text:
News: IT'S THE 2ND ANNUAL GUATEMALA LIBRARY PROJECT BOOK DRIVE!    LOOKING FOR DONATIONS OF SCIENCE BOOKS THIS YEAR.    Check it out in the "Extending the Hand of Kindness" folder or here: http://www.etiquettehell.com/smf/index.php?topic=139832.msg3372084#msg3372084

• October 20, 2017, 07:31:19 AM

### Author Topic: Your holiday hill to die on.  (Read 882926 times)

0 Members and 2 Guests are viewing this topic.

#### Ambrosia Hino

• Member
• Posts: 940
##### Re: Your holiday hill to die on.
« Reply #555 on: December 10, 2012, 02:29:27 PM »
I admit I didn't handle that part well. But she's also the type that would sit there and tell the server (repeatedly) "oh its so sad that you have to work instead of spend the day with your family" and not understand that her being there is why they're having to work. I jumped the gun there and I realize that. Now just to find a way to set up plans a different way (SIL never ever hosts, so I don't get why she's even pushing for this)

#### Outdoor Girl

• Member
• Posts: 16365
##### Re: Your holiday hill to die on.
« Reply #556 on: December 10, 2012, 02:32:48 PM »
Some people who would be considered non-emergency are needed, like folks at hotels since some of us will need hotel rooms over the holidays, either because there isn't enough room to stay at someone's house or we need our own space for our own sanity.

And a lot of restaurants that are open are owned by people who don't celebrate Christmas.  So I don't see why they should be forced to close on a day when they could make considerable money when it doesn't have spiritual significance for them.

But I agree that theme parks and the like don't need to be open.  I find the NBA games played and televised on Christmas Day to be somewhat bizarre.  The visiting team most likely had to come in the day before and thus, didn't get to spend the day with their families, not to mention all the TV folks - cameramen, producers, announcers, etc. - and all the stadium/arena workers.  I know people who go to a movie on Christmas Day.  It's a tradition in their family.  Again, I just find it strange.

But if the demand is there and people are willing to work, I guess it'll happen.  Just don't expect me to be a part of it.
After cleaning out my Dad's house, I have this advice:  If you haven't used it in a year, throw it out!!!!.
Ontario

#### faithlessone

• Member
• Posts: 2530
##### Re: Your holiday hill to die on.
« Reply #557 on: December 10, 2012, 02:33:55 PM »
I will not drive anywhere on Christmas day, and IMHO don't believe non-emergency workers should have to work.

I seem to be in the minority, at least in this area, where anyone in hospitality will most likely be working on Christmas day. The theme parks, hotels and many restaurants do not close, even for Christmas.

Not to stir the pot or anything, but you do realise that not everyone celebrates Christmas, right? People of other religions, or no religion, as well as those who just don't want to recognise a holiday that might not mean anything to them, or even actively upset them. People who might not have families to spend Christmas with, or who would prefer to spend that day being there for other people (even in such a small way as just manning the local shop!).

It's not fair to expect that the world will shut down for a day, when there are millions of people who consider it to be just another day in December.

• Member
• Posts: 11814
• Not all those who wander are lost
##### Re: Your holiday hill to die on.
« Reply #558 on: December 10, 2012, 02:51:41 PM »
I do realize that not everyone is Christian. I have Pagan friends who have celebrations that occur on December 25th as well.

I made a choice that only affects me. I never said that others should not continue with their plans.
“All that is gold does not glitter, Not all those who wander are lost; The old that is strong does not wither, Deep roots are not reached by the frost."
-J.R.R Tolkien

#### camlan

• Member
• Posts: 9273
##### Re: Your holiday hill to die on.
« Reply #559 on: December 10, 2012, 04:29:38 PM »

The way our mother explained it when we were old enough to know (actually, it was first broken to me in a rather jarring manner, so I've got mixed feelings about the whole "Santa" thing, but this actually helped), there was a man called St. Nicholas, as you said.  The legend says that he dropped bags of gold down the chimneys of the needy.  When the real man passed away a very long time ago, people honored his memory by continuing to give to the needy every Christmas, which eventually graduated to exchanging gifts.  Santa is real in that the spirit of giving can live on in all of us.  So she said.

I don't know how much of that is true, but it's a nice idea.

St. Nicholas was a Turkish saint, back when that region of the world was Christian. There's a pretty good explanation here, if anyone's interested: http://www.turkey-now.org/default.aspx?pgID=438

Now that Turkey is largely Muslim, there is still a tradition, at least in the less conservative, more modernized areas, that Santa Claus brings gifts, but on New Year's Eve instead of Christmas day. And not a huge pile of gifts, but more like a book or a pair of gloves. But you'll see Christmas trees and Santa figures in the major cities in Turkey during December.
Nothing is impossible, the word itself says, “I’m possible!” –Audrey Hepburn

#### VorFemme

• Member
• Posts: 13768
• It's too darned hot! (song from Kiss Me, Kate)
##### Re: Your holiday hill to die on.
« Reply #560 on: December 10, 2012, 05:18:55 PM »
I admit I didn't handle that part well. But she's also the type that would sit there and tell the server (repeatedly) "oh its so sad that you have to work instead of spend the day with your family" and not understand that her being there is why they're having to work. I jumped the gun there and I realize that. Now just to find a way to set up plans a different way (SIL never ever hosts, so I don't get why she's even pushing for this)

I've met your SIL - three things to consider.

Her health is not that great right now.

Would she or her husband be doing the cooking?  Or possibly her mother-in-law, if his family is coming over for a combined holiday......

Look at the bright side - dinner out can be cut off at three hours or less, so you can "take the toddler home for his nap" or "let the dogs out of the garage for their bathroom break". Going to anyone else's house is likely to get you guilt tripped if it isn't four-to-six hours or longer......or would you rather have them over at your house?  Yeah - she's that clueless - but with a narcissistic mother, it's amazing that she has any empathy at all.  She does have some empathy towards the people who are working - even if her LOGIC (as to who is making them work) isn't strong.....neither Vulcans nor Betans on that side of the family......

ETA - if anyone remembers Ambrosia Hino's earlier posts, or mine, this is CrazyMIL's DD.  The family traditions have had to be adjusted as CrazyMIL is being given the cut direct after making death threats to both her SIL and her DIL.  Since she also made death threats to her ex-DH (their father) and tried to hire someone to do the job...the situation with her would seem to warrant the etiquette version of "weapons of mass destruction".
« Last Edit: December 10, 2012, 05:57:02 PM by VorFemme »
Let sleeping dragons be.......morning breath......need I explain?

#### heathert

• Member
• Posts: 1927
##### Re: Your holiday hill to die on.
« Reply #561 on: December 10, 2012, 05:45:11 PM »
Well, I don't know your SIL but I was thinking along the same lines that when she makes those comments to servers it is meant in a "Sorry you drew the short straw" kind of way, although it is rather clueless.

#### AuntieA

• Member
• Posts: 429
• Now a great-Auntie x2!
##### Re: Your holiday hill to die on.
« Reply #562 on: December 10, 2012, 07:24:26 PM »
VorFemme - I would love to meet you + family - you all have such *interesting* dynamics!

* as in the Chinese saying "May you live in interesting times"

Seriously, its been great to see E-Hellions passing down the generations and crossing the sibling ties.
I dream of a world where chickens can cross the road without having their motives questioned.

#### KenveeB

• Member
• Posts: 8863
##### Re: Your holiday hill to die on.
« Reply #563 on: December 10, 2012, 09:05:00 PM »
My first clue should have been that we didn't leave cookies and milk, we left mincemeat tarts and a can of coke.  Which just happened to be my Dad's favourite.

After we learned the truth about Santa, we started leaving out peanut M&Ms and Diet Cokes.

#### mbbored

• Member
• Posts: 4882
##### Re: Your holiday hill to die on.
« Reply #564 on: December 10, 2012, 09:28:57 PM »
My first clue should have been that we didn't leave cookies and milk, we left mincemeat tarts and a can of coke.  Which just happened to be my Dad's favourite.

After we learned the truth about Santa, we started leaving out peanut M&Ms and Diet Cokes.

Heh, we always left Santa wine and cheese since he needed a savory snack after all those cookies.

#### jedikaiti

• Swiss Army Nerd
• Member
• Posts: 2772
• A pie in the hand is worth two in the mail.
##### Re: Your holiday hill to die on.
« Reply #565 on: December 10, 2012, 11:59:41 PM »
My first clue should have been that we didn't leave cookies and milk, we left mincemeat tarts and a can of coke.  Which just happened to be my Dad's favourite.

After we learned the truth about Santa, we started leaving out peanut M&Ms and Diet Cokes.

In my house the cookies were served with some of Dad's good brandy, to help keep Santa warm.
What part of v_e = \sqrt{\frac{2GM}{r}} don't you understand? It's only rocket science!

"The problem with re-examining your brilliant ideas is that more often than not, you discover they are the intellectual equivalent of saying, 'Hold my beer and watch this!'" - Cindy Couture

#### Shakira

• Member
• Posts: 599
##### Re: Your holiday hill to die on.
« Reply #566 on: December 11, 2012, 04:45:23 AM »
My first clue should have been that we didn't leave cookies and milk, we left mincemeat tarts and a can of coke.  Which just happened to be my Dad's favourite.

After we learned the truth about Santa, we started leaving out peanut M&Ms and Diet Cokes.

Heh, we always left Santa wine and cheese since he needed a savory snack after all those cookies.

Yeah!! My kind of Santa!
Sell crazy somewhere else, we're all stocked up here.

