.d

Leave a comment

by John Quintanilla on July 24, 2020 • Permalink

Posted in Calculus, Humor, Uncategorized

Tagged derivative

Posted by John Quintanilla on July 24, 2020

https://meangreenmath.com/2020/07/24/a-clean-calculus-joke/

Differentiation and Integration

As I tell my calculus students, differentiation is a science. There are rules to follow, but if you follow them carefully, you can compute the derivative of anything. This leads to one of my favorite classroom activities. However, integration is as much art as science; for example, see my series on different techniques for computing

$\displaystyle \int_0^{2\pi} \frac{dx}{\cos^2 x + 2 a \sin x \cos x + (a^2 + b^2) \sin^2 x}$

The contrast between differentiation and integration was more vividly illustrated in a recent xkcd webcomic:

Source: https://xkcd.com/2117/

Leave a comment

by John Quintanilla on June 22, 2020 • Permalink

Posted in Calculus

Tagged xkcd

Posted by John Quintanilla on June 22, 2020

https://meangreenmath.com/2020/06/22/differentiation-and-integration/

My Favorite One-Liners: Part 117

I absolutely love this joke. The integral looks diabolical but can be computed mentally.

For what it’s worth, while it was able to produce an answer to as many decimal places as needed, even Wolfram Alpha was unable to exactly compute this integral. Feel free to click the link if you’d like the (highly suggestive) answer.

Source: https://www.facebook.com/CTYJohnsHopkins/photos/a.323810509981/10151131549924982/?type=3&theater

Leave a comment

by John Quintanilla on February 17, 2020 • Permalink

Posted in Calculus, Humor

Tagged integral

Posted by John Quintanilla on February 17, 2020

https://meangreenmath.com/2020/02/17/free-wifi-joke/

My Favorite One-Liners: Part 115

I credit Math With Bad Drawings for this new weapon in my arsenal of awful mathematical puns.

Source: https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549/2823158341034307/?type=3&theater

Leave a comment

by John Quintanilla on February 10, 2020 • Permalink

Posted in Calculus, Humor

Posted by John Quintanilla on February 10, 2020

https://meangreenmath.com/2020/02/10/my-favorite-one-liners-part-115/

Adding by a Form of 0 (Part 2)

Often intuitive appeals for the proof of the Product Rule rely on pictures like the following:

The above picture comes from https://mrchasemath.com/2017/04/02/the-product-rule/, which notes the intuitive appeal of the argument but also its lack of rigor.

My preferred technique is to use the above rectangle picture but make it more rigorous. Assuming that the functions $f$ and $g$ are increasing, the difference $f(x+h) g(x+h) - f(x) g(x)$ is exactly equal to the sum of the green and blue areas in the figure below.

In other words,

$f(x+h) g(x+h) - f(x) g(x) = f(x+h) [g(x+h) - g(x)] + [f(x+h) - f(x)] g(x)$ ,

$f(x+h) g(x+h) - f(x+h) g(x) + f(x+h) g(x) - f(x) g(x)$ .

This gives a geometrical way of explaining this otherwise counterintuitive step for students not used to adding by a form of 0. I make a point of noting that we took one term, $f(x+h)$ , from the first product $f(x+h) g(x+h)$ , while the second term, $g(x)$ , came from the second product $f(x) g(x)$ . From this, the usual proof of the Product Rule follows:

$[(fg)(x)]' = \displaystyle \lim_{h \to 0} \frac{f(x+h) g(x+h) - f(x) g(x)}{h}$

$\displaystyle = \lim_{h \to 0} \frac{f(x+h) [g(x+h) - g(x)]}{h} + \lim_{h\ to 0} \frac{[f(x+h) - f(x)] g(x)}{h}$

$\displaystyle = \lim_{h \to 0} f(x+h) \frac{g(x+h) - g(x)}{h} + \lim_{h\ to 0} \frac{f(x+h) - f(x) }{h} g(x)$

$= f(x)g'(x) + f'(x) g(x)$

For what it’s worth, a Google Images search for proofs of the Product Rule yielded plenty of pictures like the one at the top of this post but did not yield any pictures remotely similar to the green and blue rectangles above. This suggests to me that the above approach of motivating this critical step of this derivation might not be commonly known.

