Tuesday, November 22, 2005

Solution to Episode 1: THe Cerebus

1) When is the heads of the dog not moving?

It is where the velocity function is equal to 0.
0.5x^3 - 3x^2 + 0.25x + 5 = 0
It is 0 where x = 1.5577 and 5.5906.

2) At what time does our young man have to be most careful? (*hint* Steep tangent slopes on parent funtion.)

In other words, when is the heads of the dog moving the fastest? At a very high tangent slope on the parent function. Steep tangent slopes on parent function is where the parent function have inflection points. To find inflection points of a parent function you look at its second derivative, in this case its acceleration function. Find zeroes of the acceleration function and create a line analysis to show whether it is a inflection point or not. (Changes sign at the root).
Our young man have to be most careful when x=3.9579.

3) What interval(s) is the dog's heads slowing down?
The heads are slowing down when a(x) and v(x) are opposite signs from each other. Find the zeroes of each function and create a line analysis shadowing the other. Find where a(x) and v(x) have opposite signs, this is where the dog's heads are slowing down.
This occurs when (0.0421, 1.5577) u (3.9579, 5.5906).


Blogger Ms. Armstrong said...

Congratulations on successfully creating a fascinating use for calculus. The creativity of your story will no doubt inspire your classmates to deepen their own understanding of mathematics. Well done Sarah.

9:52 PM  
Blogger Mr. Kuropatwa said...

Part 3 is an excellent question, but your solution is incorrect. When the velocity is negative, the heads are moving in the "opposite" direction; not slowing down.

To find out when they're slowing down means you need to find the intervals where the velocity and acceleration have opposite signs -- think about that. ;-)

I really like this question -- it's a real thinker!

12:43 AM  

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