True or false.
1. If F'(x) = G'(x) on the interval [a,b] then F(b)-(b)-G(a).
I think it's true but I just want to check my reasoning. F'(x) = G'(x) means that F(x) and G(x) differ by a constant (which may be 0). When the subtraction is done to find the definite integral this constant disappears so the statement is true.
2. True or false.
Integrate from -2(the number on the bottom of the integral) to 2 (number on the top of the integral) x^-2dx = -x^-1 = -(2)^-1-(-(-2)^-1) = (-1/2)-(1/2) = -1
I think this is false, but I'm not sure why.
Thanks for your help
1. The statement is true. If F'(x) = G'(x) on the interval [a,b], then by the Mean Value Theorem, there exists some c between a and b such that F'(c) = G'(c). Since the derivatives are equal, the functions F(x) and G(x) differ by a constant on the interval [a,b]. Therefore, the definite integrals of F(x) and G(x) from a to b will also differ by that constant. This means that F(b) - (b) - G(a).
2. The statement is false. Let's evaluate the integral to verify. ∫[from -2 to 2] x^(-2) dx = [x^(-1) / -1] [from -2 to 2] = [-1/x] [from -2 to 2].
Plugging in the upper limit: [-1/2] - Plugging in the lower limit: [-1/(-2)].
Simplifying gives: -(1/2) - (1/2) = -1, which contradicts your initial statement. Therefore, the statement is false.
In this case, the mistake seems to be in the calculation. The integral of x^(-2) is actually -x^(-1), not -x^(-1/2).
1. True: If F'(x) = G'(x) on the interval [a,b], it means that F(x) and G(x) have the same derivative on that interval. By the fundamental theorem of calculus, if two functions have the same derivative on an interval, their integrals will differ by a constant. Therefore, F(b)-(b)-G(a).
2. False: The integration of x^-2 is -x^-1 + C, where C is the constant of integration. When evaluating the definite integral from -2 to 2, you should subtract the value of the antiderivative at the lower limit from the value at the upper limit.
So, the correct calculation would be:
-(-2)^-1 - (-(-2)^-1) = (-1/(-2)) - (1/(-2)) = (1/2) - (1/2) = 0
Hence, the statement is false, and the value of the integral is 0, not -1.