Mathing…

Starcraft 2 Wings of Liberty and Calculus
Starcraft 2 Wings of Liberty and Calculus: UPDATE 1

This is an update to an earlier post.

Let’s see if we can get some information from the graph. Below is a screenshot of the GeoGebra applet.

Here’s a few questions:

• Which approximation gives the best estimate of total resource collected? Why?
• If the graph represents the instantaneous resource collection rate (in the context of this game), which sum will provide the best approximation?
• If the graph represents the average resource collection rate, which sum will provide the best approximation?
• The red line shows an increase followed by a decrease, what does that mean in context?
• What happens to the approximation as the number of columns is decreased? What happens when it is increased?
• Is there information that can help us determine if the graph displays instantaneous collection rate or average collection rate?
• Whats your best estimate about the amount of resources collected?

Below are 2 screenshots that show the amount of unspent resources by each team on the top right and the amount spent resources in the menu on the top left.

Let’s check our answers:

• Which sum provided the best approximation?
• What are some sources of error?

Now let’s connect the graph to what’s happening:

• Why does the graph of resource collection rate rise and fall?
• Why does the graph have humps and plateaus? What do they represent? (hint)
• How did the red player have a steeper rise on the second hump (after first plateau) than the blue player? (hint)
• What caused the drop-off in the red graph near the end? Does the drop off mean the red player is losing resources or gaining less resources?
• In the beginning why did the blue graph have dips in its graph? (hint)

Here’s something for the Calculus teachers. And no, it’s not about how Starcraft 2 is going to drop the AP scores for the boys (and some girls) this year. Even though, there is a good chance one of your students bought one of the roughly 620,000 copies sold in US on day one.

Names have been blocked to protect the innocent.

I do something similar to this when I introduce limits. It gives students a practical understanding of what limits are and how to evaluate them.

Part 1 – When Limits Exist

Evaluate the following and justify your solution

If latex does not display properly (you might have to enable javascript) or if you’re reading from Google Reader skip to the text version

1. 
2. 
3. 
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. If , what is
16. If , what is
17. If , what is
18. If , what is

Text Version

1. What is zero to the zeroth power? What is zero raised to the power of zero?
2. What is the limit as x approaches zero of x to the zeroth power?
3. What is the limit as x approaches zero of zero to the x power?
4. What is the limit as x approaches zero of x to the xth power?
5. What is zero to the power of a divided by zero to the power of a?
6. What is zero to the power of the quantity a minus a?
7. What is a to the power of zero divided by a to the power of 0?
8. What is a to the power of the quantity 0 minus 0?
9. What is the limit as x approaches zero of x to the power of a divided by x to the power of a?
10. What is zero to the infinity power?
11. What is the limit as x approaches zero of zero to the power of one divided by x?
12. What is the limit as x approaches zero of xth power of zero?
13. What is the limit as x approaches zero of x to the power of one divided by x?
14. What is the limit as x approaches zero of xth power of x?
15. If zero to the zeroth power is equal to a, what is log base zero of a?
16. If zero to the zeroth power is equal to a, what is the limit as x approaches zero of log base x of a?
17. If zero to the zeroth power is equal to a, what is zeroth root of a?
18. If zero to the zeroth power is equal to a, what is the limit as x approaches zero of xth root of a?

Note: This post is the first of a 5-part series. The original post entitled “How understanding holes in rational functions help you graph piecewise defined functions on TI graphing calculators” got a little long and probably had more information on a single post than anyone would care for. So now you will see five where there was one.

So what am I talking about?

No, I’m not talking about drilling holes into TI graphing calculators. This is not an episode of Will it Blend? or a clip of Office Space (or if you prefer the Family Guy version). I’m talking about points on the graph that are undefined where open circles are used to indicate its presence (absence?).

Before we begin, let’s discuss holes in rational functions. We typically cover this topic in Pre-Calculus when discussing difference quotients and rational functions or in Calculus when working with slope of tangent lines and working with limits. In Pre-Calculus, we typically brush over the topic (at least in the textbook we use) and focus mainly on the vertical, horizontal, and slant asymptotes produced by rational functions. At best, we discuss how indiscriminately canceling out factors is dangerous (something we do in Algebra 2 without much regard), and that the canceling out of factors is allowed only on the caveat that the factor is never 0.

But let’s get back to our topic.

Suppose you were asked to graph the following function

We typically begin by doing a little factoring

which can be rewritten as

but because

for all values of x except 1 (where it will be undefined)

it is equivalent to simply

for all values of x except 1 (where there’s a hole since it’s undefined).

In other words, the graph of is the same as the graph of at all points except at .

On a TI-83 graphing calculator, if we enter the following equation

and we get the following graph (notice the hole at x=1)

(Note: If you’re not getting the graph above, use ZOOM-> 4:ZDecimal, it’ll make every pixel correspond to 0.1 on the screen and will force 1.0, where the hole is, to be part of the graph)

If we want additional holes in the function, we simply multiply our function by an additional term where a is the place we want the holes in the function.

For example, if we wanted additional holes at x=2 and x=-3 we simply multiply our function as follows

which simplifies to

Entering the function into our calculator

produces the desired results (notice the holes at x=1, x=2, and x=-3)

So a logical next step is, can we restrict the domain of our function to only positive numbers by making all negative numbers and zero disappear from the screen?

To do that we just multiply our function by

(-4.7 is the edge of our viewscreen when using ZDecimal Zoom). This way, every point on the viewscreen to the left of 0 is essentially a hole and therefore not graphed.

This works great until you change to a different zoom where none of the points (-0.1, -0.2, … , -4.7) is represented by any of the pixels in the viewscreen. To verify this, change the zoom to ZStandard and the holes that we saw in the earlier parts disappear.

Entering each factor again and again is tedious, time consuming, and prone to errors. Plus it would take the calculator a long time to evaluate every point. Who has that kind of time?

Luckily, there is a better way.

Proceed to Domain Restrictions on TI Graphing Calculators (Part 2/5)

UPDATE: Updated clip links.