Consider Newton's Binomial Theorem series expansion for f(r) = VT –1. We use this expansion to approximate the area under the curve y = V1-1 from r = 0 to I = 1. (a) Sketch the curve y = V1-1 on the interval 0

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Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Consider Newton's Binomial Theorem series expansion for f(r) = V1-r. We use
this expansion to approximate the area under the curve y = V1-x from r = 0 to
x = 1.
(a) Sketch the curve y = V1-r on the interval 0 <a< 1.
(b) Find a provisional anti-derivative of V1-r by taking an anti-derivative of the
series term by term. Let A(r) denote this new series.
%3D
%3D
(c) Approximate
compute A(0) exactly, but for A(1) truncate the series to five terms to get an
approximation.
V1-r dx by computing A(1) – A(0). You will be able to
(d) Compute
| VI-I dr "normally" and compare your answer with part (c).
(Hint: Perform u-substitution, with u = 1 -x and du = -dr.)
Transcribed Image Text:Consider Newton's Binomial Theorem series expansion for f(r) = V1-r. We use this expansion to approximate the area under the curve y = V1-x from r = 0 to x = 1. (a) Sketch the curve y = V1-r on the interval 0 <a< 1. (b) Find a provisional anti-derivative of V1-r by taking an anti-derivative of the series term by term. Let A(r) denote this new series. %3D %3D (c) Approximate compute A(0) exactly, but for A(1) truncate the series to five terms to get an approximation. V1-r dx by computing A(1) – A(0). You will be able to (d) Compute | VI-I dr "normally" and compare your answer with part (c). (Hint: Perform u-substitution, with u = 1 -x and du = -dr.)
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