#### Ambrosia Hino

• Member
• Posts: 940
##### Re: Your holiday hill to die on.
« Reply #567 on: December 11, 2012, 07:14:18 AM »
I admit I didn't handle that part well. But she's also the type that would sit there and tell the server (repeatedly) "oh its so sad that you have to work instead of spend the day with your family" and not understand that her being there is why they're having to work. I jumped the gun there and I realize that. Now just to find a way to set up plans a different way (SIL never ever hosts, so I don't get why she's even pushing for this)

I've met your SIL - three things to consider.

Her health is not that great right now.

Would she or her husband be doing the cooking?  Or possibly her mother-in-law, if his family is coming over for a combined holiday......

Look at the bright side - dinner out can be cut off at three hours or less, so you can "take the toddler home for his nap" or "let the dogs out of the garage for their bathroom break". Going to anyone else's house is likely to get you guilt tripped if it isn't four-to-six hours or longer......or would you rather have them over at your house?  Yeah - she's that clueless - but with a narcissistic mother, it's amazing that she has any empathy at all.  She does have some empathy towards the people who are working - even if her LOGIC (as to who is making them work) isn't strong.....neither Vulcans nor Betans on that side of the family......

ETA - if anyone remembers Ambrosia Hino's earlier posts, or mine, this is CrazyMIL's DD.  The family traditions have had to be adjusted as CrazyMIL is being given the cut direct after making death threats to both her SIL and her DIL.  Since she also made death threats to her ex-DH (their father) and tried to hire someone to do the job...the situation with her would seem to warrant the etiquette version of "weapons of mass destruction".

SIL is never in charge of the cooking. Any of the cooking. Get-togethers at her house generally mean either going out to eat or ordering/bringing take-out. Her house is also a cluttered mess, and not a good place to let Toddler play. Family get-togethers, so far, are almost always at FIL & SMIL (step-mother-in-law) house. I wouldn't mind hosting, I've offered before but SMIL always wants to do it (and never shows when we do host something, but that's a different story).

This year, the dynamic is weird because we usually do Thanksgiving with them and Christmas with my family, and things got swapped. I told my DH that either we can host or we can all go out Christmas Eve and stay home just the 3 of us for Christmas Day.

*The comments that I mentioned SIL is known for, even if she doesn't mean them this way, are usually said in a gloating/condescending/snotty tone. Going out to eat with her always ends up with us picking up the bill as well. I don't have room in the budget and I doubt FIL does either, for a restaurant meal for 9 (2 college kids and a toddler...the 20 yr old boy makes up for the toddler in amount of food eaten!)

#### VorFemme

• Member
• Posts: 13768
• It's too darned hot! (song from Kiss Me, Kate)
##### Re: Your holiday hill to die on.
« Reply #568 on: December 11, 2012, 07:32:00 AM »
I admit I didn't handle that part well. But she's also the type that would sit there and tell the server (repeatedly) "oh its so sad that you have to work instead of spend the day with your family" and not understand that her being there is why they're having to work. I jumped the gun there and I realize that. Now just to find a way to set up plans a different way (SIL never ever hosts, so I don't get why she's even pushing for this)

I've met your SIL - three things to consider.

Her health is not that great right now.

Would she or her husband be doing the cooking?  Or possibly her mother-in-law, if his family is coming over for a combined holiday......

Look at the bright side - dinner out can be cut off at three hours or less, so you can "take the toddler home for his nap" or "let the dogs out of the garage for their bathroom break". Going to anyone else's house is likely to get you guilt tripped if it isn't four-to-six hours or longer......or would you rather have them over at your house?  Yeah - she's that clueless - but with a narcissistic mother, it's amazing that she has any empathy at all.  She does have some empathy towards the people who are working - even if her LOGIC (as to who is making them work) isn't strong.....neither Vulcans nor Betans on that side of the family......

ETA - if anyone remembers Ambrosia Hino's earlier posts, or mine, this is CrazyMIL's DD.  The family traditions have had to be adjusted as CrazyMIL is being given the cut direct after making death threats to both her SIL and her DIL.  Since she also made death threats to her ex-DH (their father) and tried to hire someone to do the job...the situation with her would seem to warrant the etiquette version of "weapons of mass destruction".

SIL is never in charge of the cooking. Any of the cooking. Get-togethers at her house generally mean either going out to eat or ordering/bringing takeout. Her house is also a cluttered mess, and not a good place to let Toddler play. Family get-togethers, so far, are almost always at FIL & SMIL (step-mother-in-law) house. I wouldn't mind hosting, I've offered before but SMIL always wants to do it (and never shows when we do host something, but that's a different story).

This year, the dynamic is weird because we usually do Thanksgiving with them and Christmas with my family, and things got swapped. I told my DH that either we can host or we can all go out Christmas Eve and stay home just the 3 of us for Christmas Day.

*The comments that I mentioned SIL is known for, even if she doesn't mean them this way, are usually said in a gloating/condescending/snotty tone. Going out to eat with her always ends up with us picking up the bill as well. I don't have room in the budget and I doubt FIL does either, for a restaurant meal for 9 (2 college kids and a toddler...the 20 yr old boy makes up for the toddler in amount of food eaten!)

Bean dip - if you don't have the time, money, patience, or inclination to go out to eat on Christmas Day with them you don't have the time, money, patience, etc...  Blame it on the toddler having been exposed to something at the day care and you don't want to expose her to it (health problems).  Then stay home in pajamas and have fun for a while......

For those who don't know the dynamic with this SIL, her "example" on how to behave properly was CrazyMIL or Nearly-as-Crazy-Stepmother - so her behaviour is odd at times...she can be sweet one minute then do something bizarre the next.  And she sees nothing wrong because she is still acting better than her two examples...which isn't a guarantee of "good behaviour" when those examples are so far off the mean, the norm, and even "sane" standards.  Eccentric would be not be a good description - because it would still imply that the person was closer to normal that they really are.  Does "beyond eccentric" make sense?  Because that is as close to a description as I can get without being too recognizable (small town in Georgia describes a LOT of places - but not as many once you start mentioning some of the other details - and both CrazyMIL and Nearly-as-crazy-Stepmother are still alive*).

*And a bit more than half a bubble off plumb, as well, from what I understand...
Let sleeping dragons be.......morning breath......need I explain?

#### nayberry

• Member
• Posts: 775
##### Re: Your holiday hill to die on.
« Reply #569 on: December 11, 2012, 11:34:53 AM »
VorFemme - I would love to meet you + family - you all have such *interesting* dynamics!

* as in the Chinese saying "May you live in interesting times"

Seriously, its been great to see E-Hellions passing down the generations and crossing the sibling ties.

OT  i always wondered where that came from,  especially as Terry Pratchett borrowed it
baby berry arrived june 2016


text:
## Wednesday, September 16, 2020

### What sorts of things are impossible?

Here's an interesting article in QUanta, by David Richeson. I'll be thinking about what else I might add to this post...

## Friday, September 11, 2020

### Solving Application Problems (in Trigonometry)

I started this blog in 2009, was active for about 6 years, and then not so much for the past 5 years. I wrote two posts in the spring, both related to online teaching. We were all trying to learn how to teach well as we scrambled to do it while learning. I was happy to keep seeing my students online, and Zoom was our class. I used Canvas a little but not much.

Over the summer I learned a lot about effective online teaching. (I'm still not sure it can ever be nearly as effective as in-person, but...) I developed my Canvas shells for each course, and I started the semester readier than I had expected to be. My Canvas shells are not done. I created a "module" that orients students to online learning and my course. And I created a module for our first unit. The rest is still in progress.

Today I added a page for my trig students, on solving application problems. I want to share it here. (And I may share lots of my Canvas "pages" here, sometimes with modifications.)

Years ago, I modified George Polya's wonderful outline of problem solving steps. We start with that. It's a good idea to print it out, and turn to it whenever you're stuck.

Draw a Diagram.

Always start by drawing a diagram. This step is vital, and is a major part of "Understanding the Problem".

Your diagram does not need to be artistically good. It does need to show relationships well. An artist might show my shadow going off at an angle. But for a math diagram, it is better to show the right angle involved, as a right angle.

In the diagrams on the right, the top drawing is prettier, and the shadow is more evocative, but the bottom drawing shows the right angle between a vertical object and its horizontal shadow, which is what will help you do your mathematical analysis.

Example (#22 in 2.4, page 93): If the angle of elevation of the sun is 63.4° when a building casts a shadow of 37.5 feet, what is the height of the building?

Draw your diagram now, labeling it with everything given and a variable for the value requested. (My drawing is below.)

.

.

.

.

.

.

I labeled the height of the building h.

Write a Trig Equation.

In a simple problem, with only a few pieces of information this is all you need for the "Devising a Plan" step. We are given the value of the side adjacent (next to) the given angle, and we want to find the value of the side opposite the angle. (The hypotenuse is neither given nor asked for.) Which trig function uses adjacent and opposite? (Two of them do, but the one we use most of the time is...)

.

.

.

.

... , and this gives us

Do a bit of algebra.

This is the "Carry out the Plan" step. To solve for h, we multiply both sides of the equation by 37.5:

I pulled out my calculator for that last step (making sure it was in degree mode). Since our given length was given to tenths of a foot, I round, and give my final answer as 74.9 feet.

This is the "looking back" step on the handout. If we look at our diagram, does a height of about 75 feet seem reasonable? Well, the height seems bigger than the shadow, and maybe about twice as big, so yes, it seems reasonable.

Practice.

If you get stuck on application problems, a good way to practice is to re-do problems that you've watched someone else do (perhaps on youtube). Try not to look at your notes. If you need to, go ahead and look. Do as much of the problem on your own as you can. If you looked at your notes at all, do it again the next day.

