Let G"(z) =4/z^3 - 4/z^6 with G(1)=0.
Evaluate G(3).
G(3)= _________________
To evaluate G(3), we can start by finding an expression for G(z), and then substitute z = 3 into that expression.
Given that G"(z) = 4/z^3 - 4/z^6 and that G(1) = 0, we can integrate G"(z) twice to find G(z).
First, integrate G"(z) with respect to z once. Since G"(z) is a sum of two terms, we can integrate each term separately:
∫ (4/z^3) dz = 4 ∫ z^(-3) dz = 4 * (-1/2) * z^(-2) = -2/z^2
And similarly for the second term:
∫ (4/z^6) dz = 4 ∫ z^(-6) dz = 4 * (-1/5) * z^(-5) = -4/5z^5
So, the result of the first integration is:
G'(z) = -2/z^2 - 4/5z^5
Next, integrate G'(z) with respect to z:
∫ (-2/z^2 - 4/5z^5) dz = -2 ∫ z^(-2) dz - 4/5 ∫ z^(-5) dz
∫ (-2/z^2) dz = -2 * (-1) * z^(-1) = 2/z
∫ (-4/5z^5) dz = -4/5 * (-1/4) * z^(-4) = 1/5z^4
So, the result of the second integration is:
G(z) = 2/z + 1/5z^4 + C
To find the value of C, we can use the given information that G(1) = 0:
0 = 2/1 + 1/5 * 1^4 + C
0 = 2 + 1/5 + C
C = -2 - 1/5 = -11/5
Therefore, the expression for G(z) is:
G(z) = 2/z + 1/5z^4 - 11/5
Now, we can substitute z = 3 into this expression to evaluate G(3):
G(3) = 2/3 + 1/5 * 3^4 - 11/5
G(3) = 2/3 + 1/5 * 81 - 11/5
G(3) = 2/3 + 81/5 - 11/5
G(3) = (10/15) + (81/5) - (11/5)
G(3) = (10/15) + (81-11)/5
G(3) = 2/3 + 70/5
G(3) = 2/3 + 14
G(3) = 32/3
Therefore, G(3) = 32/3.
G" = 4z^-3 - 4z^-6
G' = -2z^-2 +4/5 z^-5 + C
G = 2/z - 1/5 z^-4 + Cz + D
G(1)=0, so
2 - 1/5 + C + D = 0
C+D = -9/5
G(3) = 2/3 - 1/405 + 3C + D
Unless you have some other constraints, that's as far as we can go.
excuse me? You're in college, right?
Taking calculus, right?
you at least own a calculator, right?
or, you can use a computer, right?
and unless you have another constraint, such as G'(1)=2 or something, there is no way to get actual values for C and D.