- Dr. Jack L. Jackson II
Enter a formula for a function f(x) in the input box. Adjust the value for an x-value, a, via its slider or input box. What we want to do is to locally approximate the function with a polynomial function. We want this approximation to be perfect at (a, f(a)) and it should be a very good approximation near that point. We want the graph of the polynomial to follow the graph of the original function as closely as possible near the point (a, f(a)). This is a generalization of using the tangent line to approximate a function. The best way to do this approximation for a polynomial of degree n is to be sure that the function goes through the point: p(a) = f(a), and that all derivatives of order 1 through n at x = a are the same for both the approximating polynomial function and the original function. Here is the resulting formula for creating these polynomials. These polynomial functions are called Taylor Polynomials. In the app you can show Taylor Polynomials of degree 1-6 individually by checking their checkboxes. You can also show the Taylor Polynomial of degree n (in red) controlled by the slider or input box for n. You can check the checkbox for approximation to approximate the function at a point c with the nth degree Taylor Polynomial. Note: There seems to be a small bug in the program. When n = 1 the red Taylor polynomial formula is computed correctly, but it is graphed incorrectly. The green version found by checking the box for n = 1 gives the correct graph.