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3.03.4.Literacy - Approximation Authors Bill Davis Horacio...

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Approximation Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 3.03 Using Expansions LITERACY · L.1) When you want to study behavior near a point x = b other than x = 0, you can use expansions in powers of H x - b L . What is the expansion of a function f @ x D in powers of H x - b L ? What does it have to do with order of contact? · L.2) Explain how to find the expansion of e x in powers of H x - 3 L . · L.3) Use expansions in powers of x to calculate the limits: a. lim x Ø 0 Sin @ x D x b. lim x Ø 0 Sin A 3 x 2 E x 2 c. lim x Ø 0 Sin A 4 x 4 E Sin A 2 x 4 E d. lim x Ø 0 1 + x - e x x 2 e. lim x Ø 0 1 + x - e x 1 - Cos @ x D · L.4) Here is the expansion of f @ x D = Sin @ Tan @ x DD - Tan @ Sin @ x DD in powers of x through the x 9 term: Normal @ Series @ Sin @ Tan @ x DD - Tan @ Sin @ x DD , 8 x, 0, 9 <DD - x 7 30 - 29x 9 756 And here is the expansion of g @ x D = ArcSin @ ArcTan @ x DD - ArcTan @ ArcSin @ x DD in powers of x through the x 9 term: Normal @ Series @ ArcSin @ ArcTan @ x DD - ArcTan @ ArcSin @ x DD , 8 x, 0, 9 <DD - x 7 30 + 13x 9 756 Use what you see to calculate lim x Ø 0 f @ x D g @ x D . · L.5) Write out the expansions in powers of x of e - x 2 and Cos @ 3 x D through the x 4 terms.
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