## Loser66 one year ago For what value of b is the line y = 10x tangent to the curve $$y= e^{bx}$$ at some point in the xy plane? Please, help

1. Loser66

Options: 1) 10/e 2) 10 3) 10e 4) $$e^{10}$$ 5) e brb

2. freckles

$y=10x \\ \text{ means } y'=10 \text{ at some point } (m,n) \\ \text{ so we have the following } \\ n=10m \\ 10=be^{bm} \\ n=e^{bm} \\ \\ \text{ multiplying } n=e^{bm} \text{ on both sides by b gives us } bn=be^{bm} \\ \implies nb=10 \implies b=\frac{10}{n} \implies b=\frac{10}{10m}=\frac{1}{m} \\ \text{ going back to } n=e^{bm} \text{ we have} n=e \\ \text{ so } m=\frac{n}{10}=\frac{e}{10} \text{ and so } b=\frac{1}{m}$ one last thing to do and I leave it to you

3. freckles

http://www.ets.org/gre/subject/about/content/mathematics if anyone is curious these are gre practice questions loser got me wanting to take this practice test buy I kind of got bored with them :p

4. freckles

Not trying to imply they are easy just got bored testing myself :p

5. Loser66

Thanks a lot. :)