IMPORTANT MATH HELP. The graph of f shown consists of a semicircle and two line segments on the interval [-2,6]?

Please help, with details if possible.

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  • 2 months ago

    How many times did you check the quality of that upload?

    Guessing g(x) = int[-2 to x] f(t) dt

    g(x) is the area between the graph of f(x) and x-axis, from -2 to x.

    Where of course, regions below x-axis are taken with negative area.

    a)

    What's the area under f(x) from -2 to -2? (this one should not be that hard)

    What's the area under f(x) from -2 to 0?

    .....

    b)

    If g is increasing that means g is larger when you go right.

    Are you adding some area hence increasing g when you go right anywhere from -2 to 5? Yes hence g(x) is increasing from -2 to 5.

    After 5 you start adding negative area so the total area is being decreased: g(x) is decreasing on interval (5, 6).

    c)

    Concave up means increasing increasing.

    Not only are you adding area as you go right, you're adding MORE AND MORE area each time. (or maybe subtracting less or less). All in all, this is where rate of change is increasing.

    Suppose you start from -2. A little shift to the right will cover only a small area because f(x) is really low there. But then move a little more and now you're adding more area. As you pass the top of the circle f(t) starts decreasing so you're not adding as much area anymore. Then g is concave down.

    g(x) concave up on intervals (-2, 0) and (2, 4)

    d)

    The absolute minimum is 0 at x=0. Had the graph extended downward to the right to like x=10 then the absolute minimum would be negative at x=10. Eyeball that here it's not the case because total area is positive.

    Absolute max is at x=5. It's the area from -2 to 5.

    e)

    Points of inflection are where g turns from concave up to concave down or vice versa. These are points (0, g(0)), (2, g(2)), and (5, g(5)).

    f)

    Try it and compare

    https://www.desmos.com/calculator/gttaskhsqp

    Of course precision isn't crucial. Important are key aspects like inflections, rate of change, and no sharp corners (since f(x) is continuous g(x) needs to be differentiable at every point)

    If I missed the bounds on the integral then I hope you got the idea.

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  • ted s
    Lv 7
    2 months ago

    would love to help but this is an excellent problem to see if YOU understand some key concepts...a) talks about areas , b) talks about the meaning when g ' = f , c) about g ' ' = f ' meaning , etc....problem is very easy if you understand what you have been studying

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