Math geeks only

[Pirate Bill]

Solving for y1=x*7 at x=3; y=21.
Solving for y2=10^(x-1) at x=3; y=100.

Slope:
Derivative of y1: y1'=7 at x=3 (and all values of x)
Derivative of y2: y2'=10^(x-1); at x=3, y2'=100
Second derivative of y2: y2''=10^(x-1)

y1<y2 at x=3
y1'<y2' at x=3
y2'' is positive for all values of x indicating that y2' increases as x increases
therefore y1<y2 for all integers where x>2

That's way more complicated than just a pretty picture showing the plot of each equation on one graph, but I think it satisfies the burden of proof.

[Ghoulish Delight]

Originally Posted by Pirate Bill

That's way more complicated than just a pretty picture showing the plot of each equation on one graph, but I think it satisfies the burden of proof.

And also a slightly more formal way of saying "linear growth vs. exponential growth," got it. I think that's good enough for me.

But where do you get y2'=y2? Isn't the derivative of 10^x=10^x ln (10)? (too brain fried to figure out how the "x-1" affects that)

[Pirate Bill]

Originally Posted by Ghoulish Delight

But where do you get y2'=y2? Isn't the derivative of 10^x=10^x ln (10)? (too brain fried to figure out how the "x-1" affects that)

Honestly, I had forgotten how to do the derivative of 10^x so I looked it up. And the reference I used said it's 10^x log(10). Since log(10) is 1 I simplified. So I guess it's wrong.

Yes, it was a more convoluted way to say exponential growth, but it essentially proves that it is exponential growth.

Why did the ducks cross the bridge? Are they European or African ducks?