A useful integral for kids learning calculus

My son is in the middle of a long review for the BC Calculus exam in May. Right now he’s doing some integration practice and came across this integral today:

We didn’t get a chance to talk about it too much this morning, but we did review the problem when he got home tonight.

First we I asked him to approach the problem as he did this morning -> by trying to evaluate the integral. The practice integrating rational functions turned out to be useful:

Next we took a more geometric approach:

Finally, I wanted to show him how you could see that the integral was zero from the u-substitutions that he made in the first video. Even though this method wasn’t really the best one for this particular integral, I still wanted him to see how the algebra worked out:

Using an integral problem from the 2012 BC Calculus exam to practice with numerical integration

We had a snow day today and I was able to use some extra time with my son to talk about numerical integration techniques. I’d guess that I hadn’t seen the topic in at least 25 years, so it was fun to teach and review at the same time!

This evening I had my son review the ideas using an integration problem from the 2012 BC Calculus exam. I started by having him evaluate the integral exactly with no numerical integration techniques:

Next we used the trapezoidal rule and compared the answer to the exact answer –

The one thing I learned today was that the midpoint rule was roughly twice as accurate as the trapezoidal rule. Here my son used the midpoint rule and we do find that the answer is closer to the true value than the answer we found in the last video:

Finally, we used Simpson’s rule. Despite one little mistake in the middle, we ended up finding an answer that was surprisingly close to the exact asnwer.

Studying techniques of integration with 5 integrals

We had a 2 hour delay for school today, so we had a little extra time this morning to talk calculus. My plan was to spend all of March on techniques of integration, but we are a little ahead of schedule having already covered integration by parts and partial fractions.

Today my son moved on to a “techniques of integration” section in Stewart and was looking at a bunch of integrals without knowing what techniques to use. After he worked for a bit we talked about the first 5 problems from the section:

(1) A substitution integral with trig functions

(2) An integration by parts problem:

(3) A slightly disguised arcangent problem

(4) A sort of silly u-substitution problem

(5) A partial fractions problem

Sharing the Heaviside cover-up method for partial fractions with my son

I got an interesting question on twitter yesterday when I mentioned that my older son was studying partial fractions:

I’d not heard of this method before and though I thought it was interesting I didn’t want to cover it. Changed my mind this morning, though, because I thought it would be fun to show my son that there were interesting ideas that led to non “plug and chug” solutions to partial fraction solutions.

Here’s the introduction:

Here’s my son using the method to solve a problem that we’d worked through yesterday:

Always happy to learn about a new math idea from twitter!

Working through Jennifer Vibber’s integral problems

Last night I tweeted about starting partial fractions with my son an got a really neat response from Jennifer Vibber:

Tonight I decided to have my son work through Vibber’s integrals and talk about the technique of integration needed to solve the problem. He’s been away hiking in the White mountains for a week, so calculus wasn’t necessarily the top thing on his mind. Even with that break, though, it was interesting to see how he approached all of these problems and working through the list made for a great integration review:

Once we were done we revisited two of the problems to see if he could figure them out with a little bit more time. Here’s the first one:

This one was a bit more complicated, but he ended up finding a substitution that worked and led to a really nice solution:

Thanks to Jennifer Vibber for sharing these nice problems last night!

Sharing a fun calculus example from John Carlos Baez with my son

Saw a neat tweet from John Carlos Baez last week:

It was an nicely timed tweet for me because my son is beginning a long review of calculus ideas this year. Tonight I finally got around to sharing the idea with him.

We began by talking about some basic properties of the function:

Next we talked about how you could approach integrating the function and then used Mathematica to help with the polynomial division:

Finally, we went to the whiteboard to work through the integral and talk about the nice surprise:

I really like this integral. It is both a neat “fun fact” and a great example to share with kids learning calculus.

Sharing Jez Swanson’s amazing Fourier transformation program with kids

I saw an incredible tweet from Jez Swanson yesterday:

The program makes the ideas behind Fourier transformations accessible to kids and I decided to share the program with the boys this morning. So, I had each of them play around with it on their own for about 10 to 15 min. Here’s what they thought was interesting. (sorry for all of the sniffing – I’ve got a cold that’s been kicking my butt for the last few days):

(1) My older son who is in 9th grade:

(2) My younger son who is in 7th grade – it is really fun to hear how a younger kid describes advanced mathematical ideas:

I think Swanson’s program is a great program to share with kids – feels like at minimum it would be fantastic to share with kids learning trig.

