To find the coefficients c(sub0), c(sub1), c(sub2), c(sub3), and c(sub4) for the power series representing f(x) = tan(x), we can use the Taylor series expansion formula for the tangent function.
The Taylor series expansion for the tangent function is:
tan(x) = x + (x^3)/3 + (2x^5)/15 + (17x^7)/315 + ...
Each term in the series is obtained by taking derivatives of the function and evaluating them at x = 0. We can see that the coefficient of the n-th term in the series is given by:
c(subn) = f^(n)(0) / n!
where f^(n)(0) represents the n-th derivative of f(x) evaluated at x = 0, and n! is the factorial of n.
Let's calculate the coefficients c(sub0), c(sub1), c(sub2), c(sub3), and c(sub4) step by step:
c(sub0) = f(0) / 0! = tan(0) / 0! = 0 / 1 = 0
c(sub1) = f'(0) / 1! = sec^2(0) / 1 = 1
To find c(sub2), we need to calculate the second derivative:
f''(x) = d^2/dx^2(tan(x)) = d/dx(sec^2(x)) = 2sec^2(x)tan(x)
c(sub2) = f''(0) / 2! = 2sec^2(0)tan(0) / 2 = 0
To find c(sub3), we need to calculate the third derivative:
f'''(x) = d^3/dx^3(tan(x)) = d/dx(2sec^2(x)tan(x)) = 2sec^2(x)sec^2(x) + 2sec^4(x)
c(sub3) = f'''(0) / 3! = (2sec^4(0) + 2sec^4(0)) / 6 = 4 / 6 = 2 / 3
To find c(sub4), we need to calculate the fourth derivative:
f''''(x) = d^4/dx^4(tan(x)) = d/dx(2sec^2(x)sec^2(x) + 2sec^4(x)) = 4sec^4(x)sec^2(x) + 8sec^2(x)sec^2(x)tan^2(x)
c(sub4) = f''''(0) / 4! = (4sec^4(0) + 8sec^4(0)tan^2(0)) / 24 = 4 / 24 = 1 / 6
Now we have the coefficients:
c(sub0) = 0
c(sub1) = 1
c(sub2) = 0
c(sub3) = 2 / 3
c(sub4) = 1 / 6
To approximate the tangent of 1/4 radian using the first two nonzero terms of the series, we can use the formula:
Approximation = c(sub0) + c(sub1)x
Substituting the values of the coefficients, we have:
Approximation = 0 + 1x = x
So, the approximation of the tangent of 1/4 radian using the first two nonzero terms is simply x.
Now let's compare this approximation with the actual value of the tangent of 1/4 radian as shown on a calculator:
Using a calculator, the tangent of 1/4 radian is approximately 0.2474.
Comparing with the approximation, which is x, we can substitute x = 1/4 in the approximation:
Approximation = 1/4 = 0.25
As we can see, the approximation of the tangent of 1/4 radian using the first two nonzero terms of the series (0.25) is a relatively close approximation to the actual value (0.2474) obtained from the calculator.