Find the Taylor polynomial p3(x) for f(x)=x1/3 at x=3 .

The Taylor polynomial of degree n for a function f(x) at a point x=a is given by:

Pn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^n(a)(x-a)^n/n!

Here, we want to find the Taylor polynomial of degree 3, P3(x), for the function f(x) = x^(1/3) at x = 3.

First, we need to find the first, second, and third derivatives of f(x) = x^(1/3).

f'(x) = (1/3)x^(-2/3)

f''(x) = -(2/9)x^(-5/3)

f'''(x) = (10/27)x^(-8/3)

Now, we evaluate these at x = 3:

f(3) = 3^(1/3) = 1.44224957

f'(3) = (1/3)3^(-2/3) = 0.19245009

f''(3) = -(2/9)3^(-5/3) = -0.04246712

f'''(3) = (10/27)3^(-8/3) = 0.01282051

Now, we can plug these values into the Taylor polynomial formula:

P3(x) = f(3) + f'(3)(x-3) + f''(3)(x-3)^2/2! + f'''(3)(x-3)^3/3!

P3(x) = 1.44224957 + 0.19245009(x-3) - 0.04246712(x-3)^2/2 + 0.01282051(x-3)^3/6

This is the Taylor polynomial of degree 3 for the function f(x) = x^(1/3) at x = 3.

The Taylor polynomial of degree n for a function f(x) at a point x=a is given by:

Pn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^n(a)(x-a)^n/n!

Here, we want to find the Taylor polynomial of degree 3, P3(x), for the function f(x) = x^(1/3) at x = 3.

First, we need to find the first, second, and third derivatives of f(x) = x^(1/3).

f'(x) = (1/3)x^(-2/3)

f''(x) = -(2/9)x^(-5/3)

f'''(x) = (10/27)x^(-8/3)

Next, we evaluate these derivatives at x = 3:

f(3) = 3^(1/3) = 1.44224957

f'(3) = (1/3)3^(-2/3) = 0.19245009

f''(3) = -(2/9)3^(-5/3) = -0.04246712

f'''(3) = (10/27)3^(-8/3) = 0.01249038

Finally, we substitute these values into the Taylor polynomial formula:

P3(x) = f(3) + f'(3)(x-3) + f''(3)(x-3)^2/2! + f'''(3)(x-3)^3/3!

P3(x) = 1.44224957 + 0.19245009(x-3) - 0.04246712(x-3)^2/2 + 0.01249038(x-3)^3/6

This is the Taylor polynomial of degree 3 for the function f(x) = x^(1/3) at x = 3.