Trapezium rule applied to real-world data

7 The value of a function, \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\), and its gradient function, \(\frac { \mathrm { dy } } { \mathrm { dx } }\), when \(x = 2\), is given in Table 7.1. \begin{table}[h] \end{table}

1. Determine the approximate value of the error when \(f ( 2 )\) is used to estimate \(f ( 2.03 )\). The Newton-Raphson method is used to find a sequence of approximations to a root, \(\alpha\), of the equation \(\mathrm { f } ( x ) = 0\). The spreadsheet output showing the iterates, together with some further analysis, is shown in Table 7.2. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 7.2}

   	A	B	C	D
   1	r	Xr	difference	ratio
   2	0	12
   3	1	-13.1165572	-25.1165572
   4	2	1.76283279	14.87939004	-0.5924136
   5	3	2.18052157	0.41768878	0.02807163
   6	4	2.182419024	0.001897454	0.00454275
   7	5	2.182419066	\(4.13985 \mathrm { E } - 08\)	\(2.1818 \mathrm { E } - 05\)

   \end{table}
2. 1. Explain what the values in column D tell you about the order of convergence of this sequence of approximations.
   2. Without doing any further calculation, state the value of \(\alpha\) as accurately as you can, justifying the precision quoted.

1 The table shows some values of \(x\), together with the associated values of a function, \(\mathrm { f } ( x )\).

1. Use the information in the table to calculate the most accurate estimate of \(f ^ { \prime } ( 2 )\) possible.
2. Calculate an estimate of the error when \(f ( 2 )\) is used as an estimate of \(f ( 2.05 )\).

2 You are given that \(a = \tanh ( 1 )\) and \(b = \tanh ( 2 )\). \(A\) is the approximation to \(a\) formed by rounding \(\tanh ( 1 )\) to 1 decimal place. \(B\) is the approximation to \(b\) formed by rounding \(\tanh ( 2 )\) to 1 decimal place.

1. Calculate the following.

4 Between 1946 and 2012 the mean monthly maximum temperature of the water surface of a lake in northern England has been recorded by environmental scientists. Some of the data are shown in Table 4.1. \begin{table}[h] \end{table} Table 4.2 shows a difference table for the data. \begin{table}[h] \end{table}

1. Complete the copy of the difference table in the Printed Answer Booklet.
2. Explain why a quadratic model may be appropriate for these data.
3. Use Newton's forward difference interpolation formula to construct an interpolating polynomial of degree 2 for these data. This polynomial is used to model the relationship between \(T\) and \(t\). Between 1946 and 2012 the mean monthly maximum temperature of the water surface of the lake was recorded as \(8.9 ^ { \circ } \mathrm { C }\) for October and \(7.5 ^ { \circ } \mathrm { C }\) for November.
4. Determine whether the model is a good fit for the temperatures recorded in October and November. A scientist recorded the mean monthly maximum temperature of the water surface of the lake in 2022. Some of the data are shown in Table 4.3. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.3}

   Month	May	June	July	August	September
   \(t =\) Time in months	0	1	2	3	4
   \(T =\) Mean temperature in \({ } ^ { \circ } \mathrm { C }\)	10.3	14.7	16.9	16.9	14.8

   \end{table}
5. Adapt the polynomial found in part (c) so that it can be used to model the relationship between \(T\) and \(t\) for the data in Table 4.3.

6 Table 6.1 shows some values of \(x\) and the associated values of a function, \(y = f ( x )\). \begin{table}[h] \end{table}

1. Explain why it is not possible to use the central difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\).
2. Use the forward difference method to calculate an estimate of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 1\). A student uses the forward difference method to calculate a series of approximations to \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) with different values of the step length, \(h\). These approximations are shown in Table 6.2, together with some further analysis. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 6.2}

   \(h\)	0.8	0.4	0.2	0.1	0.05	0.025	0.0125	0.00625
   approximation	0.130452	0.138647	0.143381	0.145942	0.147277	0.147959	0.148304	0.148477
   difference		0.008195	0.004734	0.002561	0.001335	0.000682	0.000345	0.000173
   ratio			0.577633	0.541099	0.521186	0.510762	0.505424	0.502723

   \end{table}
3. 1. Explain what the ratios of differences tell you about the order of the method in this case.
   2. Comment on whether this is unusual.
4. Determine the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) when \(x = 2\) as accurately as possible. You must justify the precision quoted.

