what is the coefficient of (x+2)^3 in the cubic taylor polynomial about -2 for the function f(x)= e^x?
To find the coefficient of (x+2)^3 in the cubic Taylor polynomial for the function f(x) = e^x centered at -2, we can start by writing out the Taylor polynomial.
The Taylor polynomial for a function f(x) centered at a is given by:
P(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)^2 + (f'''(a)/3!)(x-a)^3
In this case, we are interested in the coefficient of (x+2)^3, so we want to find the coefficient of (x-a)^3, where a = -2.
First, let's find the derivatives of f(x) = e^x. The derivative of e^x is simply e^x, so the first three derivatives are:
f'(x) = e^x
f''(x) = e^x
f'''(x) = e^x
Next, we need to evaluate these derivatives at x = -2:
f'(-2) = e^(-2)
f''(-2) = e^(-2)
f'''(-2) = e^(-2)
Now, substituting these values into the Taylor polynomial formula, we have:
P(x) = f(-2) + f'(-2)(x-(-2)) + (f''(-2)/2!)(x-(-2))^2 + (f'''(-2)/3!)(x-(-2))^3
Simplifying, we get:
P(x) = f(-2) + f'(-2)(x+2) + (f''(-2)/2!)(x+2)^2 + (f'''(-2)/3!)(x+2)^3
The coefficient of (x+2)^3 is given by the term (f'''(-2)/3!). Plugging in the value e^(-2) for f'''(-2), we get:
Coefficient of (x+2)^3 = e^(-2) / 3!
So, the coefficient of (x+2)^3 in the cubic Taylor polynomial for f(x) = e^x centered at -2 is e^(-2) divided by 3 factorial.