Final answer:
To find the Taylor series for f(x) = e^2x at x = 1, we use the Taylor series formula. We find the derivatives of f(x) = e^2x and substitute the values into the formula to get the Taylor series.
Step-by-step explanation:
To find the Taylor series for f(x) = e^2x at x = 1, we will use the formula for the Taylor series expansion:
Taylor series: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
First, find the derivatives of f(x) = e^2x:
f'(x) = 2e^2x
f''(x) = 4e^2x
f'''(x) = 8e^2x
Now substitute the values into the Taylor series formula:
Taylor series: f(x) = f(1) + f'(1)(x-1) + f''(1)(x-1)^2/2! + f'''(1)(x-1)^3/3! + ...
Since f(x) = e^2x, we have:
f(1) = e^2(1) = e^2
f'(1) = 2e^2(1) = 2e^2
f''(1) = 4e^2(1) = 4e^2
f'''(1) = 8e^2(1) = 8e^2
Substituting these values into the Taylor series formula, we get:
Taylor series: f(x) = e^2 + 2e^2(x-1) + 4e^2(x-1)^2/2! + 8e^2(x-1)^3/3! + ...
This is the Taylor series for f(x) = e^2x at x = 1.