Functions and Graphs - Estimating Gradients from Tangents

The variables \(x\) and \(y\) are connected by the equation \(y = 5x - 3 + \frac{1}{2x}\).

The table below shows some values of \(x\) and the corresponding values of \(y\) correct to \(2\) decimal places.

1. Calculate the value of \(a\).
2. Using a scale of \(2\) cm to represent \(0.1\) unit, draw a horizontal \(x\)-axis for \(0 < x \le 0.7\).
   Using a scale of \(2\) cm to represent \(1\) unit, draw a vertical \(y\)-axis for \(0 \le y \le 8\).
   On your axes, plot the points given in the table and join them with a smooth curve.
3. By drawing a tangent, find the gradient of the curve at \((0.2, 0.5)\).
4. By drawing a suitable line, find the solutions of \(10x^2 - 8x + 1 = 0\) in the range \(0 < x \le 0.7\).
5. The line \(y = kx\) touches the curve \(y = 5x - 3 + \frac{1}{2x}\) at exactly one point in the range \(0 < x \le 0.7\).
   By drawing a suitable straight line on the same axes, use your graph to find the value of \(k\).