Error Bounds on Taylor Series

Description

This is an illustration of the error bounds on the Taylor Series using first and second order extrapolation. 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 but not at some other point . 1. The play button, >>, will advance the illustration one step. 2. Step 2 will show the second point where the function value is desired. Both the and points can and should be moved at each step to see what is happening. 3. Step 3 shows a linear extrapolation from to . This gives an approximation of . 4. Step 4 adds a line at the maximum slope of the function between and . The function can not rise any faster than this value between and . 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 and . 6. Step 6 shows the value of must be between these lines. Because of the Intermediate Value Theorem, the slope of the line connecting and must be the slope at a point , , between and .

Continue on Lower Graph

7. Step 7. The second order approximation for is shown. 8. Step 8. This point will be below the extrapolation using the maximum value of between and 9. Step 9. The point will be above the extrapolation using the maximum value of between and 10. Step 10. A similar argument as in the linear case will place the second derivative of the quadratic function connecting and somewhere between and which could be at a different point .