Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is
Question:
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
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).
x = 2/ Vi - f di – xVT – x² xV1 - x V1 – ² dt УА х B f(x) = V1 – x² V 1 x² х
Step by Step Answer:
a Clearly x sins because BE of length x is the side opposite the angle measured by s in a rig...View the full answer
Calculus Early Transcendentals
ISBN: 978-0321947345
2nd edition
Authors: William L. Briggs, Lyle Cochran, Bernard Gillett
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