Complete numerical table

A question is this type if and only if it provides a partially completed table of chord gradients for various h values and asks students to fill in missing entries and/or interpret the pattern.

3 questions · Moderate -1.0

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OCR MEI AS Paper 1 2020 November Q5
4 marks Moderate -0.8
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 Qchange in \(y\)gradient PQ
111
110.11.11.0488090.0488090.488088
110.011.011.0049880.0049880.498756
110.0011.0011.0005000.0005000.499875
\captionsetup{labelformat=empty} \caption{Table 5.2}
\end{table}
  1. 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.
  2. 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 \(P\).
  3. Use calculus to find the gradient of the curve at the point P .
CAIE P1 2024 June Q4
3 marks Easy -1.8
The equation of a curve is \(y = \text{f}(x)\), where f\((x) = (2x - 1)\sqrt{3x - 2} - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. \(A(2, 4)\), \(B(2.0001, k)\), \(C(2.001, 4.00625)\), \(D(2.01, 4.06261)\), \(E(2.1, 4.63566)\), \(F(3, 11.22876)\)
  1. Find the value of \(k\). Give your answer correct to 5 decimal places. [1]
The table shows the gradients of the chords \(AB\), \(AC\), \(AD\) and \(AF\).
Chord\(AB\)\(AC\)\(AD\)\(AE\)\(AF\)
Gradient of chord6.25016.25116.26087.2288
  1. Find the gradient of the chord \(AE\). Give your answer correct to 4 decimal places. [1]
  2. Deduce the value of f\('(2)\) using the values in the table. [1]
AQA AS Paper 1 2018 June Q9
8 marks Moderate -0.3
Craig is investigating the gradient of chords of the curve with equation \(\mathrm{f}(x) = x - x^2\) Each chord joins the point \((3, -6)\) to the point \((3 + h, \mathrm{f}(3 + h))\) The table shows some of Craig's results.
\(x\)\(\mathrm{f}(x)\)\(h\)\(x + h\)\(\mathrm{f}(x + h)\)Gradient
\(3\)\(-6\)\(1\)\(4\)\(-12\)\(-6\)
\(3\)\(-6\)\(0.1\)\(3.1\)\(-6.51\)\(-5.1\)
\(3\)\(-6\)\(0.01\)
\(3\)\(-6\)\(0.001\)
\(3\)\(-6\)\(0.0001\)
  1. Show how the value \(-5.1\) has been calculated. [1 mark]
  2. Complete the third row of the table above. [2 marks]
  3. State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to \(0\) [1 mark]
  4. Using differentiation from first principles, verify that your result in part (c) is correct. [4 marks]