WKS2 - D 2-q ( D + d ) 2 = q D 2 b 1-1 (1 + d D ) 2 B ....

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Worksheet 2 Math 126 1. Use the Taylor Series for the exponential function to do the following. (a) Find the Taylor Series based at zero for f ( x ) = e x 3 - 1 = e x 3 /e . What is the value of d 15 f dx 15 at x = 0? (b) Find the Taylor Series based at zero for g ( x ) = e ( x/a ) 2 - b . (c) Find the Taylor Series based at zero for h ( x ) = xe ( x/a ) 2 - b . Hint: How is h related to g in part (b)? (d) Find the Taylor Series based at 1 for f ( x ) = xe x . Here is an application of Taylor polynomials: 2. An electric dipole consists of two electric charges of equal magnitude and opposite signs. If the charges are q and - q and are located at a distance d from each other, then the electric ±eld E at the point P in the ±gure D d q - q P is given by E = q
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Unformatted text preview: D 2-q ( D + d ) 2 = q D 2 b 1-1 (1 + d D ) 2 B . Treating d/D as a variable (set x = d/D ) write down the Taylor expansion (Taylor Series) for the function inside the square brackets. Multiply the Taylor expansion by q/D 2 to obtain an expansion for E . Using the expansion you obtained, show that E is approximately 2 qd/ ( D 3 ) when P is far away from the dipole. Comment: The electric ±eld decays much more rapidly than the ±eld generated by one of the charges because the ±elds generated by each particle partially cancel one another. The approximation tells us the rate of decay....
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This note was uploaded on 05/08/2008 for the course MATH 126 taught by Professor Smith during the Spring '07 term at University of Washington.

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