Find the degree 3 Taylor polynomial approximation to the function

Question

f(x)=4ln(sec(x)) about x=0.?

Answer 1

The function is f(x) = 4ln(sec x)

Taylors Polynomial approximation of degree n for x=a is given as

Where n = 3 and x = 0

Then Taylor's Polynomial of approximation

Now f(x) = 4ln(sec x)

At x = 0,

f(0) = 4 ln(sec0) = 0

Apply derivative on each side.

Now at x = 0

Then f'(0) = 4 tan 0 = 0

Apply derivative on each side.

At x = 0,

f''(0) = 4 sec²0

f''(0) = 4

Apply derivative on each side.

At x = 0,

f'''(0) = 8 sec²0 tan 0

f'''(0) = 0.

The Taylor's Polynomial of approximation is modified at x=0 as

Therefore the third degree f(x) = 4ln(sec x) at x=0 using Taylor's Polynomial of approximation is 2x².