7.
Given that \(k\) is a positive constant and \(\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4\)
- show that \(3 k + 5 \sqrt { k } - 12 = 0\)
- Hence, using algebra, find any values of \(k\) such that
$$\int _ { 1 } ^ { k } \left( \frac { 5 } { 2 \sqrt { x } } + 3 \right) \mathrm { d } x = 4$$
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- Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 2\), find \(\mathrm { f } ( 3 + h )\)
Express your answer in the form \(h ^ { 2 } + b h + c\), where \(b\) and \(c \in \mathbb { Z }\).
- The curve with equation \(y = x ^ { 2 } - 4 x + 2\) passes through the point \(P ( 3 , - 1 )\) and the point \(Q\) where \(x = 3 + h\).
Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\).
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