Find the degree 3 Taylor polynomial T3(x) of function f(x) = (-5x+46)^(3/2) at a=6.

Question

Find the degree 3 Taylor polynomial T₃(x) of function f(x) = (-5x+46)^3/2 at a=6.

Answer 1

T₃(x) = f(6) + f'(6)(x - 6) + f"(6)/2! (x-6)² + f'''(6)/3! (x-6)³

f(x) = (-5x+46)^3/2

f'(x) = (3/2)(-5x+46)^1/2(-5) = (-15/2)(-5x+46)^1/2

f"(x) = (-15/4)(-5x+46)^-1/2(-5) = (75/4)(-5x+46)^-1/2

f'''(x) = (-75/8)(-5x+46)^-3/2(-5) = (375/8)(-5x+46)^-3/2

So, T₃(x) = 64 - 30(x - 6) + (75/32)(x - 6)² + (125/1024)(x - 6)³

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