Find the difference between the Taylor polynomial of degree 4 about the point 0 for cos(x) evaluated at x = 1, and cos(1).
A. sine of 1 over 5 factorial
B. negative sine of c over 5 factorial for some c between 0 and 1
C. the product of the sine of 1 over 5 factorial and x raised to the n plus power for some x between 0 and 1
D. sine of c over 5 factorial for some c between 0 and 1
To find the difference between the Taylor polynomial of degree 4 about the point 0 for cos(x) evaluated at x = 1 and cos(1), we need to calculate the difference between the Taylor polynomial at x = 1 and the actual value of cos(1).
The Taylor polynomial of degree 4 about the point 0 for cos(x) can be determined using the formula:
P(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + (f''''(0)/4!)x^4
For cos(x), the derivatives at 0 are as follows:
f(0) = cos(0) = 1
f'(0) = -sin(0) = 0
f''(0) = -cos(0) = -1
f'''(0) = sin(0) = 0
f''''(0) = cos(0) = 1
Substituting these values into the Taylor polynomial formula, we get:
P(x) = 1 + 0*x + (-1/2!)x^2 + (0/3!)x^3 + (1/4!)x^4
= 1 - (1/2!)x^2 + (1/4!)x^4
To find the difference between the Taylor polynomial at x = 1 and cos(1), we substitute x = 1 into both expressions:
P(1) = 1 - (1/2!) * 1^2 + (1/4!) * 1^4
= 1 - (1/2) + (1/24)
= 1 - 1/2 + 1/24
= 23/24
cos(1) ≈ 0.5403 (using a calculator)
The difference between the Taylor polynomial at x = 1 and cos(1) is:
Difference = P(1) - cos(1)
= 23/24 - 0.5403
≈ 0.0419
Comparing this result with the given options:
A. sine of 1 over 5 factorial - This is not the correct answer.
B. negative sine of c over 5 factorial for some c between 0 and 1 - This is not the correct answer.
C. the product of the sine of 1 over 5 factorial and x raised to the n plus power for some x between 0 and 1 - This is not the correct answer.
D. sine of c over 5 factorial for some c between 0 and 1 - This is not the correct answer.
Therefore, none of the given options are correct. The correct difference is approximately 0.0419.