E & P have a race - a fable about rate of change
An exponential function named E
offered to race a quadratic function name P2
.
In fact, E offered to let P2 have as large a headstart as he wished -
E still claimed that in the long run she, E, would win.
P2 always starts running slowly but speeds up as the race goes on. In fact, P2's speed is proportional to the distance he has run.
E and P2 set out to race with P2 having a headstart of 10 distance units.
After 3 units of time, E overtakes P2 at a distance of ~19 distance units.
Embarrassed, and coming from a large family with many older brothers who run [named P3, P4, P5, etc.] P2 challenged E to run against any of his older brothers. E agreed and also offered to give the older brother any headstart he wanted.
[Set the distance sliders by clicking on the arrow icon on the menu bar.
Set the distance scale by clicking on crossed arrow icon on the menu bar, and then right clicking on the distance axis and dragging it.
Set the time scale by clicking on crossed arrow icon on the menu bar, and then right clicking on the time axis and dragging it.]
What can you say about the rate of change of the functions and ?
How many roots does the equation have?
How do you know?