Taylor Series Approximation Simulation

A Taylor polynomial is built entirely from what a function is doing at a single point: its value there, its slope there, its curvature there, and so on. Each derivative fixes one more coefficient, and each new term bends the polynomial to match the curve a little further out. Two things follow. The approximation is always best at the centre and gets worse as you walk away from it, so the useful question is not whether it works but how wide the region is where it works. And for some functions that region has a hard edge: ln(1 + x) breaks at x = -1 and 1/(1 - x) blows up at x = 1, so a series centred at a converges only within that distance of a. Inside the radius the extra terms shrink and the polynomial closes in; outside it they grow, and a higher degree makes the answer worse rather than better. Students can change degree and centre and compare what happens. Use it for math inquiry with High School on a projector or student device.

Standards: Common Core CCSS.MATH.CONTENT.HSA.SSE.B.4, CCSS.MATH.CONTENT.HSF.BF.A.1

• Start at degree 0 and add one term at a time, watching how much further the match reaches each time
• Switch to 1/(1 - x) and find the point where more terms stop helping, then compare it with the radius readout
• Drag the centre towards x = -1 on ln(1 + x) and watch the radius, and the shaded band, shrink with it
• Set a wide tolerance, then a narrow one, and see how much of the band was really about how fussy you were being
• Compare sin x and 1/(1 - x) at degree 12: one is right across the whole screen, the other only near the middle

1. 1Predict how changing degree will affect the polynomial says, then test the prediction.
2. 2Use evidence from the live calculation and visual to explain the result.

Questions teachers ask