Numerical gradient deduction - Differentiation from First Principles

CAIE P1 2021 June Q3

4 marks Easy -1.2

3 The equation of a curve is $y = ( x - 3 ) \sqrt { x + 1 } + 3$. The following points lie on the curve. Non-exact values are rounded to 4 decimal places. $$A ( 2 , k ) \quad B ( 2.9,2.8025 ) \quad C ( 2.99,2.9800 ) \quad D ( 2.999,2.9980 ) \quad E ( 3,3 )$$

Find $k$, giving your answer correct to 4 decimal places.
Find the gradient of $A E$, giving your answer correct to 4 decimal places.
The gradients of $B E , C E$ and $D E$, rounded to 4 decimal places, are 1.9748, 1.9975 and 1.9997 respectively.
State, giving a reason for your answer, what the values of the four gradients suggest about the gradient of the curve at the point $E$.

CAIE P1 2020 Specimen Q1

3 marks Easy -1.2

1 The following points $$A ( 0,1 ) , \quad B ( 1,6 ) , \quad C ( 1.5,7.75 ) , \quad D ( 1.9,8.79 ) \quad \text { and } \quad E ( 2,9 )$$ lie on the curve $y = \mathrm { f } ( x )$. The table below shows the gradients of the chords $A E$ and $B E$.

Chord

$A E$

$B E$

$C E$

$D E$

Gradient of

chord

4

3

Complete the table to show the gradients of $C E$ and $D E$.
State what the values in the table indicate about the value of $\mathrm { f } ^ { \prime } ( 2 )$.

OCR MEI Further Numerical Methods 2019 June Q5

12 marks Standard +0.8

5 Fig. 5 shows spreadsheet output concerning the estimation of the derivative of a function $\mathrm { f } ( x )$ at $x = 2$ using the forward difference method. \begin{table}[h]

	A	B	C	D
1	h	estimate	difference	ratio
2	0.1	6.3050005
3	0.01	6.0300512	-0.274949
4	0.001	6.0030018	-0.027049	0.098379
5	0.0001	6.0003014	-0.0027	0.099835
6	0.00001	6.0000314	-0.00027	0.099983
7	0.000001	6.0000044	$- 2.7 \mathrm { E } - 05$	0.099994
8	1E-07	6.0000016	$- 2.71 \mathrm { E } - 06$	0.100352
9	1E-08	6.0000013	$- 3.02 \mathrm { E } - 07$	0.111457
10	1E-09	6.0000018	$4.885 \mathrm { E } - 07$	-1.61765
11	1E-10	6.0000049	$3.109 \mathrm { E } - 06$	6.363636
12	1E-11	6.0000005	$- 4.44 \mathrm { E } - 06$	-1.42857
13	1E-12	6.0005334	0.0005329	-120
14	1E-13	5.9952043	-0.005329	-10
15	1E-14	6.1284311	0.1332268	-25
16	1E-15	5.3290705	-0.799361	-6
17	1E-16	0	-5.329071	6.666667

\captionsetup{labelformat=empty} \caption{Fig. 5}

\end{table}

Write down suitable cell formulae for
The formula in cell B2 is $\quad = ( \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( ( 2 + \mathrm { A } 2 ) \wedge 3 ) ) ) - \mathrm { LN } ( \mathrm { SQRT } ( \mathrm { SINH } ( 2 \wedge 3 ) ) ) ) / \mathrm { A } 2$ and equivalent formulae are entered in cells B3 to B17.
Write $\mathrm { f } ( x )$ in standard mathematical notation. The value displayed in cell B17 is zero, even though the calculation results in a non-zero answer.
Explain how this has arisen.