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Math Help - Help me study for my test tomorrow!

  1. #1
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    Help me study for my test tomorrow!

    Tomorrow's my test and there are a few concepts that I have to work on.

    1. Sketch the region bounded by the graphs of the equations, and determine its area.
    y=\frac{4}{\sqrt x}
    y=0, x=1, x=9

    Answer: A=16

    My main problem with this is getting rid of the \sqrt x from under so I can begin to integrate.

    2. Find the particular solution of the differential equation f''(x)=6(x-1) whose graph passes through the point (2,1) and is tangent to the line 3x-y-5=0 at that point.

    I already found that f'(x)=3x^2-6x and f(x)=x^3-3x^2. What now? I don't get what the question is asking me.

    3. Use the Second Fundamental Theorem of Calculus to find F'(x).
    F(x)=(integration sign, x between 0 and x) t^2\sqrt {1+t^3} dt

    Answer= x^2+3x+2

    I know that F(x) is the antiderivative of the inside of the integration symbol, but I don't understand how to find F'(x).

    Some help on these questions would greatly be appreciated.
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  2. #2
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    Quote Originally Posted by DarkestEvil View Post
    Tomorrow's my test and there are a few concepts that I have to work on.

    1. Sketch the region bounded by the graphs of the equations, and determine its area.
    y=\frac{4}{\sqrt x}
    y=0, x=1, x=9

    Answer: A=16

    My main problem with this is getting rid of the \sqrt x from under so I can begin to integrate.

    rewrite as \textcolor{red}{4x^{-1/2}} , then integrate.

    2. Find the particular solution of the differential equation f''(x)=6(x-1) whose graph passes through the point (2,1) and is tangent to the line 3x-y-5=0 at that point.

    I already found that f'(x)=3x^2-6x and f(x)=x^3-3x^2. What now? I don't get what the question is asking me.

    f''(x) = 6x-6

    f'(x) = 3x^2 - 6x + C ... did you calculate the constant of integration?
    what information did they give you that might help you find it?

    3. Use the Second Fundamental Theorem of Calculus to find F'(x).
    F(x)=(integration sign, x between 0 and x) t^2\sqrt {1+t^3} dt

    Answer= x^2+3x+2

    I know that F(x) is the antiderivative of the inside of the integration symbol, but I don't understand how to find F'(x).

    your "answer" is incorrect.

    \textcolor{red}{\frac{d}{dx} \int_a^x f(t) \, dt = f(x)}
    ...
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