Finding the Third-Degree Taylor Polynomial for f(x) = tan^{-1}(x) Centered at x = 7
The Taylor polynomial of a function f centered at a point a is given by the formula:
T_n(x) = f(a) + f'(a)(x - a) + (f''(a)/2!)(x - a)^2 + (f'''(a)/3!)(x - a)^3 + ... + (f^{(n)}(a)/n!)(x - a)^n
In our case, we want to find the third-degree polynomial T3 for the function f(x) = tan^{-1}(x) centered at a = 7.
Step 1: Calculate f(7)
First, we need to find the value of the function at the center point:
f(7) = tan^{-1}(7)
This will give us the first term of the polynomial.
Step 2: Calculate the derivatives
Next, we need to calculate the first three derivatives of f(x):
- f'(x) = 1/(1 + x2)
- f”(x) = -2x/(1 + x2)2
- f”'(x) = rac{2(3x2 – 1)}{(1 + x2)3}
Step 3: Evaluate the derivatives at x = 7
Now we will evaluate these derivatives at x = 7:
- f'(7) = 1/(1 + 72) = 1/50
- f”(7) = -2(7)/(1 + 72)2 = -14/2500 = -0.0056
- f”'(7) = rac{2(3(7)2 – 1)}{(1 + 72)3} = rac{2(147 – 1)}{125000} = rac{292}{125000}
Step 4: Substitute into the Taylor polynomial formula
Now that we have all the necessary values, we can plug them into the Taylor polynomial formula.
T3(x) = tan^{-1}(7) + rac{1}{50}(x - 7) + rac{-0.0056}{2!}(x - 7)2 + rac{ rac{292}{125000}}{3!}(x - 7)3
Calculating the polynomial gives:
T3(x) = tan^{-1}(7) + 0.02(x – 7) – 0.0028(x – 7)2 + 0.00048667(x – 7)3
Conclusion
The third-degree Taylor polynomial for the function f(x) = tan^{-1}(x) centered at x = 7 is:
T3(x) = tan^{-1}(7) + 0.02(x - 7) - 0.0028(x - 7)2 + 0.00048667(x - 7)3
This polynomial provides a local approximation of the function tan^{-1}(x) near x = 7.