Final answer:
To find the Taylor polynomial of degree 4 for cosh(9x) near x=9π, we need to find the values of the function and its derivatives at the point and use them to build the polynomial.
Step-by-step explanation:
To find the Taylor polynomial of degree 4 for cosh(9x) near x = 9π, we need to find the values of the function and its derivatives at the point and use them to build the polynomial.
First, let's find the values of the function and its derivatives:
- f(9π) = cosh(9π) = 1
- f'(9π) = 9 sinh(9π) = 0
- f''(9π) = 81 cosh(9π) = 81
- f'''(9π) = 729 sinh(9π) = 0
- f''''(9π) = 6561 cosh(9π) = 6561
Now we can build the Taylor polynomial using these values:
T(x) = f(9π) + f'(9π)(x - 9π) + (f''(9π)/(2!))(x - 9π)^2 + (f'''(9π)/(3!))(x - 9π)^3 + (f''''(9π)/(4!))(x - 9π)^4
T(x) = 1 + 0(x - 9π) + (81/2!)(x - 9π)^2 + 0(x - 9π)^3 + (6561/4!)(x - 9π)^4.