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Math Help - Curve

  1. #1
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    Curve

    A curve y= (2x-3). Find the x coordinates of the two points on the curve which have gradient 6.
    The region under the x axis and to the right of the y axis is bounded by the curve. What is the area of the region?
    thanks
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  2. #2
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    Hello, gracey!

    Given: . y\:=\: (2x-3)^3

    (a) Find the x-coordinates of the two points on the curve which have gradient 6.

    We have: . y' \:=\:6(2x-3)^2 \:=\:6 \quad\Rightarrow\quad(2x-3)^2 \:=\:1 \quad\Rightarrow\quad 2x-3 \:=\:\pm1

    . . \begin{array}{cccccc}2x-3 \:=\:1 & \Rightarrow & 2x\:=\:4 & \Rightarrow & \boxed{x \:=\:2} \\ \\[-3mm] 2x-3\:=\:\text{-}1 & \Rightarrow & 2x \:=\:2 & \Rightarrow & \boxed{x \:=\:1} \end{array}




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    (b) The region under the x-axis and to the right of the y-axis
    is bounded by the curve. .What is the area of the region?

    \text{Area} \;=\;\int^{\frac{3}{2}}_0 \text{-}\,(2x-3)^3\,dx \;=\;\text{-}\,\frac{1}{8}(2x-3)^4\,\bigg]^{\frac{3}{2}}_0 \quad\hdots\quad\text{etc.}

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  3. #3
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    thanks so much
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