## Thursday, June 25, 2020

### Playful Math Education Carnival #139 (formerly known as Math Teachers at Play or MT@P)

"It’s like a free online monthly magazine of mathematical adventures." (Denise Gaskins)

Black Lives Matter. How does that idea and movement intersect with math and play?  It's hard to imagine play intersecting with the painful history of racism in the U.S.  We can collect data to show how pervasive anti-Blackness has been and is. We can discuss how math courses have been used to filter out students from desirable professions (doctors, engineers, lawyers).  We can discuss how Black people are more involved in the history of math than you'd guess from the Eurocentric naming. (Check out who knew Pascal's triangle before Pascal!) None of that is playful. But celebration can be playful. Let's celebrate Juneteenth!

## 139

Every number is cool.* Here are some ways 139 is cool:
• 139 is the sum of 5 consecutive prime numbers (19 + 23 + 29 + 31 + 37).
• 139 is the smallest prime before a prime gap of length 10.
• 137 and 139 form the 11th pair of twin primes.
• 139 is the 34th prime number.

Puzzle: The digit sum is the result after adding the digits repeatedly until you get down to one digit. 139’s digit sum is 4. If you write 139 in base two, you get 100 1011, which still has a digit sum of 4. Does this always happen? If not, does it happen in any other bases?

New Homeschoolers
I have a hunch the quarantine has moved lots of families from school to homeschooling. If you’re new to homeschooling, get ready to have fun playing with math. Most mathematicians are in it at least partially for the fun of it. We like to play with numbers, shapes, and logic. The more you play with math with your kids, the more likely they are to enjoy it.

There are vast resources online to help you. Until 3rd grade, just play games, cook, measure, read mathy stories, and have fun with it all. If your kid wants a curriculum before that because they love math, then check out Beast Academy. It has levels 2 to 5 (topics correspond to grades 2 to 5, difficulty levels are a grade or two higher). Some families never use a curriculum; if you’re interested, you may want to explore unschooling. Math lovers eventually want to take classes, which you can do either through your local community college (I’ll be teaching trigonometry, pre-calculus, and calculus I online this fall) or Art of Problem Solving. There are lots of other great resources; these are just my personal favorites.

You might find ideas that work for you in my book, Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers. Or from other books from my publisher, Natural Math. I also highly recommend Denise Gaskins’ blog (especially this post on homeschooling math), website, and booksDan and Christopher have some good ideas about playing mathematically with kids too.

Talking Math With Your Kids (#TMWYK)

Math & Language Play
One of my favorite math educators, Marilyn Burns, invented a game where students look for $1 words. A=1¢, B=2¢, etc. You could combine math and any other subject by making$1 phrases. Sometimes kids like the simplest games. This might be a craze at your house. (My son used to love Shut the Box, a simple dice game that did nothing for me. It sure was good number practice for him.)

π-ku, a competition, in which all their favorites will be posted at the Aperiodical blog. I'll try:
Three One Four.
Hmm.
Not very hard.

Games
So much of math is based on logic, any logic games you play will deepen your students' affinity for math. Here are a few others:
• Set Tic Tac Toe, described by Tanya Khovanova, invented by her students. You may want to play the basic game of Set for a few months before attempting this. But if I could figure out a way to do this at a distance, I'd love to try this out.
• Planarity game. (This is connected to a field of math called graph theory.)
• Play with wallpaper symmetries.

Math History
Podcasts aren't my thing. Yet. But if this series is as good as it sounds, I'll just have to  figure this newfangled genre out. Opinionated History of Mathematics. With an interview and glowing review at Aperiodical.

Online Events
This summer Art of Inquiry is hosting free science webinars on space, astrobiology, and AI for school children and their families. The webinars are led by university professors and industry experts. You can register for the events on Eventbrite.  Here is their June-July 2020 schedule:
• Living Through a Revolution: Multi-messenger Astrophysics - Dr. Roopesh Ojha, GSFC NASA, June 26th
• Figuring out the Earth from inside out - Dr. Kanani Lee, Lawrence Livermore National Laboratory, June 30th
• Mars Rovers - Dr. Allan Treiman, Lunar and Planetary Institute, July 3rd
• The search for life on Mars in XXI century - Dr. Alex Pavlov, GSFC NASA, July 10th
• Where in the Universe did we come from? - Dr. Ethan Siegel, science author, "Starts with a Bang" Forbes contributor, July 23rd
• Why we should build a Moon base - Dr. Ian Crawford, University of London, July 31st
If you know of other math-related online events, please mention them in the comments.

This series of blog carnivals was founded and is kept going by the fabulous Denise Gaskins. You can find out more at her blog. Last month's carnival was hosted by John Golden, the Math Hombre. Check it out!

-----
*Well, sometimes their coolness is in their bad reputation (sounds like a few people I knew in high school) ...

## Thursday, June 18, 2020

The Math Teachers at Play Blog Carnival (aka Playful Math Education Carnival) will be a bit late this month.

I am looking for good posts now. If you can send me any links by Saturday, that would be great. I am hoping to put it together on Sunday.

Want to know what a blog carnival is? Check out last month's, by my pal John Golden.

## Sunday, April 19, 2020

### Corona Post #2: Teaching Online

[#2 because my previous post in March on my online math circle was due to people needing to take their math circles online when the shelter-in-place orders were just starting.]

I've been teaching online for 4 weeks now, two before our spring break and two after. At first I was just trying to learn how to manage teaching on zoom. I bought a whiteboard that's still sitting on two chairs in my living room, and I sit in a tiny chair while I write on it. Not ideal, but I get to see my students, and I feel like I'm still working with them where they are, not just lecturing.

(Some day I'll finally install it on my living room wall. I procrastinate with tasks like that. I'm not sure why it feels intimidating...)

A few weeks ago I made a google slides presentation for my Discrete Math course to explain a way of counting possibilities called Stars & Bars. I had fun doing it. You're welcome to modify it and use it in your teaching.

Just now I made another. This one is for Calculus II, on Taylor & Maclaurin Series (really just a Maclaurin series). I was motivated by knowing that there would be too much writing for my little whiteboard. This presentation has a handout to go with it.

I'm also teaching Calculus I. I haven't made any cool new materials for that course yet. But I will...

## Monday, March 23, 2020

### Online Math Circle: Pythagorean Triples

The Pythagorean theorem tells us that if a and b are the legs, and c the hypotenuse, of a right triangle, then a2+b2 = c2. Usually that makes at least one side something ugly like square root of 2. But a few combinations make all three sides whole numbers. Those are called Pythagorean triples. Here are a few of them: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 20-12-29.

Are there patterns to this? Let's play, and see what we can figure out! (We will use some algebra.)

This online math circle happened on Friday, March 27, at 10am PDT (1pm EDT). [This link is to the zoom recording, along with its automatically produced (therefore hilariously bad) audio transcript.]

I promised to write up some of it here.

Way back in 2007, I read Bob and Ellen Kaplan's book, Out of the Labyrinth: Setting Mathematics Free, about the math circles they lead. It was such a discovery for me! I went to their first Summer Math Circle Teacher Training Institute, held at Notre Dame, and fell in love with this community. I kept going back for years, craving a discussion of math among equals, figuring out new ways of seeing. One summer we discussed Pythagorean triples, and that December I tried to rebuild what I had learned. I am blessed with a very bad memory, so what I did in December looked very different from what we had done in the summer.

I was also exploring online, and ended up putting together a book that collected some of the best resources I had found: Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers.

Our circle was prompted by Rodi Steinig's request for help learning how to use zoom for online math circles. I offered one of my favorite topics, and off we went. Participants came from as far away as Colombia (and farther?).

We proved a few things, and explored a bunch more. I hope some participants went home eager to prove more on their own.

## Saturday, January 4, 2020

### Multiplication Chart with Pictures

In the story I'm writing, Althea remembers a multiplication chart that was posted in their bathroom. It had cool pictures around the edges for many of the facts.

• 2x3 was a 6-pack of soda.
• 2x6 was a carton of eggs.
• 8x8 was a chessboard.
• The fives were sometimes collections of nickels, but 5x12 was the 60 minutes on a clock, and 5x6 was time too.

I thought I knew of more iconic sets like these, but I can't think of any more as good as these. I'm hoping for help. Do you have images in your head for any of the multiplication facts?

Maybe threes will be 3-leaf clovers. 6 of them have 18 petals. That doesn't seem nearly as iconic as the ones above, though.

Fours could be legs on dogs. 6 dogs have 24 legs. Twos could be eyes on friends...

What are your favorite images for multiplication facts?

format: plain


text:
# What is the equation that expresses one-half of a certain number n is 95?

Aug 7, 2016

$95 = \frac{1}{2} n \leftarrow \text{ equation}$

For this to work the actual value of $n$ is $190$

#### Explanation:

$\textcolor{g r e e n}{\text{Solved by thinking it out}}$

Given that:$\text{ } 95 = \frac{1}{2} n$

If half a number is 95 then the number must be two lots of 95.

