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Math Help - function with derivative

  1. #1
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    function with derivative

    Right I was given this problem to solve. Some of the questions I can do, others i am stuck with and need a maths guru to help.

    So here goes, I am given on a grid the drawing of the graph y = xsquared - 2x -4

    1 - I am asked to write down the coordinates of the minimum point of the curve - THIS I CAN DO, NO HELP NEEDED

    2 - Use the graph to find estimates of the solutions to the equation xsquared - 2x - 4 = 0. Give your answer to 1d.p. THIS I CAN DO, NO HELP NEED

    3 - Draw a suitable straight line on the grid to find estimates of the solutions of the equation xsquared - 3x - 6 = 0 THIS I CANNOT DO, AS IT IS A QUADRATIC AND THEREFORE WOULD PRODUCE A BELL SHAPE CURVE AND NOT A STRAIGHT LINE AS THEY SAY, NEED HELP PLEASE

    4 - For y = xsquared - 2x - 4

    4a) Find dy/dx CAN DO THIS, NO HELP NEEDED

    4b) find the gradient of the curve at the point where x = 6 NEED HELP WITH THIS PLEASE

    Many thanks in advance,

    Nat
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  2. #2
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    Quote Originally Posted by Natasha1 View Post
    Right I was given this problem to solve. Some of the questions I can do, others i am stuck with and need a maths guru to help.

    So here goes, I am given on a grid the drawing of the graph y = xsquared - 2x -4

    1 - I am asked to write down the coordinates of the minimum point of the curve - THIS I CAN DO, NO HELP NEEDED

    2 - Use the graph to find estimates of the solutions to the equation xsquared - 2x - 4 = 0. Give your answer to 1d.p. THIS I CAN DO, NO HELP NEED

    3 - Draw a suitable straight line on the grid to find estimates of the solutions of the equation xsquared - 3x - 6 = 0 THIS I CANNOT DO, AS IT IS A QUADRATIC AND THEREFORE WOULD PRODUCE A BELL SHAPE CURVE AND NOT A STRAIGHT LINE AS THEY SAY, NEED HELP PLEASE

    4 - For y = xsquared - 2x - 4

    4a) Find dy/dx CAN DO THIS, NO HELP NEEDED

    4b) find the gradient of the curve at the point where x = 6 NEED HELP WITH THIS PLEASE

    Many thanks in advance,

    Nat
    For (3)

    x^2-3x-6=\left(x^2-2x-4\right)-(x+2)

    This is zero where x^2-2x-4=x+2

    Hence draw the line y=x+2 and find where it intersects x^2-2x-4
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  3. #3
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    Not sure I fully understand this, how about 4b?
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  4. #4
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    Quote Originally Posted by Natasha1 View Post
    Not sure I fully understand this, how about 4b?
    y=x+2 is a line.

    If you subtract the linear equation from the quadratic, you get the new quadratic.
    Subtracting 2 things that are equal give zero, and they are equal at the points of intersection.

    (4)

    \displaystyle\frac{dy}{dx} is the slope of the tangent to the curve (the gradient of the curve) over the domain of x.

    So you first calculate the derivative (instantaneous gradient of the curve) which is \displaystyle\frac{dy}{dx} and evaluate it for x=6
    (the derivative will be in terms of x).
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  5. #5
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    Ok so dy/dx = 2x - 2

    but how do you evaluate it for x = 6, I have never done this. Just need clear explanation to hopefully understand.
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  6. #6
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    Quote Originally Posted by Natasha1 View Post
    Ok so dy/dx = 2x - 2

    but how do you evaluate it for x = 6, I have never done this. Just need clear explanation to hopefully understand.
    Ok,

    you have evaluated the derivative.
    This is a formula for the gradient of the curve, which varies with x
    (imagine the tangent to the curve changing slope as you move from one value of x to another along the x axis).

    If you now place any real value of x into the derivative (formula for calculating the gradient),
    you will evaluate the curve gradient at that value of x.

    So, if you substitute x with 6 in 2x-2, to get 2(6)-2, that will be the curve gradient when x is 6.
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  7. #7
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    So the gradient of the curve is 10
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  8. #8
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    Quote Originally Posted by Natasha1 View Post
    So the gradient of the curve is 10
    Yes, when x is 6 only.
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  9. #9
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    Thanks for clearly explaining
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  10. #10
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    You're welcome Natasha,

    here is a graph of the situation
    Attached Thumbnails Attached Thumbnails function with derivative-quadratic-gradients.jpg  
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  11. #11
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    Here is a graph for part (3)
    Attached Thumbnails Attached Thumbnails function with derivative-2-quadratics.jpg  
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