Page 23 - mathematics

Example Example

Find the Taylor series generated by f  x   1/ x at x = 2.  Find the Macluarin series generated by f x  e 3x

Solution 1 Solution
2
f x   x 1 f  2  f x  e3x f 0 1

f  x   x 2 f 2  1 f  x   3e 3x f 0  3
4
1
f  x   2 x 3 f   2   4 f x   9e 3x f 0  9
f  x   27e 3x
f  x   6 x 4 f   2   3 f 0  27
8
9 x2 27 x 3  ....
f x   1  1 x  2  1  x  22  1 x  23  .... f x  1 3x  2!  3!
2 4 8 16

Example

Find the Taylor series generated by f  x   sin x at x=π/4 .

Solution

f  x   sin x f  / 4 1/ 2

f x   cos x f 1  / 4  1/ 2

f x   sin x f 2  / 4  1/ 2

f  x    cos x f 3  / 4  1/ 2

f x  1  1   x     1  x   2  1  x   3 
2   4  2!  4  3!  4   ....



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