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Math Help - Integration question

  1. #1
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    Integration question

    Let f(x) be a continous function such that its first two derivatives f'(x) and f''(x) are continous. The tangents to the graph of f(x) at the points with abscissa x=a and x=b make with the X-axies angles π / 3 and π /4 respectively. then the value of the integral

    Integrate [ f'(x) * f''(x) dx ] from a to b equals?

    a. 1-root3
    b. 0
    c. 1
    d. -1

    Thanks
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  2. #2
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    Hello,
    Quote Originally Posted by champrock View Post
    Let f(x) be a continous function such that its first two derivatives f'(x) and f''(x) are continous. The tangents to the graph of f(x) at the points with abscissa x=a and x=b make with the X-axies angles π / 3 and π /4 respectively. then the value of the integral

    Integrate [ f'(x) * f''(x) dx ] from a to b equals?

    a. 1-root3
    b. 0
    c. 1
    d. -1

    Thanks
    I don't know how to solve the final stuff (I guess you have notes about values of the derivative if the tangent makes a certain angle with the x-axis), but I hope this can help you :

    \int_a^b f'(x) f''(x) ~dx=\frac 12 \int_a^b 2 f'(x) [f'(x)]' ~dx

    And this integrates into :
     \left.\frac 12 ~[f'(x)]^2 \right|_a^b
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  3. #3
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    no notes here.

    What I had initially attempted was take f'(x) = t so f''(x) dx = dt therefore the integral reduces to Integrate[t d(t)]

    But dont know exactly how to interprete (a,b) here.
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  4. #4
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    Quote Originally Posted by champrock View Post
    no notes here.

    What I had initially attempted was take f'(x) = t so f''(x) dx = dt therefore the integral reduces to Integrate[t d(t)]

    But dont know exactly how to interprete (a,b) here.
    Yes, your attempt is correct.
    But please next time, write down what you've tried;...

    \int_a^b t ~dt=\left. \frac 12 t^2 \right|_a^b=\left. \frac 12 [f'(x)]^2 \right|_a^b=\frac 12 [f'(b)]^2-\frac 12 [f'(a)]^2

    Now you know that the tangent to the curve at point a, has a slope of f'(a).
    If it forms an angle of pi/3 with the x axis, it means that its slope is tan(pi/3) ---> f'(a)=tan(pi/3) and f'(b)=tan(pi/4)

    There is an example here : Vectors Aligned with X-Y Axes, but I can't find a good link (and it's time to eat ^^)
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  5. #5
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    actually, I was not at all sure whether this was the right way or not, so I didnt bother to write it. Will keep it in mind.

    Enjoy! and thanks once again for boththe questions!
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