7. The curve with equation
$$y = - x + \tanh ( 36 x ) , \quad x \geq 0$$
has a maximum turning point \(A\).
- Find, in exact logarithmic form, the \(x\)-coordinate of \(A\).
- Show that the \(y\)-coordinate of \(A\) is
$$\frac { \sqrt { 35 } } { 6 } - \frac { 1 } { 36 } \ln ( 6 + \sqrt { 35 } )$$
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The function \(f\) is defined by \(f ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\).
- Find \(\mathrm { f } ^ { \prime \prime \prime } ( \mathrm { x } )\) (i.e. the third derivative of \(f\) ) showing all your intermediate steps. Hence, find the Maclaurin series for \(f ( x )\) up to and including the \(x ^ { 3 }\) term.
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[8 marks] - Use the expansion of \(e ^ { x }\) together with the result in part (a) to show that, up to and including the \(x ^ { 3 }\) term,
$$e ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } = 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$
where \(k\) is a rational number to be found.
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[3 marks]
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