Once students have been introduced to the idea of adding by a form of 0, my experience is that the proof of the Quotient Rule is much more palatable. I’m unaware of a geometric proof that I would be willing to try with students (a description of the best attempt I’ve seen can be found here), and so adding by a form of 0 becomes unavoidable. The proof begins

$\left[\left( \displaystyle \frac{f}{g} \right)(x) \right]' = \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h)}{ g(x+h)} - \frac{f(x)}{ g(x)}}{h}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x) g(x+h)}{ g(x) g(x+h)}}{h}$

$= \displaystyle \lim_{h \to 0} \frac{f(x+h) g(x) - f(x) g(x+h)}{ h g(x) g(x+h)}$ .

At this point, I ask my students what we should add and subtract this time to complete the derivation. Given the previous experience with the Product Rule, students are usually quick to chose one factor from the first term and another factor from the second term, usually picking $f(x) g(x)$ . In fact, they usually find this step easier than the analogous step in the Product Rule because this expression is more palatable than the slightly more complicated $f(x+h) g(x)$ . From here, the rest of the proof follows:

$[(fg)(x)]' = \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x) g(x+h)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x)g(x) + f(x)g(x) - f(x) g(x+h)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x)g(x)}{h} + \frac{f(x)g(x) - f(x) g(x+h)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x)g(x)}{h} - \frac{f(x) g(x+h) - f(x)g(x)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{[f(x+h) - f(x)] g(x)}{h} - \frac{f(x) [g(x+h) - g(x)]}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) - f(x) }{h} g(x) - f(x) \frac{ g(x+h) - g(x)}{h }}{g(x) g(x+h)}$

$= \displaystyle \frac{ f'(x) g(x) - f(x) g'(x)}{g(x)^2}$

P.S.

The website https://mrchasemath.com/2017/04/02/the-product-rule/ also suggests an interesting pedagogical idea: before giving the formal proof of the Product Rule, use a particular function and the limit definition of a derivative so that students can intuitively guess the form of the rule. For example, if $g(x) = x^2$ :

See also this previous post for another way of deriving the Product Rule without adding or subtracting the same quantity.

3 Comments

by John Quintanilla on January 24, 2020 • Permalink

Posted in Calculus, Theorems

Tagged derivative

Posted by John Quintanilla on January 24, 2020

https://meangreenmath.com/2020/01/24/adding-by-a-form-of-0-part-2/

Adding by a Form of 0 (Part 1)

Adding by a form of 0, or adding and subtracting the same quantity, is a common technique in mathematical proofs. For example, this technique is used in the second step of the standard proof of the Product Rule in calculus:

$[(fg)(x)]' = \displaystyle \lim_{h \to 0} \frac{f(x+h) g(x+h) - f(x) g(x)}{h}$

$\displaystyle = \lim_{h \to 0} \frac{f(x+h) g(x+h) - f(x+h) g(x) + f(x+h) g(x) - f(x) g(x)}{h}$

$\displaystyle = \lim_{h \to 0} \left[ \frac{f(x+h) g(x+h) - f(x+h) g(x)}{h} + \frac{f(x+h) g(x) - f(x) g(x)}{h} \right]$

$\displaystyle = \lim_{h \to 0} \frac{f(x+h) g(x+h) - f(x+h) g(x)}{h} + \lim_{h\ to 0} \frac{f(x+h) g(x) - f(x) g(x)}{h}$

$\displaystyle = \lim_{h \to 0} \frac{f(x+h) [g(x+h) - g(x)]}{h} + \lim_{h\ to 0} \frac{[f(x+h) - f(x)] g(x)}{h}$

