Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is

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Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point.

a. Referring to the figure, show that x = sin s or s = sin-1 x.

b. The area of a circular sector of radius r subtended by an angle θ is 1/2 r2 θ. Show that the area of the circular sector APE is s/2, which implies that

x = 2/ Vi - f di – xVT – x² xV1 - x V1 – ² dt

c. Use the binomial series for f(x) = √1 - x2 to obtain the first few terms of the Taylor series for s = sin-1 x.

d. Newton next inverted the series in part (c) to obtain the Taylor series for x = sin s. He did this by assuming that sin s = ∑aksk and solving x = sin (sin-1 x) for the coefficients ak. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well).

УА х B f(x) = V1 – x² V 1 x² х

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Calculus Early Transcendentals

ISBN: 978-0321947345

2nd edition

Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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