That is: $95 + 95 = 190$

,~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{g r e e n}{\text{Solved by using algebra}}$

Given that:$\text{ } 95 = \frac{1}{2} n$

Determine the value of $n$

Multiply both sides by $\textcolor{b l u e}{2}$

$\textcolor{b r o w n}{\textcolor{b l u e}{2 \times} 95 = \textcolor{b l u e}{2 \times} \frac{1}{2} \times n}$

$\textcolor{b r o w n}{\textcolor{b l u e}{2} \times 95 = \frac{\textcolor{b l u e}{2}}{2} \times n}$

But $\frac{2}{2} = 1$ giving:

$190 = 1 \times n$

$\implies n = 190$


text:
The determination of absorbed doses for sterilization of shu-ganwan and evaluation for effects of irradiated shu-ganwan on liver function of rat

YAO Qingwan1,2  LIU Qingfang1

1. 1 (School of Public Health, Soochow University, Suzhou 215123, China)
2 (Center for Disease Control and Prevention of Kunshan, Kunshan 215301, China)
• Received:2011-11-17 Revised:2011-12-30 Online:2012-08-20 Published:2014-09-11

Abstract:

The absorbed dose of irradiation sterilization for Shu-Gan Wan and the effects of irradiated Shu-Gan Wan on rat liver function was investigated to provide the scientific basis for its clinical safety applications. The amount range of contaminating bacteria in original Shu-Gan Wan was determined before irradiation and the D10 & SD were calculated after irradiation. The irradiated Shu-Gan Wan were given to the rats and the biochemical items of liver were measured before and after-irradiation. The experimental results show that the amount range of contaminating bacteria count of the Shu-Gan Wan is 50-1800 CFU/g, the D10 value is 1.81 kGy and the minimum bactericidal dose has been calculated to be 11.322 kGy. The 60Co radiation dose for sterilization of Shu-Gan Wan has been suggested to be 11.322 kGy and irradiated Shu-Gan Wan has no significant damage to liver function of the rat.

CLC Number:

• R817，R144.1


text:
# Math Help - Simplifying Another Rational Expression

1. ## Simplifying Another Rational Expression

Hey there, I can't seem to simplify these rational expressions to save my life.

I know all about bedmas, like terms, factoring, or at least I'd like to think so, but I'm stumped again, here. Is there any oder of steps I can take when simplifying a rational expression to ensure it turns out good?

Here's my attempt:

If anyone could solve and explain, that would be super-helpful.

- Cam

2. You went from:

$\frac{3a (a-2) - (a-4)(a+3)}{(a-2)(a+3)}$

to

$\frac{3a (a-2) - a+4(a+3)}{(a-2)(a+3)}$

This is wrong. You also need to multiply the a of a+4 by the (a+3), you lost some terms.

$\frac{3a (a-2) - (a-4)(a+3)}{(a-2)(a+3)}$

$= \frac{3a (a-2) - (a^2-4a +3a-12)}{(a-2)(a+3)}$ (multiplying out that term before doing anything else)

$= \frac{3a^2 -6 - (a^2-a -12)}{(a-2)(a+3)}$

$= \frac{3a^2 -6 - a^2 + a +12}{(a-2)(a+3)}$ (now you can unpack the minus-thing)

$= \frac{2a^2 + a + 6}{(a-2)(a+3)}$

The thing on top probably factorises. I'll leave it up to you to continue.

3. Thanks, Matt

$\frac{2a^2 + a + 6}{(a-2)(a+3)}
$

The top 'a' variable will factor to be two numbers with a product of 12, and a sum of 1, then.

I can't even think of it...

1 and 12 = 13, so no.
2 and 6 = 8, so no.
3 and 4 = 7, so no.
-3 and 4 = 1, but their product is -12, so that can't be right...

4. Well if it doesn't factorise, it doesn't. Never mind.


text:
## Alexandrian Integers

### Problem 221

We shall call a positive integer A an "Alexandrian integer", if there exist integers p, q, r such that:

$$A = p \cdot q \cdot r$$ and $$\dfrac{1}{A} = \dfrac{1}{p} + \dfrac{1}{q} + \dfrac{1}{r}$$

For example, 630 is an Alexandrian integer ($p = 5, q = -7, r = -18$). In fact, 630 is the 6th Alexandrian integer, the first 6 Alexandrian integers being: 6, 42, 120, 156, 420, and 630.

Find the 150000th Alexandrian integer.


text:
# The Zebra Of Riemann

July 13, 2014
By

(This article was first published on Ripples, and kindly contributed to R-bloggers)

Mathematics is the art of giving the same name to different things (Henri Poincare)

Many surveys among experts point that demonstration of the Riemann Hypothesis is the most important pending mathematical issue in this world. This hypothesis is related to Riemann zeta function, which is supossed to be zero only for those complex whose real part is equal to 1/2 (this is the conjecture of Riemann itself). Confirming the conjecture would imply deep consequences in prime numbers teory and also in our knowledge of their properties. Next plot represents the argument of zeta function over complex with real part between -75 and 5 (x axis) and imaginary part between -40 and 40 (y axis). An explanation of this kind of graph can be found here. Does not it remind you of something?

Solving the hypothesis of Riemann is one of the seven Clay Mathematics Institute Millennium Prize Problems. Maybe zebras keep the secret to solve it in their skins.

This is the code to plot the graph:

require(pracma)
z=outer(seq(-75, 5, by =.1),1i*seq(-40, 40, by =.1),'+')
z=apply(z, c(1,2), function(x) Arg(zeta(x)))
opt=theme(legend.position="none",
panel.background = element_blank(),
panel.grid = element_blank(),
axis.ticks=element_blank(),
axis.title=element_blank(),
axis.text =element_blank())
z=data.frame(expand.grid(x=seq(ncol(z)), y=seq(nrow(z))), z=as.vector(z))
require(ggplot2)
ggplot(z, aes(x=x, y=y, color=z)) + geom_tile() + scale_colour_gradientn(colours=c("black","white", "gray80", "gray40")) + opt


text:
# Quark–lepton complementarity

The quark–lepton complementarity (QLC) is a possible fundamental symmetry between quarks and leptons. First proposed in 1990 by Foot and Lew,[1] it assumes that leptons as well as quarks come in three "colors". Such theory may reproduce the Standard Model at low energies, and hence quark–lepton symmetry may be realized in nature.

## Possible evidence for QLC

Recent[when?] neutrino experiments confirm that the Pontecorvo–Maki–Nakagawa–Sakata matrix UPMNS contains large[clarification needed] mixing angles. For example, atmospheric measurements of particle decay yield
θPMNS
23
≈ 45°, while solar experiments yield
θPMNS
12
≈ 34°. These results should be compared with
θPMNS
13
which is small,[2] and with the quark mixing angles in the Cabibbo–Kobayashi–Maskawa matrix UCKM. The disparity that nature indicates between quark and lepton mixing angles has been viewed in terms of a "quark–lepton complementarity" which can be expressed in the relations

${\displaystyle \theta _{12}^{PMNS}+\theta _{12}^{CKM}\approx 45^{\circ }\ ,}$
${\displaystyle \quad \quad \theta _{23}^{PMNS}+\theta _{23}^{CKM}\approx 45^{\circ }\ .}$

Possible consequences of QLC have been investigated in the literature and in particular a simple correspondence between the PMNS and CKM matrices have been proposed and analyzed in terms of a correlation matrix. The correlation matrix VM is simply defined as the product of the CKM and PMNS matrices:

${\displaystyle V_{\mathrm {M} }=U_{\mathrm {CKM} }\cdot U_{\mathrm {PMNS} }\ ,}$

Unitarity implies:

${\displaystyle U_{\mathrm {PMNS} }=U_{\mathrm {CKM} }^{\dagger }V_{\mathrm {M} }\ .}$

## Open questions

One may ask where do the large lepton mixings come from? Is this information implicit in the form of the VM matrix? This question has been widely investigated in the literature, but its answer is still open. Furthermore, in some Grand Unification Theories (GUTs) the direct QLC correlation between the CKM and the PMNS mixing matrix can be obtained. In this class of models, the ${\displaystyle V_{M}}$ matrix is determined by the heavy Majorana neutrino mass matrix.

Despite the naive relations between the PMNS and CKM angles, a detailed analysis shows that the correlation matrix is phenomenologically compatible with a tribimaximal pattern, and only marginally with a bimaximal pattern. It is possible to include bimaximal forms of the correlation matrix VM in models with renormalization effects that are relevant, however, only in particular cases with tanβ > 40 and with quasi-degenerate neutrino masses.

## References

1. ^ R. Foot, H. Lew (1990). "Quark-lepton-symmetric model". Physical Review D. 41 (11): 3502–3505. Bibcode:1990PhRvD..41.3502F. doi:10.1103/PhysRevD.41.3502.
2. ^ F. P. An et al. [DAYA-BAY Collaboration], Phys. Rev. Lett. 108, 171803 (2012) [arXiv:1203.1669 [hep-ex]] https://arxiv.org/abs/arXiv:1203.1669


text:
# Effect of population growth on Solow steady state

My textbook says that:

Ratio of capital per capita to income per capita in the steady state is a positive function of s and an inverse function of η and δ. Thus, k*/y* is a constant. This means when saving increase, the ratio does not change as both capital per capita and income per capita increase at the same rate. However, both capital per capital and income per capital decrease at the same rate if the depreciation and population growth rate are higher.

It also says that:

When you get to the steady state of output per worker, it means that your economy will still grow, just that it will only grow at the rate of growth of the labour force i.e. equals to the rate of population growth)

Doesn't both statement contradict each other? I mean if the population growth rate grows, why could the capital and income per capita decreases which the first statement seems to state. Am I reading the statements wrongly?

• Which textbook is it? Does it show the equations for these statements? Can you interpret these equations? If not, type them in your post it is easier if you learn it by interpreting the equation parameters etc. Sep 27 '17 at 10:29
• Sandra @Jane, why did you change your name? Sep 27 '17 at 11:35
• @denesp, was thinking the same aftering reading the other similar post, but, note that the uni has just started and you may get many new members... Sep 27 '17 at 15:58
• @london Indeed, perhaps I am wrong. Though I don't think the $\eta$ notation is very common. Also we don't get that many new Solow questions. Then two on the same day...? Sep 27 '17 at 16:32
• @denesp, yeah. and the other post was donwvoted Sep 27 '17 at 17:35

if the population growth rate grows, why could the capital and income per capita decreases

This is basically asking "why does higher population growth lower the steady state capital per worker"?

