Find the function G(x) whose graph passes through (pi/38,-12)and has f(x) as its derivative: G(x)=
I already found which is: F(x)=76(1/-19)cos(19x)+C
To find the function G(x) that satisfies the given conditions, we can integrate f(x) to find F(x), and then use the point (pi/38, -12) to determine the value of the constant C.
Given: F(x) = 76(1/-19)cos(19x) + C
To find G(x), we need to determine C using the point (pi/38, -12).
Substituting x = pi/38 and y = -12 into the equation, we have:
-12 = 76(1/-19)cos(19(pi/38)) + C
Let's simplify the equation first.
cos(19(pi/38)) can be simplified further by recognizing that pi/38 = 2(pi/76). Using the double-angle formula for cosine, we have:
cos(2x) = 2cos^2(x) - 1
cos(2(pi/76)) = 2cos^2(pi/76) - 1
Now, substitute this value back into our equation:
-12 = 76(1/-19) * (2cos^2(pi/76) - 1) + C
Simplifying further:
-12 = (-4) * (2cos^2(pi/76) - 1) + C
-12 = -8cos^2(pi/76) + 4 + C
-8cos^2(pi/76) + C = -16
Now, we can solve for C:
C = -16 + 8cos^2(pi/76)
Therefore, the function G(x) is:
G(x) = 76(1/-19)cos(19x) + (-16 + 8cos^2(pi/76))
So, the final function G(x) that passes through (pi/38, -12) and has f(x) as its derivative is:
G(x) = 76(1/-19)cos(19x) + (-16 + 8cos^2(pi/76))