Let f(x) = e^x and g(x) = sin(lnx). Find equations for two distinct parallel lines l1 and l2 where; l1 is tangent to f at x = 0 and l2 is tangent to g.

Answer 1

f'(x) = e^x
f'(0) = 1

g'(x) = cos(lnx)/x
cos(lnx)/x=1
cos(lnx)=x
x = 0.0089,0.2745

So, you want lines with slope=1 passing through
(0,1),(0.0089,0.9933),(0.2745,-0.9610)

y-1 = x
y-0.9933 = x-0.0089
y+0.9610 = x-0.2745

See the graphs at

http://www.wolframalpha.com/input/?i=plot+y%3Dsin(lnx),y%3De%5Ex,y%3Dx%2B0.9910,+y%3Dx-1.2355+for+0.0080%3Cx%3C1

Answer 2

oops. there is also another solution at x=1 since cos(log1)/1 = cos(0)/1 = 1/1 = 1