So I’m gearing up to take a calculus 1 exam, and this question is on the sample test. My initial thought was that since we are looking for F(9), and F(x) is an antiderivative of f(x), I can just use the integral of the equation of f(x) at 9, which is f(x) = -2x/3 + 5, which, when integrated, becomes -x^2/3 + 5x + 2 (C = 2 because F(0) = 2). Thing is, though, that won’t give me any of the answers listed. And even after taking the integral of all of the equations of f(x), I still have no idea how to produce any of the answers in the multiple choice.
I’m super stumped on this one. Any help would be welcome!
I recommend the math discord server. You might get faster responses there.
Some clarifications: f(x) = -2x/3 + 5 isn’t technically correct. It happens to equal that when x is between 6 and 9, but the function is different outside of that range. Similarly, your equation for F(x) is only correct when x is between 6 and 9. The reason this matters is because F(0) = 2 doesn’t mean C = 2. That only works if the function is the same all the way to x = 0, which it’s not.
If you want to solve by integrating, you would have to integrate each section and find the right C for each section that makes the integrals all connect to each other.
Alternatively, you can use the property that F(b) - F(a) = the area under f(x) from a to b. I think that region from x = 4 to 6 is supposed to be a semicircle, so each section is a standard shape and you can calculate the area using geometry.
This is a question designed to test your understanding of the underlying concepts as opposed to just being able to mechanically manipulate the functions.
If you have a value of the anti derivative F(x) at some point x=a and you want to find F(x) at x=b, you can find it with the equation F(b) = F(a) + <the area under f(x) between x=a and x=b>.
This problem gives you F(a) where a=0 and gives you f(x) between 0 and 9.