Fourth Degree Taylor's Polynomial Approximation to f(x)=e^(2x) about x=0:

To obtain the fourth degree Taylor's polynomial approximation to f(x)=e^(2x) about x=0, we need to find the first four derivatives of f(x) at x=0.

f(x) = e^(2x)

f'(x) = 2e^(2x)

f''(x) = 4e^(2x)

f'''(x) = 8e^(2x)

f''''(x) = 16e^(2x)

Now, we can write the fourth degree Taylor's polynomial approximation to f(x) as:

P4(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + f''''(0)x^4/4!

P4(x) = 1 + 2x + 2x^2 + 8x^3/3 + 16x^4/24

P4(x) = 1 + 2x + x^2(2 + 4x/3) + x^4(2/3)

Maximum Error:

To find the maximum error when 0 ≤ x ≤ 0.5, we need to use the remainder term of the Taylor's polynomial approximation:

R4(x) = f^(5)(c)x^5/5!

where c is some value between 0 and x.

Since f^(5)(x) = 32e^(2x), we have:

|R4(x)| ≤ (32/5!)*|x|^5*e^(2c)

Now, we need to find the maximum value of e^(2c) when 0 ≤ c ≤ 0.5.

Since e^(2x) is an increasing function, its maximum value on the interval [0, 0.5] occurs at x=0.5. Therefore, e^(2c) ≤ e^(2*0.5) = e.

Substituting this value in the remainder term, we get:

|R4(x)| ≤ (32/5!)*|x|^5*e

For 0 ≤ x ≤ 0.5, the maximum value of |x|^5 occurs at x=0.5. Therefore, we have:

|R4(x)| ≤ (32/5!)*(0.5)^5*e

|R4(x)| ≤ 0.000266e

Therefore, the maximum error when 0 ≤ x ≤ 0.5 is approximately 0.000266 times the value of e.