Find the degree 3 Taylor polynomial approximation to the function f(x)=8ln(sec(x)) about the point a=0 .
Go to
http://www.wolframalpha.com/input/?i=8ln%28sec%28x%29%29+
and scroll down a bit.
Look at that link. It gives you the series expansion about zero explicitly.
To find the degree 3 Taylor polynomial approximation to the function f(x) = 8ln(sec(x)) about the point a = 0, we will need to find the first three derivatives of the function at that point.
Step 1: Find the first derivative.
f'(x) = (8 / cos(x)) * (-sec(x) * tan(x))
= -8tan(x)
Step 2: Find the second derivative.
f''(x) = -8 * sec^2(x)
Step 3: Find the third derivative.
f'''(x) = -16sec(x) * tan(x)
Now we have the derivatives, we can use them to find the coefficients of the Taylor polynomial.
The degree 3 Taylor polynomial approximation is given by:
P3 (x) = f(0) + f'(0)(x - 0) + (f''(0) / 2!)(x - 0)^2 + (f'''(0) / 3!)(x - 0)^3
Step 4: Evaluate the derivatives at the point a = 0.
f(0) = 8ln(sec(0)) = 8ln(1) = 0
f'(0) = -8tan(0) = 0
f''(0) = -8 * sec^2(0) = -8
f'''(0) = -16sec(0) * tan(0) = 0
Step 5: Substitute the values into the equation for P3(x).
P3(x) = 0 + 0(x - 0) + (-8 / 2!)(x - 0)^2 + (0 / 3!)(x - 0)^3
Simplifying, we get:
P3(x) = -4x^2
Therefore, the degree 3 Taylor polynomial approximation to the function f(x) = 8ln(sec(x)) about the point a = 0 is P3(x) = -4x^2.