Reviewing 7 multiple choice questions from the 2012 BC Calculus exam

I’ve been teaching my son calculus this school year with the goal of having him take the BC Calculus exam in May of 2019. We’ve finished most of the course material (the main gap is techniques of integration which I’ve put off a bit) and I had him take the publicly available 2012 BC Calculus exam this past weekend.

He missed 7 questions on the multiple choice part of the exam, and we went over those questions this morning.

The test itself is here on the AP website:

The 2012 Public BC Calculus Exam

Our discussion of the 7 multiple choice problems he missed is below . . . and, dang, the actual questions are a little out of focus in the videos. I’ll upload pics before each question to clear up that little technical glitch:

Question #4:


Here’s our talk through this problem – I think the error here was simply being a little careless during the exam:

Question #14:


Again, I think the error was a little careless during the exam.

Question #20


This question is asking about integration using the technique of partial fractions, so a topic we have not yet covered. I showed him the basic idea of partial fractions and also how he could have estimated the value of this definite integral.

Question #24


I like this question. It touches on integration by parts and also the fundamental theorem of calculus. Even though he left it blank on the exam, he was able to find the solution here:

Question #85


This question is a “related rates” question. It still gave him a little trouble today.

Question #90


I like this question – it hits on several important properties of convergent / non-convergent series. Although he got this problem wrong on the exam, he does a nice job talking through the ideas in the problem here:

Question #92


This question might be my personal favorite on the exam. The underlying concepts are basically even / odd functions and the definition of the derivative.

My son is able to recognize why two of the properties listed in the question do not have to be true, but recognizing why the 2nd property has to be true was still a little out of reach for him. Definitely a fun problem to talk through, though (and listening to my explanation for the solution as I publish this blog post . . . I definitely wish my explanation was a little less careless):

Sharing two of Patrick Honner’s calculus ideas with my son

I’ve been looking forward to sharing two calculus ideas from Patrick Honner with my son for the last week. We were, unfortunately, a little rushed when we sat down and there are a couple of mistakes in the videos below. Even though things didn’t go perfectly, I really enjoyed talking through these ideas.

Here’s the first idea – a twist on integration by parts that Honner learned from the British mathematician Tim Gowers:

Here’s the second idea – a fun surprise when a student made a creative substitution in a integration problem:

So, I stared the project by talking about how to integrate arctangent without using integration by parts:

In the last video we found a possibly surprising connection between arctan(x) and ln(x). Here I introduced the integral from the 2nd Patrick Honner tweet above and showed my son how you solve that integral using partial fractions. The point here wasn’t so much the integral, but rather to show that ln(x) showed up in an integral similar to the one we looked at in the first part of the project:

How I showed the technique that Honner’s student used (though I goofed up the substitution, unfortunately, using u = ix rather than x = iu. By dumb luck, that mistake doesn’t completely derail the problem because it only introduces an incorrect minus sign):

Now that we’ve found two connections between arctan(x) and ln(x), we went to Mathematica to see if the two anti-derivatives were really the same. It turns out the are (!) and we got an even bigger surprise when we found that Mathematica uses the same technique that Patrick Honner’s student used 🙂

Also, in this video I find a new way to introduce a minus sign by reversing the endpoints of an integral . . . . .

A fun introductory Taylor series exercise with Sin(x) I saw on Twitter last week

Last week I saw a neat “fun math fact” via a Matt Enlow tweet:

By happy coincidence my older son is studying Taylor Series this week, so I thought it would be fun to talk through the problem.

Here’s the introduction:

My son had some nice ideas about how to approach the problem in the last video, so next we went to the white board to work out the details:

Finally, I asked my son to finish up the details and then asked him for a sort of number theory proof of why 180 multiplied by an integer with all digits equal to 5 was always close to a power of 10:

Definitely a fun little problem – definitely accessible to students learning some introductory calculus.