posted by Michael .
Use linear approximation to show that each function below can be approximated by the given expression when |x| is small
i.) sinx = x
ii.) e^x = 1+x
using Taylor series,
sinx = x - x^3/3! + x^5/5! - ...
e^x = 1 + x + x^2/2! + ...
when |x| is small, the higher powers vanish.
Or, just using calculus,
y' = cosx
at x=0, y=0, y'=1
so y can be approximated by the line through (0,0) with slope 1, or y=x
similarly for e^x
The line through (0,1) with slope 1 is y = 1+x