Expert: Scotto - 5/15/2010

Question
let P5(x) be the quintic Maclaurin polynomial for the function f defined by       x
          f(x)=e cos x , x e R

1.  Compute P5(x)

2. Use the result of the previous part to compute the approximation f(1)~P5(1)

3. Use Taylor's Theorm to approximate /f(1)-P5(1) /

Answer
The Maclaurin polynomial is a special case of the Taylor polynomial about 0.
I assume the quintic goes out to the power on x is 5, so there are six possible terms.
If the function is suppose to be cos x, the derivative of cos is -sin and the derivative of sin is cos.  Since it is about 0, 2, and 4 exist, but terms 1, 3, and 5 are zero.
It is cos x = 1 - x^2/2 + x^4/24.

It is best to put all characters on the same line, for I see and x straggling out at the end way far away from the function, so it looks like f(x) = (e^x)cos(x).
The derivaitve of e^x is 3^x, so the first five derivatives of f(x) are
f'(x) = e^x(cos(x) - sin(x));
f"(x) = -2e^x(sin(x));
f"'(x) = -2e^x(sin(x) + cos(x));
f""(x) = -4e^x(cos(x)); and
f"'"(x) = -4e^x(cos(x) - sin(x)).

1. Evaluate the function and the first five derivatives at 0, then put them in front of x^n
for the nth derivative, and there's the Maclaurin polynomial for degree 5.

2. Put in 1 for x to find P5(1).

3. Taylor's theore is just like the Maclaurin polynomial, except the series is found at 1 for each term and is multiplied by (x-1)^n instead of x^n.