we used Taylor series expansions of f(xo+h) and f(xo - h) along with f(xo+h)-f(xo - h) +0(h²). 2h Assuming f satisfies the conditions of Taylor's theorem on a sufficiently large interval around to, show that Taylor's theorem to show that f '(xo) = f'(x) = = f(xo - 2h) - 6f(xo − h) +3ƒ (xo) + 2ƒ(xo + h) 6h + 0(h³). Your proof should include appropriate Taylor series expansions of f(xo-2h), f(xo-h), and f(xo + h) as well as a clear indication of when you use Taylor's theorem. The following table shows the position (displacement), x(t), at selected times, t, of a 400 metre runner who is running along a straight road. Time (seconds) 0 5 10 15 20 30 40 50 Position (metres) 0 37 82 130 175 280 340 397 The derivative of position (displacement) with respect to time gives velocity. Use the formula in part (i) and the data in the table to estimate the runner's velocity (in metres per second) when time t = 15 seconds, rounding your answer to two decimal places. What is another time from the table for which one could estimate the runner's velocity using the formula from part (i) and the data from the table? What is the problem with using the formula in part (i) to estimate the runner's velocity at an arbitrary time between 0 and 50 seconds?

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