Suppose f(x) is a function with continous Derivatives...

Question

Suppose f(x) is a function with continous Derivatives and..

f(8) = 4   f'(8)= 5   f''(8)=-4   f'''(8)= 1

 

a.) Write the Taylor Polynomial of degree 3 (first four terms) of f(x) centered at x=8

 

b.) Use the polynomial found in part a to approximate f(8.1)

 

Calc II Homework question, plz help!

Answer 1

Taylor expansion :

 

f(8)(x-8)^0/0!  +

f'(8) (x-8)^1/1! +

f''(8)(x-8)^2 / 2! +

f'''(8)(x-8)^3/3!

 

= 4 + 5*(x-8) + -4(x-8)^2/2 + (x-8)^3/6 

 

= 4 + 5(x-8) - 2(x-8)^2 + (1/6)(x-8)^3

 

 

The approximation is

 

 4 + 5(0.1) - 2(0.1)^2 + (1/6)(0.1)^3 =

 

4 + 0.5 - 0.02 + (1/6)(0.001) = 4.48016