need help with taylor series 2 problemsf(x)=(1/x) about center c=1 poly Psub4(x) for f(x)=1/(sqrt x) c=1?

Question

i just have no clue how to do these types of problems so a little explanation would be very helpful. maybe a little step by step could help too. thx

Answer 1

Taylor series takes a function f(x) and expands it about a point by estimating the function's value at that point plus the values of derivatives about that point.

f(x = x0) = f(x0)+ (x-x0)*f'(x)|x0 + 1/2!(x-x0)^2*f"(x)|x0 + 1/3! (x-x0)^3*f"'(x)|x0 +...

where n! = 1*2*3*...*n, and "|x0" means evaluated at x=x0.

So take your first function f(x) = 1/x. The expansion is:

f(x=1) = 1 + (x-1)*(-1/x^2)|x=1+1/2*(x-1)^2(2/x^3)|x=1 + 1/6*(x-1)^3*(-6/x^4)|x=1

f(1) = 1 -(x-1) +(x-1)^2 - (x-1)^3

You can simplify this by exanding the terms of (x-1). I won't do the drudge work here.

Do the other function the same way