Find the order 3 Taylor polynomial T3(x) of the given function f(x) = (3x+4)^(5/4) at a=4. Use exact values.

Question

Find the order 3 Taylor polynomial T₃(x) of the given function f(x) = (3x+4) ^5/4 at a=4. Use exact values.

Answer 1

Hey there, recall that the Taylor expansion of a differentiable function f around a point 'a' is given by

f(x) = sum_(k = 0)^(infty) (f^{(k)} (a))/k! * (x - a)^k

In our case, we only want up to the k = 3 term. We first want to compute our derivatives

f(4) = (3(4) + 4)^(5/4) = 32 ------> f(4)/0! = 32

f'(4) = 15/4 * (3(4) + 4)^(1/4) = 15/2 ------> f'(4)/1! = 15/2

f''(4) = 45/16 * (3(4) + 4)^(-3/4) = 45/128 ------> f''(4)/2! = 45/256

f'''(4) = -145/64 * (3(4) + 4)^(-7/4) = -145/8192. ------> f'''(4)/3! = -45/16384

Therefore, our answer should be the polynomial

32 + 15/2*(x - 4) + 45/256 * (x - 4)^2 - 45/16384*(x - 4)^3

Let me know if there is any part of this answer that you would like clarification on.