Sure, let's solve this step by step.
Step 1: Define Variables
We are given that the value of x is -5.52 and n equals 4.
Step 2: Taylor Series Approximation
First, we will use the Taylor series to approximate the value of \( e^{x} \). The Taylor series for \( e^{x} \) is given as:
\[ e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + ... \]
Substitute the given value of x and solve till the nth term (n equals 4 in this case). This will give us the approximation value of \( e^{x} \). In our case, the approximation is roughly 21.367651839999994.
Step 3: Calculate Actual Value
The actual value of \( e^{x} \) can be calculated using the mathematical function exp(x). For x equals -5.52, the exact value of \( e^{x} \) is approximately 0.00400584794209042.
Step 4: Remainder Term
Next, we estimate the remainder term in the Taylor series approximation which is given by:
\[ R_{n} = |f^{(n+1)}(c) * x^{(n+1)} / (n+1)!| \]
where c is some value between x and 0.
In our case, the remainder term is approximately 0.17108368754705888.
Step 5: Absolute Error
Finally, the absolute error can then be computed by taking the absolute difference between the actual value and the approximation. We find the absolute error in this case to be roughly 21.363645992057904.
In conclusion, the approximation of \( e^{-5.52} \) using the 4th degree Taylor series centered at 0 is roughly 21.367651839999994. The actual value is approximately 0.00400584794209042 and the absolute error associated with this approximation is roughly 21.363645992057904. The remainder term is around 0.17108368754705888.