Mathematics

Let us assume a Cobb-Douglas production function with constant returns to scale. This is,

$$Y=K^{\alpha}L^{1-\alpha}$$

It can be shown that in the steady state optimal capital per capita is:

$$k^* = \left(\frac{s}{\delta+n}\right)^\frac{1}{1-\alpha}$$

(for example, see here)

Therefore, an increase in $n$ lowers $k^*$.

Similarly, in the steady state, output per capita is $y^*=k^*{^{\alpha}}$. Thus, a fall in $k^*$ leads to a fall in $y^*$, given that $\alpha \in (0,1)$.

Intuition

To start simple, think of an economy with a fixed level of workers ($n=0$). This economy has an exogenous saving rate $s$. Thus, for an initial level of output $Y$, workers save $s$ percentage of their income $Y$. These savings become investment, which then contributes to expanding capital. Meanwhile, there is a positive depreciation rate $\delta$, which in a given year renders a $\delta$% of the capital stock unusable. These two forces go in the opposite direction, moving capital stock up or down, depending on which force is stronger. Will there be a point in which these forces are in equilibrium? Necessarily there is one. Why? Because $\alpha \in (0,1)$ implies there is decreasing marginal returns to capital, case in which the Inada Conditions hold. In other words, if capital is very low, adding more capital will expand output and investment by very much. Conversely, when capital is very low, an identical increase in capital expands output and investment by very little. In the middle, there is a point where such expansion is equal to the depreciation (which is constant). In the Solow diagram with axes $K$ and $Y$ (in capital letters, not in per capita terms), that point is represented by the intersection between the depreciation function (a straight line) and the $sY$ line (concave, in line with the Inada Conditions).

Now, let us move to the case of positive population growth ($n>0$). Here, employment is increasing every period. Therefore, $Y$ is also increasing. In consequence, savings, investment and therefore capital is also increasing. In this economy, there is no steady state in terms of $K$ (unlike before), because $K$ is growing every period with the expansion of the population size. In terms of the same Solow diagram of above (with axes $K$ and $Y$, in capital letters, not in per capita terms), the function $sY$ is shifting upwards every period, pushing the never-reachable point where $sY$ intersects $\delta K$ further to the right.

Since there is no steady state in terms of $K$, the next question to ask is, is there a steady state in terms of $K/L$? Because of the same logic than before, necessarily there is one. To see this, notice that capital per capita terms has decreasing marginal returns in the production function (see the function above). This is, an increase in $K/L$ by 1% leads to an increase in $Y/L$ by less than 1%. Therefore, for a very low $K/L$, an increase in $K/L$ means a relatively large increase in $Y/L$ and therefore in savings $sY$ and in investment, and therefore in capital. Conversely, for a very high $K/L$, an increase in $K/L$ means a relatively small increase in $Y/L$ and therefore in savings $sY$ and in investment, and therefore in capital. The Inada Conditions ensure there is a middle point by which the forces that lower $K/L$ (in this case, depreciation and population growth) equilibrate the expansion of capital.

Finally, why does an increase in $n$ lowers that optimal $K/L$? On the one hand, the first force in the equilibrium (pushing $K/L$ up) is given by both the savings rate $s$ and the degree of marginal returns ($\alpha$). Neither of these have changed. On the other hand, the second force in the equilibrium (pushing $K/L$ down) is given by both the depreciation rate $\delta$ (which makes a certain proportion of $K$ obsolete) and by the exogenous increase in $L$, which itself depends on $n$. In consequence, an increase in $n$ makes the second force stronger. This pushes the equilibrium to a lower level of $K/L$.

Notice that a way to compensate this is to make the first force stronger, for instance, by increasing the savings rate. This can be easily seen in the Solow diagram, where it is possible to revert to the original $K/L$ steady state by increasing $s$.


text:
Note: I have no intention of discrediting the project or anyone involved with it. The beautiful part about crypto and smart contract development is all of the code is on display. I’m just commentating on a monumental screwup and a very interesting situation.

Legolas Exchange (LGO) raised ~$30M USD early in 2018 and deployed an ERC-20 token on Ethereum. The token has a peculiar feature built into the code that is supposed to reward hodlers (sic). For every 6 month period in the first 2 years, all of the accounts that haven’t moved their funds yet should receive an additional 5% in tokens as a reward for their loyalty. The reward has the potential to be higher than 5% — increasing as others abandon their rewards. Sounds great right? Unfortunately the immutable contract was coded such that the extra 20% of tokens, that should have been used to pay out the bonuses, is virtually guaranteed to be lost. I say “virtually guaranteed” because there is a way to retrieve them — it just requires ungodly levels of cooperation. More on this soon. The Bug There are several parts of the contract that I think should not have made it through any serious audit, such as the perverse use of percentages to assign token amounts. Nevertheless, the problematic bug that led to this article can be found on line 326 of the contract source: // calculate the bonus for one holded LGO uint256 bonusByLgo = (BONUS_AMOUNT / 4)/unspentAmounts[_bonusDate];  I’m not going to go deep into this — you can easily check what I’m saying for yourself. All you need to know is that this variable, bonusByLgo is expecting to be assigned a percentage. Something like 0.05. If you’re a developer (from any background, not just smart contract development!) you know how absurd this is. A uint256 can only be assigned 0, 1, or any other whole integer. So when the bonus percentage to dish out is calculated, the user claiming is always going to receive a 0% reward due to natural rounding-down. Besides this mistake — the bigger mistake is this: Any testing of the claimBonus method would have found this problem.$30 million US Dollars were put into a contract where the core feature was completely untested.

How They “Resolved” It

To their credit, the LGO team started paying out the bonuses manually, from their own reserve, which was also 20% of tokens. You can see these transactions happening over a long period of time — if you want to do in-depth chain analysis you can find one LGO-owned address dispensing a round of bonuses here.

Potentially holders of LGO will be reading this and have no idea something was wrong, but this mistake cost the company ~\$7M USD at the worst, if we were to value the tokens at LGO’s highest historical market cap.

A Really Weird Way Out

So as I mentioned when discussing the bug, the variable bonusByLgo will almost always evaluate to 0. Except in one case. When (BONUS_AMOUNT/4) is bigger than unspentAmounts[_bonusDate] . When this happens, the remaining holders will receive one bonus token for every LGO token they hold — a 100% bonus!!

What can trigger this condition? Only if 95% of the holders move their tokens. Only when 5% of the tokens are still eligible for bonuses, they would all receive a 100% bonus at each of the remaining bonus dates.

The really crazy part is this! If the community came together and all decided to move their tokens — completely the opposite of what the contract intended — they could potentially deliver LGO back at least some of their tokens and save LGO money. You could even imagine a situation where the LGO team or another rogue developer builds a smart contract that allows the bonus to be claimed and distributed amongst the remaining holders — though good luck getting the required >95% participation.

The Lesson

Be skeptical of any token contracts with overly complex functionality. “Smart” smart contract developers should be staying away from any such gimmickry, and when it is absolutely needed, it should be tested and audited to extremes.

For more on interesting aspects of smart contracts find me on Twitter: https://twitter.com/codingupastorm


text:
Press J to jump to the feed. Press question mark to learn the rest of the keyboard shortcuts
1.2k
Posted byCombinatorics15 days ago

## 33=8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3

Found by Timothy Browning. He did the Numberphile video. 42=a3 + b3 + c3 is the smallest unsolved case.
Myself, I recently did a blog on Substitution Tilings Using 2, φ, ψ, χ, ρ.

The hard cases
24 = -29010966943 + -155505555553 + 155841398273
30 = -2830599653 + -22188885173 + 22204229323
33 = 88661289752875283 + -87784054428622393 + -27361114688070403
42 = ??????????? ??????????? ???????????
52 = 239612924543 + 607029013173 + -619227128653
74 = −2846502925558853 + 662298321905563 + 2834501056977273
75 = 43811593 + 4352030833 + -435203233
81 = 396184514443 + -87284087913 + -394767274183 84 = -82411913 + -415317263 + 416396113
110 = 1099389193 + 165402900303 + -165402916493
114 = ??????????? ??????????? ???????????

Other large solutions.
The paper for 33: Andrew R. Booker, Cracking the Problem With 33.
The paper for 74: Sander G. Huisman, Newer sums of three cubes.

205 comments
97% Upvoted
What are your thoughts? Log in or Sign uplog insign up
Sort by
level 1
198 points · 15 days ago

Wait actually?

level 3
74 points · 15 days ago

I assume it's legit based on who it's coming from but I'm curious whether Wolfram alpha would actually be accurate enough to tell or if it's just within the bounds of floating point error

level 4
Control Theory/Optimization173 points · 15 days ago

Fun-fact: you can also do this in Python 3. Python 3 has no maximum integer; the limit of exact-arithmetic with integers in Python is limited only by the amount of memory on your computer.

Last login: Mon Mar  4 13:07:56 on ttys000
Rileys-MacBook-Pro:~ RJMurray$source activate cvx36 (cvx36) Rileys-MacBook-Pro:~ RJMurray$ python
Python 3.6.8 |Anaconda, Inc.| (default, Dec 29 2018, 19:04:46)
[GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)] on darwin
Type "help", "copyright", "credits" or "license" for more information.
>>> 8866128975287528**3+(-8778405442862239)**3+(-2736111468807040)**3
33

And it spat out 33 instantaneously.

level 5
54 points · 15 days ago

I didn't know, that! I just started learning python for computing and data analysis type applications pretty recently. That's cool.

level 6
67 points · 15 days ago

When picking a data analysis tool five years ago, I tested them by creating a 10 GB matrix then did some basic operations with it.

