Question

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

Answer 1

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=sinx
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