The Maclaurin series is used to provide a polynomial approximation to complicated functions. Move the slider to change the number of terms in the series (1=constant, 2=linear, 3=quadratic and so on). The blue line shows the mathematical function to be approximated; the red line shows the Maclaurin series approximation and the black line shows the difference between the original function and the approximation.[br][br]Type your own functions into the input box to see how well the Maclaurin series approximates them with a given number of terms.[br][br]

See how in general, if you increase the number of terms that the approximation becomes more applicable over a wider range of x.[br][br]Can you explain why increasing the number of terms from 2 to 3, or from 4 to 5, for sin(x), produces no improvement?[br][br]Is this also true if you approximate exp(x) ?