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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

  • Math -

    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

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