Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
(a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of
$$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$
(b) Given that this approximate value is equal to 16 , find the value of \(k\).