$\displaystyle = \lim_{h \to 0} f(x+h) \frac{g(x+h) - g(x)}{h} + \lim_{h\ to 0} \frac{f(x+h) - f(x) }{h} g(x)$

$= f(x)g'(x) + f'(x) g(x)$

Or the proof of the Quotient Rule:

$\left[\left( \displaystyle \frac{f}{g} \right)(x) \right]' = \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h)}{ g(x+h)} - \frac{f(x)}{ g(x)}}{h}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x) g(x+h)}{ g(x) g(x+h)}}{h}$

$= \displaystyle \lim_{h \to 0} \frac{f(x+h) g(x) - f(x) g(x+h)}{ h g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x) g(x+h)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x)g(x) + f(x)g(x) - f(x) g(x+h)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x)g(x)}{h} + \frac{f(x)g(x) - f(x) g(x+h)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) g(x) - f(x)g(x)}{h} - \frac{f(x) g(x+h) - f(x)g(x)}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{[f(x+h) - f(x)] g(x)}{h} - \frac{f(x) [g(x+h) - g(x)]}{h }}{g(x) g(x+h)}$

$= \displaystyle \lim_{h \to 0} \frac{ \displaystyle \frac{f(x+h) - f(x) }{h} g(x) - f(x) \frac{ g(x+h) - g(x)}{h }}{g(x) g(x+h)}$

$= \displaystyle \frac{ f'(x) g(x) - f(x) g'(x)}{g(x)^2}$

This is a technique that we expect math majors to add to their repertoire of techniques as they progress through the curriculum. I forget the exact proof, but I remember that, when I was a student in honors calculus, we had some theorem that required an argument of the form

$|x - y| = |x - A + A - B + B - C + C - D + D - E + E - F + F - y|$

$\le |x - A| + |A - B| + |B - C| + |C - D| + |D - E| + |E - F| + |F - y|$

$\le \displaystyle \frac{\epsilon}{7} + \frac{\epsilon}{7} +\frac{\epsilon}{7} +\frac{\epsilon}{7} +\frac{\epsilon}{7} +\frac{\epsilon}{7} +\frac{\epsilon}{7}$

$= \epsilon$

But while this is a technique that expect students to master, there’s no doubt that this looks utterly foreign to a student first encountering this technique. After all, in high school algebra, students would simplify something like $x - A + A - B + B - C + C - D + D - E + E - F + F - y$ into $x-y$ . If they were to convert $x-y$ into something more complicated like $x - A + A - B + B - C + C - D + D - E + E - F + F - y$ , they would most definitely get points taken off.

In this brief series, I’d like to give some thoughts on getting students comfortable with this technique.

1 Comment

by John Quintanilla on January 20, 2020 • Permalink

Posted in Calculus, Theorems

Tagged derivative

Posted by John Quintanilla on January 20, 2020

https://meangreenmath.com/2020/01/20/adding-by-a-form-of-0-part-1/

C is for constant of integration, and that’s good enough for me

Source: https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549/2248966371786843/?type=3&theater

Leave a comment

by John Quintanilla on March 25, 2019 • Permalink

Posted in Calculus, Humor

Posted by John Quintanilla on March 25, 2019

https://meangreenmath.com/2019/03/25/c-is-for-constant-of-integration-and-thats-good-enough-for-me/

Derivative of 1/x

All Posts

Source: https://www.facebook.com/MathematicalMemesLogarithmicallyScaled/photos/a.1605246506167805/2700740763285035/?type=3&theater

Leave a comment

by John Quintanilla on March 22, 2019 • Permalink

Posted in Calculus, Humor

Tagged derivative

Posted by John Quintanilla on March 22, 2019

https://meangreenmath.com/2019/03/22/derivative-of-1-x/

Richard Feynman’s Integral Trick

“I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. [It] showed how to differentiate parameters under the integral sign — it’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. [If] guys at MIT or Princeton had trouble doing a certain integral, [then] I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me.” (Surely you’re Joking, Mr. Feynman!)

