Find x-coordinates with given gradient

2. In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} A curve has equation $$y = 3 x ^ { 5 } + 4 x ^ { 3 } - x + 5$$ The points \(P\) and \(Q\) lie on the curve.
The gradient of the curve at both point \(P\) and point \(Q\) is 2
Find the \(x\) coordinates of \(P\) and \(Q\).

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1. Find the gradient of the curve \(y = 2 x ^ { 2 }\) at the point where \(x = 3\).
2. At a point \(A\) on the curve \(y = 2 x ^ { 2 }\), the gradient of the normal is \(\frac { 1 } { 8 }\). Find the coordinates of \(A\). Points \(P _ { 1 } \left( 1 , y _ { 1 } \right) , P _ { 2 } \left( 1.01 , y _ { 2 } \right)\) and \(P _ { 3 } \left( 1.1 , y _ { 3 } \right)\) lie on the curve \(y = k x ^ { 2 }\). The gradient of the chord \(P _ { 1 } P _ { 3 }\) is 6.3 and the gradient of the chord \(P _ { 1 } P _ { 2 }\) is 6.03.
3. What do these results suggest about the gradient of the tangent to the curve \(y = k x ^ { 2 }\) at \(P _ { 1 }\) ?
4. Deduce the value of \(k\).

1. In this question you should show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point \(P ( 5 , - 13 )\) lies on \(C\) The line \(l\) is the tangent to \(C\) at \(P\)

1. Use differentiation to find the equation of \(l\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.
2. Hence verify that \(l\) meets \(C\) again on the \(y\)-axis. The finite region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\) and the line \(l\).
3. Use algebraic integration to find the exact area of \(R\).