Python's numpy package didn't even flinch, which is why I picked it.

level 7
18 points · 15 days ago

Keep in mind though that numpy uses fixed-precision integers because they are faster and can be plugged into SIMD. If you are doing operarations on a giant matrix then you cannot have numbers bigger than 263 - 1 in the regular case.

level 8
4 points · 15 days ago

I believe you can setup a matrix to use dtype float128. I haven't had a need to go to that large of numbers though. Aircraft usually live in a world of small numbers.

level 9
3 points · 15 days ago

Oh neat. float128 allows you to have number until around 2215 if you don't care about precision.

If you need anything slightly larger, you can also represent infinity inside your matrices.

level 8
Applied Math2 points · 15 days ago

it seems to seamlessly switch between Int64 and BigInt as far as single integers are concerned: 8866128975287528**3 = 696950821015779435648178972565490929714876221952

but in Julia, which is more clear with types, I get 88661289752875283 = -1277230016499447296

level 7
13 points · 15 days ago

That’s good to know! Thank you

level 7
Physics2 points · 15 days ago

Wow that’s impressive.

I’m currently using numpy for my master’s project, which involves exact diagonalisation of some rather large matrices (up to ~10000x10000, so about 800 MB) and it seems to handle those fairly well.

I’m surprised it turns out to be the best though, I would’ve thought something dedicated like MATLAB or Mathematica, or something low level like C++, might be faster. I just picked it because it was easy to use but still capable.

level 8
4 points · 15 days ago

Python binds to C, so the overhead is really only in the interpretation. Most of the libraries are pretty good.

level 9
5 points · 15 days ago

Python binds to C, so the overhead is really only in the interpretation. Most of the libraries are pretty good.

If you're using something like Numpy (and doing so using actual Numpy calls), then sure. If you're using pure Python, then very much no. Native python code is usually very slow, it's only reasonably fast if you use a library like Numpy that is really C/C++ underneath (although good code written in something like C/C++ will still be faster to run, though typically not to write).

level 7
Mathematical Physics2 points · 15 days ago

Yeah, a skilled numpy user can do anything wolfram can do but 50 times faster - accounting for programming time overhead.

level 8
38 points · 15 days ago · edited 12 days ago

You do know that Wolfram's Mathematica covers a huge amount of math besides linear algebra, right?

level 9
Mathematical Physics1 point · 15 days ago

Yes, I do programming for much of my work. I use Wolfram relatively often, and for what it does well, it is great (if a little frustrating sometimes). It is still very very slow for a lot of things that shouldn't be very very slow.

And at the end of the day, I was kind of joking.

level 9
Algebraic Geometry-1 points · 15 days ago

Linear algebra is applicable to way more problems than you probably think. Also SageMath is built on top of numpy among other widely available python packages. Much of the functionality in Wolfram alpha and Mathematica can be had in python.

level 10
Algebraic Topology19 points · 15 days ago

Linear algebra is applicable to way more problems than you probably think

The person you replied to is a math professor, so is aware of the utility of linear algebra in mathematics. And his point is true: there are questions Wolfram Alpha can solve, such as symbolic integration, that are not included in NumPy.

level 10
4 points · 15 days ago · edited 15 days ago

I agree with you, but still disagree with the grandparent comment. Yes, the python ecosystem for mathematics is great (I love Sage!), and no, numpy doesn't do everything Wolfram alpha can do, not by a long shot!

level 8
2 points · 15 days ago

Last I checked numpy didn't do symbolic integration lol

level 9
Mathematical Physics-1 points · 15 days ago

True, but I was a) joking, and b) speaking generally, of course, there are specific things that Wolfram does better, just not everything, or most things.

Community Details

747k

Subscribers

596

Online

Welcome to r/math!

This subreddit is for discussion of mathematical links and questions. Please read the FAQ and the rules below before posting.

If you're asking for help understanding something mathematical, post in the Simple Question thread or /r/learnmath. This includes reference requests - also see our lists of recommended books and free online resources. Here is a more recent thread with book recommendations.

If you are asking for a calculation to be made, please post to /r/askmath or /r/learnmath.

If you are asking for advice on choosing classes or career prospects, please post in the stickied Career & Education Questions thread.

Please be polite and civil when commenting, and always follow reddiquette.

r/math Rules
1.
No homework problems
2.
Stay on-topic
3.
Be excellent to each other
4.
No low-effort image posts
5.
No career or education related questions
Recurring Threads & Resources

Everything about X - every Wednesday

What Are You Working On? - posted Mondays

Career and Education Q&A - Every other Thursday

Simple Questions - Posted Fridays

Click here to chat with us on IRC!

Using LaTeX

To view LaTeX on reddit, install one of the following:

MathJax userscript (install Greasemonkey or Tampermonkey first)

[; e^{\pi i} + 1 = 0 ;]

Post the equation above like this:

[; e^{\pi i}+1=0 ;]

Useful Symbols

Basic Math Symbols

≠ ± ∓ ÷ × ∙ – √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °

Geometry Symbols

∠ ∟ ° ≅ ~ ‖ ⟂ ⫛

Algebra Symbols

≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘∏ ∐ ∑ ⋀ ⋁ ⋂ ⋃ ⨀ ⨁ ⨂ 𝖕 𝖖 𝖗

Set Theory Symbols

∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟

Logic Symbols

¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ↔ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣

Calculus and Analysis Symbols

∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ

Greek Letters

𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔

Other math subreddits
r/learnmath

110k subscribers

r/mathbooks

9.9k subscribers

r/cheatatmathhomework

37.1k subscribers

r/matheducation

14.5k subscribers

r/CasualMath

9.1k subscribers

r/puremathematics

10.7k subscribers

r/mathpics

18.1k subscribers

r/mathriddles

8.6k subscribers

Related subreddits
r/Mathematica

4.6k subscribers

r/matlab

18.8k subscribers

r/sagemath

725 subscribers

r/actuary

13.3k subscribers

r/algorithms

45.6k subscribers

r/compsci

481k subscribers

r/interdisciplinary

1.7k subscribers

r/statistics

71.6k subscribers

Moderators
u/yesmanapple
Geometry/Topology
u/inherentlyawesome
Algebraic Topology
u/amdpox
Geometric Analysis
u/AngelTC
Algebraic Geometry
u/dogdiarrhea
Dynamical Systems
u/AcellOfllSpades
Undergraduate
u/functor7
Number Theory
© 2019 Reddit, Inc. All rights reserved


text:
# Q.7.(ii)    If in a stick of chalk, carbon is 3g, what is the weight of the chalk stick?

$\Rightarrow \frac{3}{3}\times 25g = 25g$


text:
# nuclear_reaction_energy¶

plasmapy.particles.nuclear_reaction_energy(*args, **kwargs)

Return the released energy from a nuclear reaction.

Parameters
Returns

energy – The difference between the mass energy of the reactants and the mass energy of the products in a nuclear reaction. This quantity will be positive if the reaction is exothermic (releases energy) and negative if the reaction is endothermic (absorbs energy).

Return type

Quantity

Raises
• ParticleError – If the reaction is not valid, there is insufficient information to determine an isotope, the baryon number is not conserved, or the charge is not conserved.

• TypeError – If the positional input for the reaction is not a string, or reactants and/or products is not of an appropriate type.

nuclear_binding_energy

finds the binding energy of an isotope

Notes

This function requires either a string containing the nuclear reaction, or reactants and products as two keyword-only lists containing strings representing the isotopes and other particles participating in the reaction.

Examples

>>> from astropy import units as u

>>> nuclear_reaction_energy("D + T --> alpha + n")
<Quantity 2.8181e-12 J>

>>> triple_alpha1 = '2*He-4 --> Be-8'
>>> triple_alpha2 = 'Be-8 + alpha --> carbon-12'
>>> energy_triplealpha1 = nuclear_reaction_energy(triple_alpha1)
>>> energy_triplealpha2 = nuclear_reaction_energy(triple_alpha2)
>>> print(energy_triplealpha1, energy_triplealpha2)
-1.471430e-14 J 1.1802573e-12 J
>>> energy_triplealpha2.to(u.MeV)
<Quantity 7.3665870 MeV>

>>> nuclear_reaction_energy(reactants=['n'], products=['p+', 'e-'])
<Quantity 1.25343e-13 J>


text:
# Rate & Accumulation (Type 1)

### The Free-response Questions

There are ten general types of AP Calculus free-response questions. This and the next nine posts will discuss each of them.

NOTE: The numbers I’ve assigned to each type DO NOT correspond to the CED Unit numbers. Many AP Exam questions intentionally have parts from different Units. The CED Unit numbers will be referenced in each post.

## AP Questions Type 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some quantities are changing. There are usually two rates acting in opposite ways (sometimes called an in-out question). Students are asked about the change that the rates produce over a time interval either separately or together.

The rates are often fairly complicated functions. If the question is on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any graphing, integration, or differentiation that may be necessary.

The main idea is that over the time interval [a, b] the integral of a rate of change is the net amount of change

$\displaystyle \int_{a}^{b}{{{f}'\left( t \right)dt}}=f\left( b \right)-f\left( a \right)$

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{{x}_{0}}} \right)+\int_{{{{x}_{0}}}}^{x}{{{f}'\left( t \right)dt}}$

where $\displaystyle {{x}_{0}}$ is the initial time, and $\displaystyle f\left( {{{x}_{0}}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

• Be ready to read and apply; often these problems contain a lot of words which need to be carefully read and understood.
• Understand the question. It is often not necessary to do as much computation as it seems at first.
• Recognize that rate = derivative.
• Recognize a rate from the units given without the words “rate” or “derivative.”
• Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{{f}'\left( t \right)dt}}=f\left( b \right)-f\left( a \right)$

• Find the final amount by adding the initial amount to the amount found by integrating the rate. If $\displaystyle {{x}_{0}}$ is the initial time, and $\displaystyle f\left( {{{x}_{0}}} \right)$  is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{{x}_{0}}} \right)+\int_{{{{x}_{0}}}}^{x}{{{f}'\left( t \right)dt}}$,

• Write an integral expression that gives the amount at a general time. BE CAREFUL, the dt must be included in the correct place. Think of the integral sign and the dt as parentheses around the integrand.
• Find the average value of a function
• Use FTC to differentiate a function defined by an integral.
• Explain the meaning of a derivative or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) what the numerical argument means in the context of the question.
• Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in the context of the question.
• Store functions in their calculator recall them to do computations on their calculator.
• If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids. Also, be ready to approximate a derivative using a quotient from the numbers in the table.
• Do a max/min or increasing/decreasing analysis.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

The Rate – Accumulation question may cover topics primarily from Unit 4, Unit 5, Unit 6 and Unit 8 of the CED.