I read Surely You’re Joking, Mr. Feynman! dozens of times when I was a teenager, and I was always curious about exactly what this integration technique actually was. So I enjoyed reading this article about the Leibniz Integration Rule: https://medium.com/dialogue-and-discourse/richard-feynmans-integral-trick-e7afae85e25c

Leave a comment

by John Quintanilla on March 8, 2019 • Permalink

Posted in Calculus, News Clips

Tagged Feynman

Posted by John Quintanilla on March 8, 2019

https://meangreenmath.com/2019/03/08/richard-feynmans-integral-trick/

Search for:
Follow Blog via Email

Enter your email address to follow this blog and receive notifications of new posts by email.

Email Address:
Top Posts & Pages
Archives
- April 2021
- March 2021
- February 2021
- January 2021
- December 2020
- November 2020
- October 2020
- September 2020
- August 2020
- July 2020
- June 2020
- May 2020
- April 2020
- March 2020
- February 2020
- January 2020
- December 2019
- November 2019
- October 2019
- September 2019
- August 2019
- July 2019
- June 2019
- May 2019
- April 2019
- March 2019
- February 2019
- January 2019
- December 2018
- November 2018
- October 2018
- September 2018
- August 2018
- July 2018
- June 2018
- May 2018
- April 2018
- March 2018
- February 2018
- January 2018
- December 2017
- November 2017
- October 2017
- September 2017
- August 2017
- July 2017
- June 2017
- May 2017
- April 2017
- March 2017
- February 2017
- January 2017
- December 2016
- November 2016
- October 2016
- September 2016
- August 2016
- July 2016
- June 2016
- May 2016
- April 2016
- March 2016
- February 2016
- January 2016
- December 2015
- November 2015
- October 2015
- September 2015
- August 2015
- July 2015
- June 2015
- May 2015
- April 2015
- March 2015
- February 2015
- January 2015
- December 2014
- November 2014
- October 2014
- September 2014
- August 2014
- July 2014
- June 2014
- May 2014
- April 2014
- March 2014
- February 2014
- January 2014
- December 2013
- November 2013
- October 2013
- September 2013
- August 2013
- July 2013
- June 2013
Categories
- Algebra I (209)
- Algebra II (303)
- Bases (44)
- Calculus (235)
- Chemistry (11)
- Computer science (10)
- Discrete mathematics (272)
- Elementary (152)
- Engagement (808)
- Geometry (242)
- Guest presenter (479)
- History (54)
- Humor (276)
- Linear Algebra (14)
- News Clips (130)
- Physics (42)
- Popular Culture (271)
- Pre-Algebra (180)
- Precalculus (523)
- Preparing for college (12)
- Probability (89)
- Statistics (87)
- Tales from the Classroom (228)
- Technology (80)
- Theorems (165)
- Uncategorized (64)
Tags
angle arccosine arcsine arctangent area circle combinatorics Common Core complex numbers compound interest conceptual barrier confidence interval convex polygons cosine decimal De Moivre's Theorem derivative differential equation distributive law divisibility division algorithm e exponent exponential factoring fraction function geometric series graph high-stakes testing hypothesis test index to a series of posts induction inequalities integral intercepts inverse function law of cosines limit linear equation logarithm logic MAA magic trick matrices multiplication parallel Pascal's triangle perpendicular pi polar coordinates polynomial primes PRIMUS proof Pythagorean theorem Pythagorean trig identities quadratic formula residue sequence series sine slope sports square root system of equations tangent Taylor series textbook problems triangle trigonometry unsolved problems vector volume xkcd
Meta

All posts in category Calculus

Follow Blog via Email

Top Posts & Pages

Archives

Categories

Tags

Meta