Find the order 3 Taylor polynomial T3(x) of the given function f(x) = (7x-6)^(4/3) at a=2. Use exact values.

Question

Find the order 3 Taylor polynomial T₃(x) of the given function f(x) = (7x-6) ^4/3 at a=4. Use exact values.

Answer 1

Remember that the Taylor polynomial of degree 3 around a point a is

f(a)+f'(a)(x-a)+f''(a)/2 (x-a)^2+f'''(a)/3! (x-a)^3

so we need to calculate the first 3 derivatives of f. We can just use the power and chain rules here:

f'(x) = 4/3*7(7x-6)^1/3

f''(x) = 4*49/9(7x-6)^(-2/3)

I bet you can find f'''(x) from here. Once you finish with that, you can calculate T3(x) by plugging in the value of a (not sure if you wanted a=2 as in the title or a=4 as in the subtitle).