5 Fig. 5.1 shows part of the curve \(y = x ^ { \frac { 1 } { 2 } }\). P is the point \(( 1,1 )\) and \(Q\) is the point on the curve with \(x\)-coordinate \(1 + h\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a1b6c827-7d74-4527-9b60-58872e3d5ef7-4_451_611_991_242}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
\end{figure}
Table 5.2 shows, for different values of \(h\), the coordinates of P , the coordinates of Q , the change in \(y\) from P to Q and the gradient of the chord PQ .
\begin{table}[h]
| \(x\) for P | \(y\) for P | \(h\) | \(x\) for Q | \(y\) for Q | change in \(y\) | gradient PQ |
| 1 | 1 | 1 | | | | |
| 1 | 1 | 0.1 | 1.1 | 1.048809 | 0.048809 | 0.488088 |
| 1 | 1 | 0.01 | 1.01 | 1.004988 | 0.004988 | 0.498756 |
| 1 | 1 | 0.001 | 1.001 | 1.000500 | 0.000500 | 0.499875 |
\captionsetup{labelformat=empty}
\caption{Table 5.2}
\end{table}
- Fill in the missing values for the case \(h = 1\) in the copy of Table 5.2 in the Printed Answer Booklet. Give your answers correct to 6 decimal places where necessary.
- Explain how the sequence of values in the last column of Table 5.2 relates to the gradient of the curve \(y = x ^ { \frac { 1 } { 2 } }\) at the point \(P\).
- Use calculus to find the gradient of the curve at the point P .