An exponential function named E [br][math]f\left(t\right)=D\left(e^{\frac{t}{t_0}}-1\right),.D-distance.unit,.t_0-time.unit[/math] [br]offered to race a quadratic function name P2[br][math]g\left(t\right)=D\left(\frac{t}{t_0}\right)^2,.D-distance.unit,.t_0-time.unit[/math]. [br]In fact, E offered to let P2 have as large a headstart as he wished - [br]E still claimed that in the long run she, E, would win.[br][br]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.[br][br]E and P2 set out to race with P2 having a headstart of 10 distance units.[br][br]After 3 units of time, E overtakes P2 at a distance of ~19 distance units.[br][br]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.[br][br][[i][b][size=85]Set the distance sliders by clicking on the arrow icon on the menu bar. [br][br]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. [br][br]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.[/size][/b][/i]][br] [br][br]What can you say about the rate of change of the functions[math]e^x-1[/math][size=200] [size=100]and [math]x^n[/math] ?[/size][/size][br][br]How many roots does the equation [size=100][math]e^x-1=x^n[/math][/size] have?[br]How do you know?