Find the Taylor polynomial p2(x) for f(x)=5ex at x=1 .

The Taylor polynomial of degree 2 for a function f(x) at x=a is given by:

P2(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2!

First, we need to find the first and second derivatives of f(x) = 5e^x.

f'(x) = 5e^x f''(x) = 5e^x

Now, we substitute x = 1 into f(x), f'(x), and f''(x):

f(1) = 5e^1 = 5e f'(1) = 5e^1 = 5e f''(1) = 5e^1 = 5e

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

P2(x) = f(1) + f'(1)(x-1) + f''(1)(x-1)^2/2! = 5e + 5e(x-1) + 5e(x-1)^2/2 = 5e + 5e(x-1) + 2.5e(x-1)^2

So, the Taylor polynomial of degree 2 for f(x) = 5e^x at x = 1 is P2(x) = 5e + 5e(x-1) + 2.5e(x-1)^2.