## 102 – Power Series on Desmos

Power Series work in AP Calculus BC.

$\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}-...$

Process: Since it’s a infinite series, look at partial sums to get an idea what this graph looks like.
So look at
$y_1=\frac{x}{1!}$
$y_2=\frac{x}{1!}-\frac{x^3}{3!}$
$y_2=\frac{x}{1!}-\frac{x^3}{3!}+\frac{x^5}{5!}$

Perfect time to use technology.

## Texas Instruments Method

Go to y1. Enter in $y=x$.
Graph.
Wait 3-5 seconds.
Go to y1. Subtract a $\frac{x^3}{3!}$.
Graph.
Wait 3-5 seconds.
Go to y1. Add on a $\frac{x^5}{5!}$.
Graph.
Wait 3-5 seconds.
Go to y1. Subtract a $\frac{x^7}{7!}$.

REALLY crappy resolution. Awful zoom system. Where’s the factorial sign? Hopefully you remember what the previous graphs looked like. Lots of waiting. Ugh.

## Desmos: version 1

Graph $y=x$
(NO WAIT STEP)
Subtract a $\frac{x^3}{3!}$.
(STILL NO WAIT STEP)
Add on a $\frac{x^5}{5!}$.
Subtract a $\frac{x^7}{7!}$.

Great!

## Desmos: version 2

Students teaching teachers: Have one your students find this out for himself, and remark that they can enter in the entire series, but he’s having trouble finding the infinity sign. SLIDERS!
Whoa. (That seemed like it should be much harder to type in. Took me 5 minutes to type in the original latex code at the beginning of this post. Took me < 1 minute to actually enter the series into desmos. I had to help 1 student out of 15. That’s it. Wow easy.)

# Watch the power series create sin(x) step by step by moving a slider. LIVE!

We live in good times.

## 101 – Reciprocal Trig Functions

I’m trying to build some intuition (is that even possible??) with reciprocal trig functions, so they took the graph of sine (cosine and tangent) and found the 17 exact function values that they know from 0 to 2pi. I let them know that their calculators were now broken and they can only perform division. With the broken calculators they had to find the graph of cosecant. Pretty basic stuff, but I like all the different skills that are being reinforced by this task.
Here’s an example of cosine and secant:

Some of them broke their brains (in a good way) when they were creating the graph of cotangent. By then, they understood that 1/0 was undefined, so there was a vertical asymptote at that location (not that a vertical asymptote always happens with 1/0). But when they worked together and got to 1/undefined = 0, you could see the look of confusion. All the division by 0 and undefined’s was a great preview for limits. Neat.

## 100 – Chaos Game

More to be reported on later, but did the chaos game with a couple of classes.

## 99 – IB Sector Area and Arc Length Questions

Worked on some IB Sector Area and Arc Length Questions in Pre Calc H. These kids are one year off from the start of our IB program, they won’t be able to take the IB tests. But they offer some really interesting questions. I doubt IB will get upset, so here is the link to the 5 questions that we worked on: these are some great questions. Applying right triangle trig, arithmetic sequences, arc lengths, inverse trig, radians, and tangent lines to circles all in one question (first question).

Here’s the kids working… with some interesting choices for variables :-).

## 98 – Monster Induction Proof

Love the whiteboard in the back. This was a tough proof for the pre-calc h group. Sticking points: understanding the summation form with the three variables, i,n, and k, and manipulating the algebra to get what was necessary.

## 97 – Binomial Expansion (Avoiding off by one error)

Here what some students staying after came up with to avoid the typical “off-by-one” error that many students get tripped up on.

## 96 – Series Convergence and Divergence Tests

There are a boatload of convergence/divergence tests in BC Calc. The students ask me when to use one test over another. I have no solid answer for them (I don’t mind that too much). “Whatever works” is the best I can come up with. There are some tips here and there, but I don’t know if there are enough patterns to bother. Thoughts?

## 95 – Hexagon inscribed in a Circle

Warmup after winter break in PreCalc H. Find the area and perimeter of one section of a hexagon inscribed in a circle. Sector Area, and Arc Length.
Whiteboarding.

Gallery Walk of whiteboards.

Started with this activity with the pre-calculus students. The group with the closest measurement to 1 radian got a prize, as measured by one of Jen Silverman’s Radian Protractor.
Here’s my entry (done with a straight edge and rubber band in less than 30 sec).

We had a lot of discussion regarding how degrees are a human created measurement, while radian measure is a measured quantity.

## 93 – Slope Fields

/* field for y'=xy (click on a point to get an integral curve) */
plotdf( x*y, [x,-2,2], [y,-2,2]);