- In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
The curve \(C\) has equation \(y = \mathrm { f } ( x )\) where \(x \in \mathbb { R }\)
Given that
- \(\mathrm { f } ^ { \prime } ( x ) = 2 x + \frac { 1 } { 2 } \cos x\)
- the curve has a stationary point with \(x\) coordinate \(\alpha\)
- \(\alpha\) is small
- use the small angle approximation for \(\cos x\) to estimate the value of \(\alpha\) to 3 decimal places.
The point \(P ( 0,3 )\) lies on \(C\)
Find the equation of the tangent to the curve at \(P\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.