Typical free-response examples:

• 2013 AB1/BC1
• 2015 AB1/BC1
• 2018 AB1/BC1
• 2019 AB1/BC1
• 2022 AB1/BC1 – includes average value, inc/dec analysis, max/min analysis
• One of my favorites Good Question 6 (2002 AB 4)

Typical multiple-choice examples from non-secure exams:

• 2012 AB 8, 81, 89
• 2012 BC 8 (same as AB 8)

Updated January 31, 2019, March 12, 2021, March 11, 2022

# Posts on Accumulation

One of the main uses of the definite integral is summed up (pun intended) in the idea of accumulation. When you integrate a rate of change you get the (net) amount of change. This important idea is often treated very lightly, if at all, in textbooks.

Here are a series of past posts that use, explain, and illustrate that concept.

Accumulation – Need an Amount? The Fundamental Theorem of Calculus says that the integral of a rate of change (a derivative) is the net amount of change. This post shows how that works in practice.

AP Accumulation Questions and Good Question 7 – 2009 AB 3 the “Mighty Cable Company” show how accumulation is tested on the AP Calculus exams. The “Mighty Cable Company” question is a particularly good and difficult example.

The next two posts show how to use the concept of accumulation to analyze a function and its graph without reference to the derivative. The graphical idea of a Riemann sum rectangle moving across the interval of integration makes the features of function much more intuitive than the common approach. You will not find these ideas in textbooks. Nevertheless, a lesson on this idea may help your students.

Graphing with Accumulation 1 explains how to analyze the derivative to determine when a function is increasing or decreasing and finding the locations of extreme values. By thinking of the individual Rieman sum rectangles moving across the interval the features of the function are easy to see and easier to remember. Once understood, this method will help students with their graph analysis work.

Graphing with Accumulation 2 continues the idea of using accumulation to determine information about the concavity of a function.

# Unit 6 – Integration and Accumulation of Change

Unit 6 develops the ideas behind integration, the Fundamental Theorem of Calculus, and Accumulation. (CED – 2019 p. 109 – 128). These topics account for about 17 – 20% of questions on the AB exam and 17 – 20% of the BC questions.

### Topics 6.1 – 6.4 Working up to the FTC

Topic 6.1 Exploring Accumulations of Change Accumulation is introduced through finding the area between the graph of a function and the x-axis. Positive and negative rates of change, unit analysis.

Topic 6.2 Approximating Areas with Riemann Sums Left-, right-, midpoint Riemann sums, and Trapezoidal sums, with uniform partitions are developed. Approximating with numerical methods, including use of technology are considered. Determining if the approximation is an over- or under-approximation.

Topic 6.3 Riemann Sums, Summation Notation and the Definite Integral. The definition integral is defined as the limit of a Riemann sum.

Topic 6.4 The Fundamental Theorem of Calculus (FTC) and Accumulation Functions Functions defined by definite integrals and the FTC.

Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area Graphical, numerical, analytical, and verbal representations.

Topic 6.6 Applying Properties of Definite Integrals Using the properties to ease evaluation, evaluating by geometry and dealing with discontinuities.

Topic 6.7 The Fundamental Theorem of Calculus and Definite Integrals Antiderivatives. (Note: I suggest writing the FTC in this form $displaystyle int_{a}^{b}{{{f}'left( x right)}}dx=fleft( b right)-fleft( a right)$ because it seem more efficient then using upper case and lower case f.)

### Topics 6.5 – 6.14 Techniques of Integration

Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation. Using basic differentiation formulas to find antiderivatives. Some functions do not have closed-form antiderivatives. (Note: While textbooks often consider antidifferentiation before any work with integration, this seems like the place to introduce them. After learning the FTC students have a reason to find antiderivatives. See Integration Itinerary

Topic 6.9 Integration Using Substitution The u-substitution method. Changing the limits of integration when substituting.

Topic 6.10 Integrating Functions Using Long Division and Completing the Square

Topic 6.11 Integrating Using Integration by Parts (BC ONLY)

Topic 6.12 Integrating Using Linear Partial Fractions (BC ONLY)

Topic 6.13 Evaluating Improper Integrals (BC ONLY) Showing the work requires students to show correct limit notation.

Topic 6.14 Selecting Techniques for Antidifferentiation This means practice, practice, practice.

### Timing

The suggested time for Unit 6 is  18 – 20 classes for AB and 15 – 16 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Previous posts on these topics include:

### Introducing Integration

Integration Itinerary

The Old Pump and Flying to Integrationland   Two introductory explorations

Working Towards Riemann Sums

Riemann Sums

The Definition of the Definite Integral

Foreshadowing the FTC

The Fundamental Theorem of Calculus

More About the FTC

Y the FTC?

Area Between Curves

Under is a Long Way Down

Properties of Integrals

Trapezoids – Ancient and Modern  On Trapezoid sums

Good Question 9 – Riemann Reversed   Given a Riemann sum can you find the Integral it converges to?  A common and difficult AP Exam problem

Adapting 2021 AB 1 / BC 1

Adapting 2021 AB 4 / BC 4

### Accumulation

Accumulation: Need an Amount?

Good Question 7 – 2009 AB 3

Good Question 8 – or Not?  Unit analysis

AP Exams Accumulation Question    A summary of accumulation ideas.

Graphing with Accumulation 1

Graphing with Accumulation 2

Accumulation and Differential Equations

Painting a Point

### Techniques of Integrations (AB and BC)

Antidifferentiation

Why Muss with the “+C”?

Good Question 13  More than one way to skin a cat.

### Integration by Parts – a BC Topic

Integration by Parts 1

Integration by Part 2

Parts and More Parts

Good Question 12 – Parts with a Constant?

Modified Tabular Integration

Improper Integrals and Proper Areas

Math vs the Real World Why $displaystyle int_{{-infty }}^{infty }{{frac{1}{x}}}dx$ does not converge.

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series

# Integration and Accumulation of Change – Unit 6

Unit 6 develops the ideas behind integration, the Fundamental Theorem of Calculus, and Accumulation. (CED – 2019 p. 109 – 128 ). These topics account for about 17 – 20% of questions on the AB exam and 17 – 20% of the BC questions.

### Topics 6.1 – 6.4 Working up to the FTC

Topic 6.1 Exploring Accumulations of Change Accumulation is introduced through finding the area between the graph of a function and the x-axis. Positive and negative rates of change, unit analysis.

Topic 6.2 Approximating Areas with Riemann Sums Left-, right-, midpoint Riemann sums, and Trapezoidal sums, with uniform partitions are developed. Approximating with numerical methods, including use of technology are considered. Determining if the approximation is an over- or under-approximation.

Topic 6.3 Riemann Sums, Summation Notation and the Definite Integral. The definition integral is defined as the limit of a Riemann sum.

Topic 6.4 The Fundamental Theorem of Calculus (FTC) and Accumulation Functions Functions defined by definite integrals and the FTC.

Topic 6.5 Interpreting the Behavior of Accumulation Functions Involving Area Graphical, numerical, analytical, and verbal representations.

Topic 6.6 Applying Properties of Definite Integrals Using the properties to ease evaluation, evaluating by geometry and dealing with discontinuities.

Topic 6.7 The Fundamental Theorem of Calculus and Definite Integrals  Antiderivatives. (Note: I suggest writing the FTC in this form $\displaystyle \int_{a}^{b}{{{f}'\left( x \right)}}dx=f\left( b \right)-f\left( a \right)$ because it seem more efficient then using upper case and lower case f.)

### Topics 6.5 – 6.14 Techniques of Integration

Topic 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation. Using basic differentiation formulas to find antiderivatives. Some functions do not have closed-form antiderivatives. (Note: While textbooks often consider antidifferentiation before any work with integration, this seems like the place to introduce them. After learning the FTC students have a reason to find antiderivatives. See Integration Itinerary

Topic 6.9 Integration Using Substitution The u-substitution method. Changing the limits of integration when substituting.

Topic 6.10 Integrating Functions Using Long Division and Completing the Square

Topic 6.11 Integrating Using Integration by Parts  (BC ONLY)

Topic 6.12 Integrating Using Linear Partial Fractions  (BC ONLY)

Topic 6.13 Evaluating Improper Integrals (BC ONLY) Showing the work requires students to show correct limit notation.

Topic 6.14 Selecting Techniques for Antidifferentiation This means practice, practice, practice.

### Timing

The suggested time for Unit 6 is  18 – 20 classes for AB and 15 – 16 for BC of 40 – 50-minute class periods, this includes time for testing etc.

Previous posts on these topics include:

### Introducing Integration

Integration Itinerary

The Old Pump and Flying to Integrationland   Two introductory explorations

Working Towards Riemann Sums

Riemann Sums

The Definition of the Definite Integral

Foreshadowing the FTC

The Fundamental Theorem of Calculus

More About the FTC

Y the FTC?

Area Between Curves

Under is a Long Way Down

Properties of Integrals

Trapezoids – Ancient and Modern  On Trapezoid sums

Good Question 9 – Riemann Reversed   Given a Riemann sum can you find the Integral it converges to?  A common and difficult AP Exam problem

### Accumulation

Accumulation: Need an Amount?

