Find the Taylor polynomial p3(x) for f(x)=ln(5x) at x=1 .

The Taylor polynomial of degree n for a function f(x) at 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!

We want to find the Taylor polynomial of degree 3, P3(x), for f(x)=ln(5x) at x=1.

First, we need to find the derivatives of f(x) at x=1.

f(x) = ln(5x) f'(x) = 1/x f''(x) = -1/x^2 f'''(x) = 2/x^3

Now, we evaluate these at x=1:

f(1) = ln(5*1) = ln(5) f'(1) = 1/1 = 1 f''(1) = -1/1^2 = -1 f'''(1) = 2/1^3 = 2

Now, we can plug these into the formula for P3(x):

P3(x) = f(1) + f'(1)(x-1) + f''(1)(x-1)^2/2! + f'''(1)(x-1)^3/3! = ln(5) + 1*(x-1) - 1*(x-1)^2/2 + 2*(x-1)^3/6 = ln(5) + (x-1) - (x-1)^2/2 + (x-1)^3/3

So, the Taylor polynomial of degree 3 for f(x)=ln(5x) at x=1 is P3(x) = ln(5) + (x-1) - (x-1)^2/2 + (x-1)^3/3.