How can you find the value of sin x without using the sine feature on a calculator? There is obviously a need to find approximations for functions and differentiation provides the methods. The tangent gives an approximation close to the point where the tangent is drawn. Better approximations require a knowledge of Taylor polynomials.

activity page - Numerical methods - small changes - Taylor polynomials

calculator for Taylor polynomials - Mathserv Calculus Toolkit
Enter the function, number of terms and centre. A graph is available showing the function and the polynomial approximation.

calculator for Taylor polynomials - WIMS
Enter the function and centre. This function calculator produces the Taylor polynomial of order 5, 8, 10, 15, 20, 30, 50 or 100.

Linear Approximation and Error Estimation - Finite Math & Calculus
notes and examples

Taylor Series - James Sellers
notes and examples