By sketching the curves \(y = \ln x\) and \(y = 8 - 2 x ^ { 2 }\) on a single diagram, show that the equation
$$\ln x = 8 - 2 x ^ { 2 }$$
has exactly one real root.
Explain how your diagram shows that the root is between 1 and 2 .
Use the iterative formula
$$x _ { n + 1 } = \sqrt { 4 - \frac { 1 } { 2 } \ln x _ { n } } ,$$
with a suitable starting value, to find the root. Show all your working and give the root correct to 3 decimal places.
The curves \(y = \ln x\) and \(y = 8 - 2 x ^ { 2 }\) are each translated by 2 units in the positive \(x\)-direction and then stretched by scale factor 4 in the \(y\)-direction. Find the coordinates of the point where the new curves intersect, giving each coordinate correct to 2 decimal places.
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The diagram shows the curve with equation
$$x = ( y + 4 ) \ln ( 2 y + 3 ) .$$
The curve crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
Find the gradient of the curve at each of the points \(A\) and \(B\), giving each answer correct to 2 decimal places.