Chord gradient estimation

5 A is the point \(( 2,1 )\) on the curve \(y = \frac { 4 } { x ^ { 2 } }\).
B is the point on the same curve with \(x\)-coordinate 2.1.

1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places.
2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A .
3. Use calculus to find the gradient of the curve at A .

3 The points \(\mathrm { P } ( 2,3.6 )\) and \(\mathrm { Q } ( 2.2,2.4 )\) lie on the curve \(y = \mathrm { f } ( x )\). Use P and Q to estimate the gradient of the curve at the point where \(x = 2\).

5 The equation of a curve is \(y = \sqrt { 1 + 2 x }\).

1. Calculate the gradient of the chord joining the points on the curve where \(x = 4\) and \(x = 4\). Give your answer correct to 4 decimal places.
2. Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when \(x = 4\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f540b962-ee6b-409a-a2a1-cd7ad4945514-2_1031_1113_273_499} \captionsetup{labelformat=empty} \caption{Fig. 5}
   \end{figure} Fig. 5 shows the graph of \(y = 2 ^ { x }\).

3 The equation of a curve is \(y = \sqrt { 1 + 2 x }\).

1. Calculate the gradient of the chord joining the points on the curve where \(x = 4\) and \(x = 4.1\). Give your answer correct to 4 decimal places.
2. Showing the points you use, calculate the gradient of another chord of the curve which is a closer approximation to the gradient of the curve when \(x = 4\).

4 The diagram shows points \(A\) and \(B\) on the curve \(y = \left( \frac { x } { 4 } \right) ^ { - x }\).
The \(x\)-coordinate of A is 1 and the \(x\)-coordinate of B is 1.1 . \includegraphics[max width=\textwidth, alt={}, center]{a0d9573f-8273-4562-a2d3-07f15d9da1af-4_522_707_1758_278}

1. Find the gradient of chord AB . Give your answer correct to 2 decimal places.
2. Give the \(x\)-coordinate of a point C on the curve such that the gradient of chord AC is a better approximation to the gradient of the tangent to the curve at A .

In Fig. 5, A and B are the points on the curve \(y = 2^x\) with \(x\)-coordinates 3 and 3.1 respectively. \includegraphics{figure_5}

1. Find the gradient of the chord AB. Give your answer correct to 2 decimal places. [2]
2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2^x\) at A. [2]