Question

for the functions In(x) and e x, calculate separately each of the first non-zero terms of the Taylor series for the function, expanded around the point a 1

Answer 1

Taylor series of function f(x) at a is defined as:

Find the derivatives of fx)In(x), at a 1 in(x)1) d(Inx))1) (x-1)3- In( 1)- YC 3

now we find the derivatives individually,

$\therefore$ln(1)=0

FIND THE FIRST DERIVATIVE (In (x) The derivative of natural logarithm is (in (x)), (in(z)), = z Thus, (In () THE 2ND DERIVATIVE IS THE DERIVATIVE OF THE 1ST DERIVATIVE: (In (x)) _ 1: Apply the power rule (z"), = n-z-1+n with n- Thus, ) =

THE 3RD DERIVATIVE IS THE DERIVATIVE OF THE 2ND DERIVATIVE: (In (x))", Apply the constant multiple rule (c . f (z))-c-(f (z) )' with c =-1 and f (z) =- Apply the power rule (z"), = n-z-1+n with n =-2 Thus,
put this results in above eq we get

+.................

2 2

+..................

Refine =(x-1)-如一1)2 (z-1)3

now we calculate for e^x

Taylor series of function f(x) at a is defined as:

Find the derivatives off(x) 1 at a (ei) 1e (r-1)3- l. (1-1)-의 1! 2! 3!

now we find the derivatives individually,

FIND THE FIRST DERIVATIVE(e) The derivative of exponential is (e)e" (ey = ez Thus, (e*), = ez THE 2ND DERIVATIVE IS THE DERIVATIVE OF THE 1ST DERIVATIVE: (e*)" (e"), The derivative of exponential is (e ez : (ey = ez Thus, (e e THE 3RD DERIVATIVE IS THE DERIVATIVE OF THE 2ND DERIVATIVE: (e*)",-(ez) The derivative of exponential is (ez ), e Thus, (e*), = ez

put this results in above eq we get

+..................

2 1! 2! 3!

+.................

Refine =e-e(x-1)-음(x-1)2--(x-1)