Good Question 7 – 2009 AB 3

Good Question 8 – or Not?  Unit analysis

AP Exams Accumulation Question    A summary of accumulation ideas.

Graphing with Accumulation 1

Graphing with Accumulation 2

Accumulation and Differential Equations

Painting a Point

### Techniques of Integrations (AB and BC)

Antidifferentiation

Why Muss with the “+C”?

Good Question 13  More than one way to skin a cat.

### Integration by Parts – a BC Topic

Integration by Parts 1

Integration by Part 2

Parts and More Parts

Good Question 12 – Parts with a Constant?

Modified Tabular Integration

Improper Integrals and Proper Areas

Math vs the Real World Why $\displaystyle \int_{{-\infty }}^{\infty }{{\frac{1}{x}}}dx$ does not converge.

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series

# Rate & Accumulation (Type 1)

### The Free-response Questions

There are ten general categories of AP Calculus free-response questions.

NOTE: The type number I’ve assigned to each type DO NOT correspond to the 2019 CED Unit numbers. Many AP Exam questions have parts from different Units. The CED Unit numbers will be referenced in each post.

### AP  Questions Type 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some things are changing. There are usually two rates acting in opposite ways (sometimes called an in-out question). Students are asked about the change that the rates produce over some time interval either separately or together.

The rates are often fairly complicated functions. If they are on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any graphing,  integration, or differentiation that may be necessary.

The main idea is that over the time interval [a, b] the integral of a rate of change is the net amount of change

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

where ${{x}_{0}}$ is the initial time, and  $f\left( {{x}_{0}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

• Be ready to read and apply; often these problems contain a lot of writing which needs to be carefully read.
• Recognize that rate = derivative.
• Recognize a rate from the units given without the words “rate” or “derivative.”
• Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$.

• Find the final amount by adding the initial amount to the amount found by integrating the rate. If $x={{x}_{0}}$ is the initial time, and $f\left( {{x}_{0}} \right)$  is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

• Write an integral expression that gives the amount at a general time. BE CAREFUL, the dt must be included at the correct place. Think of the integral sign and the dt as parentheses around the integrand.
• Find the average value of a function
• Understand the question. It is often not necessary to as much computation as it seems at first.
• Use FTC to differentiate a function defined by an integral.
• Explain the meaning of a derivative or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how numerical argument applies in context.
• Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in context.
• Store functions in their calculator recall them to do computations on their calculator.
• If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids.
• Do a max/min or increasing/decreasing analysis.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

The Rate – Accumulation question may cover topics primarily from Unit 4, Unit 5, Unit 6 and Unit 8 of the 2019 CED.

Typical free-response examples:

Typical multiple-choice examples from non-secure exams:

• 2012 AB 8, 81, 89
• 2012 BC 8 (same as AB 8)

Updated January 31, 2019, March 12, 2021

# 2019 CED Unit 8: Applications of Integration

This unit seems to fit more logically after the opening unit on integration (Unit 6). The Course and Exam Description (CED) places Unit 7 Differential Equations before Unit 8 probably because the previous unit ended with techniques of antidifferentiation. My guess is that many teachers will teach Unit 8: Applications of Integration immediately after Unit 6 and before Unit 7: Differential Equations. The order is up to you.

Unit 8 includes some standard problems solvable by integration (CED – 2019 p. 143 – 161). These topics account for about 10 – 15% of questions on the AB exam and 6 – 9% of the BC questions.

### Topics 8.1 – 8.3 Average Value and Accumulation

Topic 8.1 Finding the Average Value of a Function on an Interval Be sure to distinguish between average value of a function on an interval, average rate of change on an interval and the mean value

Topic 8.2 Connecting Position, Velocity, and Acceleration of Functions using Integrals Distinguish between displacement (= integral of velocity) and total distance traveled (= integral of speed)

Topic 8. 3 Using Accumulation Functions and Definite Integrals in Applied Contexts The integral of a rate of change equals the net amount of change. A really big idea and one that is tested on all the exams. So, if you are asked for an amount, look around for a rate to integrate.

### Topics 8.4 – 8.6 Area

Topic 8.4 Finding the Area Between Curves Expressed as Functions of x

Topic 8.5 Finding the Area Between Curves Expressed as Functions of y

Topic 8.6 Finding the Area Between Curves That Intersect at More Than Two Points Use two or more integrals or integrate the absolute value of the difference of the two functions. The latter is especially useful when do the computation of a graphing calculator.

### Topics 8.7 – 8.12 Volume

Topic 8.7 Volumes with Cross Sections: Squares and Rectangles

Topic 8.8 Volumes with Cross Sections: Triangles and Semicircles

Topic 8.9 Volume with Disk Method: Revolving around the x– or y-Axis Volumes of revolution are volumes with circular cross sections, so this continues the previous two topics.

Topic 8.10 Volume with Disk Method: Revolving Around Other Axes

Topic 8.11 Volume with Washer Method: Revolving Around the x– or y-Axis See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

Topic 8.12 Volume with Washer Method: Revolving Around Other Axes. See Subtract the Hole from the Whole for an easier way to remember how to do these problems.

### Topic 8.13  Arc Length BC Only

Topic 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled  BC ONLY

### Timing

The suggested time for Unit 8 is  19 – 20 classes for AB and 13 – 14 for BC of 40 – 50-minute class periods, this includes time for testing etc.

### Previous posts on these topics for both AB and BC include:

Average Value and Accumulation

Half-full or Half-empty

Accumulation: Need an Amount?

AP Accumulation Questions

Good Question 7 – 2009 AB 3 Accumulation, explain the meaning of an integral in context, unit analysis

Good Question 8 – or Not Unit analysis

Graphing with Accumulation 1 Seeing increasing and decreasing through integration

Graphing with Accumulation 2 Seeing concavity through integration

Area

Area Between Curves

Under is a Long Way Down  Avoiding “negative area.”

Math vs. the “Real World”  Improper integrals  BC Topic

Volume

Volumes of Solids with Regular Cross-sections

Volumes of Revolution

Why You Never Need Cylindrical Shells

Visualizing Solid Figures 1

Visualizing Solid Figures 2

Visualizing Solid Figures 3

Visualizing Solid Figures 4

Visualizing Solid Figures 5

Painting a Point

Other Applications of Integrals

Density Functions have been tested in the past, but are not specifically listed on the CED then or now.

Who’d a Thunk It? Some integration problems suitable for graphing calculator solution

Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.

2019 CED – Unit 1: Limits and Continuity

2019 CED – Unit 2: Differentiation: Definition and Fundamental Properties.

2019 CED – Unit 3: Differentiation: Composite , Implicit, and Inverse Functions

2019 CED – Unit 4 Contextual Applications of the Derivative  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 5 Analytical Applications of Differentiation  Consider teaching Unit 5 before Unit 4

2019 – CED Unit 6 Integration and Accumulation of Change

2019 – CED Unit 7 Differential Equations  Consider teaching after Unit 8

2019 – CED Unit 8 Applications of Integration   Consider teaching after Unit 6, before Unit 7

2019 – CED Unit 9 Parametric Equations, Polar Coordinates, and Vector-Values Functions

2019 CED Unit 10 Infinite Sequences and Series

# Type 1 Questions: Rate and Accumulation

The Free-response Questions

There are ten general categories of AP Calculus free-response questions, listed below. These are usually the subject of individual free-response questions. Keep in mind that two or more types may be included in the same free-response question and are also the topics of shorter multiple-choice questions. There are links to all the types here.

• Type 1 questions – Rate and accumulation questions
• Type 2 questions – Linear motion problems
• Type 3 questions – Graph analysis problems
• Type 4 questions – Area and volume problems
• Type 5 questions – Table and Riemann sum questions
• Type 6 questions – Differential equation questions
• Type 7 questions – miscellaneous
• Type 8 questions – Parametric and vector questions (BC topic)
• Type 9 questions – Polar equations
• Type 10 questions – Sequences and Series

AP Type Questions 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some things are changing. There are usually two rates acting in opposite ways (sometimes called in-out question). Students are asked about the change that the rates produce over some time interval either separately or together.

The rates are often fairly complicated functions. If they are on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any graphing,  integration, or differentiation that may be necessary.

The main idea is that over the time interval [a, b] the integral of a rate of change is the net amount of change

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

where ${{x}_{0}}$ is the initial time, and  $f\left( {{x}_{0}} \right)$ is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

• Be ready to read and apply; often these problems contain a lot of writing which needs to be carefully read.
• Recognize that rate = derivative.
• Recognize a rate from the units given without the words “rate” or “derivative.”
• Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

$\displaystyle \int_{a}^{b}{{f}'\left( t \right)dt}=f\left( b \right)-f\left( a \right)$.

• Find the final amount by adding the initial amount to the amount found by integrating the rate. If $x={{x}_{0}}$ is the initial time, and $f\left( {{x}_{0}} \right)$  is the initial amount, then final accumulated amount is

$\displaystyle f\left( x \right)=f\left( {{x}_{0}} \right)+\int_{{{x}_{0}}}^{x}{{f}'\left( t \right)}\,dt$,

• Understand the question. It is often not necessary to as much computation as it seems at first.
• Use FTC to differentiate a function defined by an integral.
• Explain the meaning of a derivative or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how numerical argument applies in context.
• Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in context.
• Store functions in their calculator recall them to do computations on their calculator.
• If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids.
• Do a max/min or increasing/decreasing analysis.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

Typical free-response examples:

Typical multiple-choice examples from non-secure exams:

• 2012 AB 8, 81, 89
• 2012 BC 8 (same as AB 8)

Updated January 31, 2019, March 12, 2021