Solutions for Numerical and Analytical Methods with MATLAB for Electrical Engineers, 1st Edition Bober (All Chapters included)
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Complete Solutions Manual for Numerical and Analytical Methods with MATLAB for Electrical Engineers, 1st Edition by William Bober ; Andrew Stevens ; ISBN13: 9781439854297. (Full Chapters included Chapter 2 to 11), ch 1 solutionsn not available....1.Numerical Methods for Electrical Engineers
2.MATL...
Show that the Taylor Series expansion of cos( x) about x = 0 is:
x2 x4 x6
cos ( x) = 1 − + − +
2! 4! 6!
Solution:
Taylor series expansion of f (x ) about x = 0 is:
f ′′(0) 2 f ′′′(0) 3 f IV 4
f ( x) = f (0) + f ′(0) x + x + x + x +
2! 3! 4!
For cos( x) ,
f ( x ) = cos ( x ) , f (0) = 1,
f ′( x ) = − sin( x ), f ' (0) = 0,
f ′′( x) = − cos( x), f ′′(0) = − 1,
f ′′′( x) = + sin( x), f ′′′(0) = 0,
f IV ( x) = + cos( x), f IV (0) =1
We can see that
x 2 x 4 x 6 x8
cos( x) = 1 − + − + − +
2! 4! 6! 8!
Exercise 2.2
Show that the Taylor Series expansion of sin( x) about x = 0 is:
x3 x5 x 7
sin ( x) = x − + − +
3! 5! 7!
1
, Solution:
For sin( x) ,
f ( x ) = sin ( x ) , f (0) = 0
f ′( x) = cos ( x), f ′(0) = 1
f ′′( x) = − sin ( x), f ′′(0) = 0
f ′′′( x) = − cos ( x), f ′′′(0) = − 1
f IV ( x) = sin ( x), f IV (0) = 0
f V ( x ) = cos ( x ), f V (0) = 1
We can see that
x3 x5 x7 x9
sin ( x) = x − + − + − +
3! 5! 7! 9!
Exercise 2.3
Using the Taylor Series expansion of e x about x = 0 (see Example 2.1) , show that
e jx = cos x + j sin x and that e − jx = cos x − j sin x , where j = − 1.
Solution:
For e jx ,
f ( x) = e jx , f (0) = 1
f ′( x) = je jx , f ′(0) = j
f ′′( x) = − e jx , f ′′(0) = −1
f ′′′( x) = − je jx , f ′′′(0) = − j
f IV ( x) = e jx , f IV (0) = 1
2
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