Deja Vu All Over Again

Today is the birthday of Franz Aepinus born in Rostock, Germany, 1724. Aepinus made contributions in the area of electricity and magnetism.

Today's quote is by Niccolo Tartaglia. This quote is a poem written to Jerome Cardan. In this poem, Tartaglia reveals to Cardan the secret to solving a cubic equation.

"When the cube and the things together

Are equal to some discrete number,

Find two other numbers differing by this one,

Then you will keep this as a habit,

That their product will always be equal,

Exactly to the cube of a third of the things.

The remainder then as a general rule

Of their cube roots subtracted

Will be equal to your principal thing."

In my calculus classes, I talk about my dog, Rosie, and she has become quite a celebrity. Last year and again this year, I showed them a graph that I entitled My Walk With Rosie. I assigned them the task of detailing the calculus concepts that are embedded in the graph. I have listed those responses. The responses are both mathematical and creative. One student included cartoons of Rosie and myself.

Rosie patiently waiting for a walk.

Rosie spending some quality time with my son.

The graph of My Walk With Rosie.

1) The limit of f(t) as t approaches 45 from the right is 150.

2) At 45 minutes there is a relative maxima (vertex), where the derivative is zero. This represents the point at which you turned around to return home because for a moment you were not moving away from or toward you house.

3) There are x-intercepts at (0, 0) and (60, 0). These points represent times when you were at your house. Because there are only two points, we can infer that you left at (0, 0) and returned at (60, 0) and did not go to your house at anytime between those two points.

4) There are two points on this graph (30, 50) and (45, 150). We can find a secant line with those two points. The slope of this secant line is 20/3 or 6 2/3 feet per minute. Because his rate (slope) is positive we can tell that you’re moving away from your house.

5) After coming home from a grueling day of teaching, Kruger came home and decided to take his dog, Rosie, on a walk. Starting at the end of his driveway, he started walking at an average rate of 3 1/3 feet per minute^* for a duration of 15 minutes. This took him a total of 50 feet. Being too tired to continue, he had to stop and sit down on the curb. They sat on the curb for 15 minutes when Rosie got up and started chasing a passing car. Alarmed, Kruger got up and pursued his runaway dog at the slow average rate of 6 2/3 feet per minute^*. Even though this was double his previous pace you would think that if he was concerned about Rosie, he would pick up the pace. He grew discouraged after 100 feet and 15 minutes and turned around to go home to call the police and report Rosie missing. On the return trip of 150 feet he averaged a speed of 10 feet per minute^* and arrived home after 15 minutes. When he got to his front door he found Rosie there waiting for him.

^* Secant lines can be used to find his average speeds at different intervals of his walk. The part of the graph with a slope of zero means he is not walking.

6) I can tell you took a break from 15 minutes to 30 minutes because you stayed at 50 feet the whole time. The slope is zero because the tangent line is a horizontal.

7) I can tell your started off slow and once you got closer to 15 minutes you moved faster. The slope of the tangent line is steeper towards the end.

8) 150 feet is the furthest you walked. This shown at the top of the parabola.

9) It took you 45 minutes to walk Rosie the first half of your walk and 15 minutes to walk her back. The graph is increasing the it starts decreasing that's how I can tell.

10) It was a find day outside an on October 12, 2014. The sun was out and the chill of autumn was in the air. Being that math nerd that he was, Mr. Kruger decided to make a very well drawn graph of his little adventure. With his handy dandy calculator in hand and his sidekick Rosie by his side, Kruger began his adventure.

Mr. Kruger started off his walk at a good pace. After 15 minutes, he had already gone 50 feet! This fast pace was due to the fact that Rosie saw a squirrel ahead and she would stop at nothing to chase it. Well, Rosie did stop for the squirrel ran up an oak tree to seek refuge.

Rosie's sudden outburst almost caused Mr. Kruger to drop his calculator right down the storm drain near by. Thankfully, Rosie was jumping after the squirrel at the same time the calculator air born and their trajectories collided and the calculator deflected off of Rosie's side. The calculator then soared back into Mr. Kruger's waiting hands.

Mr. Kruger wept at the sight of the mathematical miracle that had happened here. He tried to go on with his walk but the heat from his tears had caused the poor man's glasses to fog to the point where he couldn't see. While waiting for his glasses to defog, he decided to figure out the equation for the first 15 minutes of his walk. After fiddling with this calculator for 12 minutes, he did some math like this: Start point (0, 0), End point (15, 50), m (slope) = (50 - 0)/(15 - 0) = 50/15 = 10/3, 50 = 10/3(15) + b, 50 = 50 which means the secant line of line for the first 15 minutes of his journey was: y = 10/3x. 

Kruger was about to continue on his walk when he realized he had just wasted 15 minutes standing still, finding no challenge in determining an equation to the next section of his walk. The equation was y = 50. For those 15 minutes he was nothing but a sad constant.

Mr. K quickly got over the fact that he was a constant and continued on his merry way. After another 15 minutes, Kruger stopped to pick up a coin on the ground - after all it was heads up - and stuck the coin in his pocket. Little did he know that this move would alter the equation of his walk for the day.

Mr. Kruger, being the math genius he is, quickly figured out the equation to this piece of his walk was y = -2/9(x - 45)² + 150.

Kruger then started to increase his speed. After all, he had been walking Rosie for 45 minutes now. Mr. Kruger pushed with all his might to go another 15 minutes. He was out of breath and huffing and puffing and wheezing and squeezing and Rosie may have had to drag hime a bit but he made it a whole hour.

Mr. Kruger sat in his kitchen wondering what else he could conclude from his walk. He noticed his graph was continuous. Since time never stops, there was no holes or asymptotes. Mr. Kruger then decided to dig a little deeper and find the derivative portions of his graph.