HW4 - b = 1. 7. The sinh and cosh functions are used, for...

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Homework #4 Math 126 These problems use the techniques of section 5 except for diFerentiation and inte- gration of series. Each problem can be derived from the basic series given in Examples 4.2. (a) In problems 1-6, ±nd the Taylor series for f ( x ) based at b . Your answer should have one Sigma (Σ) sign. On some problems you might want to describe the coe²cients using a multi-part notation as in Example 5.5. (b) Then write the solution in expanded form: a 0 + a 1 ( x - b ) + a 2 ( x - b ) 2 + . . . where you write at least the ±rst three non-zero terms explicitly. (c) Then give an interval I where the Taylor series converges. Note that there are some hints below. 1. f ( x ) = cos(3 x 2 ) based at b = 0. 2. f ( x ) = sin 2 ( x ) based at b = 0. 3. f ( x ) = e 4 x - 5 based at b = 2. 4. f ( x ) = sin( x ) based at b = π 6 . 5. f ( x ) = 1 4 x - 5 - 1 3 x - 2 based at b = 0. 6. f ( x ) = x (2 x + 1)(3 x - 1) based at
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Unformatted text preview: b = 1. 7. The sinh and cosh functions are used, for example, in electrical engineering, and are dened by sinh( x ) = ( e x-e-x ) / 2 , and cosh( x ) = ( e x + e-x ) / 2 . Do questions (a) and (b) above for the function h ( x ) = 2 sinh(3 x )-4 cosh(3 x ) based at b = 0. 8. ind the 6 th degree Taylor polynomial for f ( x ) = sin(3 x-5) based at b = 0, without diFerentiating. Hints: Change the base from b to 0 by substituting u = x-b . Be sure that the terms in your answers are numbers (coecients) times powers of x-b . Use the double angle formula in problem 2. Use partial fractions in problem 6. Use the addition formulae for sin( A B ) in problems 4 and 8. 1...
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