Find the Taylor polynomial of degree n=4 for each function expanded about the given value of x 0 . f ( x ) = cos ⁡ ( x ) , x 0 = 0

Question

Find the Taylor polynomial of degree n=4 for each function expanded about the given value of       x          0      .  f  (  x  )  =  cos  ⁡  (  x  )  ,      x          0        =  0

Bradley Sherman · Accepted Answer

f  (  x  )  =  f  (  x  0  )  +            x      f      ∗      (      x      0      )              1      !        +                    x                  2                    f      ∗      (      x      0      )              2      !        +  .  .  .  f  (  x  )  =  cos  ⁡  0  =  1      f    ′    (  x  )  =  −  sin  ⁡  x  ⇒      f    ′    (  0  )  =  −  sin  ⁡  0  =  0      f    ″    (  x  )  =  −  cos  ⁡  x  ⇒      f    ″    (  0  )  =  −  cos  ⁡  0  =  −  1      f    ‴    (  x  )  =  sin  ⁡  x  ⇒      f    ‴    (  0  )  =  sin  ⁡  0  =  0      f          (      i      v      )        (  x  )  =  cos  ⁡  x  ⇒      f          (      i      v      )        (  0  )  =  cos  ⁡  0  =  1      f          (      v      )        (  x  )  =  −  sin  ⁡  x  ⇒      f          (      v      )        (  0  )  =  −  sin  ⁡  0  =  0      f          (      v      i      )        (  x  )  =  −  cos  ⁡  x  ⇒      f          (      v      i      )        (  0  )  =  −  cos  ⁡  0  =  −  1so on  f  (  x  )  =  1  −            x              2                    2      !        +            x              4                    4      !        −            x              6                    6      !        +  .  .  .  .  =      ∑          n      =      0              ∞                  (      −      1              )                  n                            x                  2          n                            (      2      n      )      !

Find the Taylor polynomial of degree n=4 for each function expanded about the given value of x_0 f(x)=cos(x), x_0 =0

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