### 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).

## 2 Comments:

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.

Part 3 is an

excellentquestion, 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 downmeans 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!

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