The only notation there that might be unfamiliar is f([-2,3]). That just means the set of all values that the function takes when evaluated at all numbers in the interval [-2,3]. So, for example, if f is a strictly increasing continuous function, then f([-2,3]) = [f(-2), f(3)].
I like the work you did with option D, but I'm confused about how you came up with the inequality with the minimun of the function, integral of the function, and maximum of the function. Is that some truth or property of integrals? I've taken a calculus course, but its been a bit.
If you have two functions, f and g, and f(x) is less than or equal to g(x) for all x, then the integral of f will be less than or equal to the integral of g. Obviously the minimum value of a function is less than or equal to the function itself, so the integral of that constant is less than the integral of the function. Similar for the maximum.
I think all of them are true. A) since the function is continuous it will be bounded B) from above the surface under curve will be finite C) another way of stating intermediate values all exist D) the average value of f(X) is between f(-2) and f(3) E) the function is continuous but necessarily derivable on the interval so f'(0) may not exist I thought all are true when I read it lol
As I read them one by one, I thought maybe D was not true. But in reading them all it’s obvious that option E is not true. So then I just had to convince myself that D is true, but that wasn’t too difficult with a little bit of work.
didn’t understand the notation for D however i settled on E after remembering the cube root function
The only notation there that might be unfamiliar is f([-2,3]). That just means the set of all values that the function takes when evaluated at all numbers in the interval [-2,3]. So, for example, if f is a strictly increasing continuous function, then f([-2,3]) = [f(-2), f(3)].
I like your explanation for option D .. keep the good work
Thanks!
I like the work you did with option D, but I'm confused about how you came up with the inequality with the minimun of the function, integral of the function, and maximum of the function. Is that some truth or property of integrals? I've taken a calculus course, but its been a bit.
If you have two functions, f and g, and f(x) is less than or equal to g(x) for all x, then the integral of f will be less than or equal to the integral of g.
Obviously the minimum value of a function is less than or equal to the function itself, so the integral of that constant is less than the integral of the function. Similar for the maximum.
I think all of them are true.
A) since the function is continuous it will be bounded
B) from above the surface under curve will be finite
C) another way of stating intermediate values all exist
D) the average value of f(X) is between f(-2) and f(3)
E) the function is continuous but necessarily derivable on the interval so f'(0) may not exist
I thought all are true when I read it lol
As I read them one by one, I thought maybe D was not true. But in reading them all it’s obvious that option E is not true. So then I just had to convince myself that D is true, but that wasn’t too difficult with a little bit of work.