This is an illustration of the error bounds on the Taylor Series using first and second order extrapolation. [br]Initially a function is shown with a point on the x axis that can be moved. The assumption for the Taylor Series is that the function values and derivatives can be easily calculated at the point [math]a[/math] but not at some other point [math]b[/math].[br]1. The play button, >>, will advance the illustration one step.[br]2. Step 2 will show the second point where the function value is desired. Both the [math]a[/math] and [math]b[/math] points can and should be moved at each step to see what is happening.[br]3. Step 3 shows a linear extrapolation from [math]a[/math] to [math]b[/math] . This gives an approximation of [math]f(b)[/math].[br]4. Step 4 adds a line at the maximum slope of the function between [math]a[/math] and [math]b[/math]. The function can not rise any faster than this value between [math]a[/math] and [math]b[/math].[br]5. Step 5 adds a line at the minimum slope of the function between a and b. The function can not fall any faster than this value between [math]a[/math] and [math]b[/math].[br]6. Step 6 shows the value of [math]f(b)[/math] must be between these lines. Because of the Intermediate Value Theorem, the slope of the line connecting [math]f(a)[/math] and [math]f(b)[/math] must be the slope at a point , [math]c[/math], between [math]a[/math] and [math]b[/math][br].

7. Step 7. The second order approximation for [math]f(b)[/math] is shown.[br]8. Step 8. This point will be below the extrapolation using the maximum value of [math]f''[/math] between [math]a[/math] and [math]b[/math][br]9. Step 9. The point will be above the extrapolation using the maximum value of [math]f''[/math] between [math]a[/math] and [math]b[/math][br]10. Step 10. A similar argument as in the linear case will place the second derivative of the quadratic function connecting [math]f(a)[/math] and [math]f(b)[/math] somewhere between [math]f''_{max}[/math] and [math]f''_{min}[/math] which could be at a different point [math]c[/math].