1 Fig. 1 shows some spreadsheet output. \begin{table}[h] \end{table}

1. Write the value displayed in cell A3 in standard mathematical notation. The formula in cell A3 is \(= \mathrm { A } 2 - \mathrm { A } 1\)
2. Explain why the value displayed in cell A3 is non zero.
3. Write down the value of the number stored in cell A2 to the highest precision possible.
4. Explain why your answer to part (c) may be different to the actual value stored in cell A2.

2 Fig. 2 shows 3 values of \(x\) and the associated values of a function, \(\mathrm { f } ( x )\). \begin{table}[h] \end{table} Find a polynomial \(p ( x )\) of degree 2 to approximate \(\mathrm { f } ( x )\), giving your answer in the form \(p ( x ) = a x ^ { 2 } + b x + c\), where \(a\), \(b\) and \(c\) are constants to be determined.

3 At Heathwick airport each passenger's luggage is weighed before being loaded into the hold of the aeroplane. Each weight is displayed digitally in kg to 1 decimal place. Some examples are given in Fig. 3. \begin{table}[h] \end{table} On each flight, the total weight of luggage is calculated to ensure compliance with health and safety regulations. Winston models this situation by assuming that the displayed weights are rounded to 1 decimal place, and that the total weight of luggage is calculated using the displayed values. On a flight to Athens, there are 154 items of passengers' luggage.

1. Determine the maximum possible error, according to Winston's model, when the total weight of luggage is calculated for the flight to Athens. Piotre models this situation by assuming that the displayed weights are chopped to 1 decimal place, and that the total weight of luggage is calculated using the displayed values.
2. Determine the maximum possible error, according to Piotre's model, when the total weight of luggage is calculated for the flight to Athens. A health and safety inspector notes that the total of the displayed weights is 3080.2 kg . However, when the luggage is all weighed together in the loading bay, the total weight is found to be 3089.44 kg .
3. Determine whether Winston's model or Piotre's model is a better fit for the data.

5 You are given that \(g ( x ) = \frac { \sqrt [ 3 ] { x ^ { x } + 25 } } { 2 }\). Fig. 5.1 shows two values of \(x\) and the associated values of \(\mathrm { g } ( x )\). \begin{table}[h] \end{table}

1. Use the central difference method to calculate an estimate of \(\mathrm { g } ^ { \prime } ( 1.5 )\), giving your answer correct to 3 decimal places. The equation \(x ^ { x } - 8 x ^ { 3 } + 25 = 0\) has two roots, \(\alpha\) and \(\beta\), such that \(\alpha \approx 1.5\) and \(\beta \approx 4.4\).
2. Obtain the iterative formula \(x _ { n + 1 } = g \left( x _ { n } \right) = \frac { \sqrt [ 3 ] { x _ { n } ^ { X _ { n } } + 25 } } { 2 }\).
3. Use your answer to part (a) to explain why it is possible that the iterative formula \(x _ { n + 1 } = g \left( x _ { n } \right) = \frac { \sqrt [ 3 ] { x _ { n } ^ { X _ { n } } + 25 } } { 2 }\) may be used to find \(\alpha\).
4. Starting with \(x _ { 0 } = 1.5\), use the iterative formula to find \(x _ { 1 } , x _ { 2 } , x _ { 3 } , x _ { 4 } , x _ { 5 }\), and \(x _ { 6 }\).
5. Use your answer to part (d) to state the value of \(\alpha\) correct to 8 decimal places. Starting with \(x _ { 0 } = 4.5\) the same iterative formula is used in an attempt to find \(\beta\). The results are shown in Fig. 5.2. \begin{table}[h]

   \(n\)	\(x _ { n }\)
   0	4.5
   1	4.81826433
   2	6.27473453
   3	23.2937196
   4	\(2.0654 \mathrm { E } + 10\)
   5	\#NUM!

   \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{table}
6. Explain why \#NUM! is displayed in the cell for \(x _ { 5 }\).
7. On the diagram in the Printed Answer Booklet, starting with \(x _ { 0 } = 4.5\), illustrate how the iterative formula works to find \(x _ { 1 }\) and \(x _ { 2 }\).
8. Determine what happens when the relaxed iteration \(x _ { n + 1 } = ( 1 - \lambda ) x _ { n } + \lambda g \left( x _ { n } \right)\) is used to try to find \(\beta\) with \(x _ { 0 } = 4.5\), in each of the following cases.

7 Fig. 7.1 shows two values of \(x\) and the associated values of \(\mathrm { f } ( x )\). \begin{table}[h] \end{table}

1. Use the forward difference method to calculate an estimate of the gradient of \(\mathrm { f } ( x )\) at \(x = 3\), giving your answer correct to 4 decimal places. Fig. 7.2 shows some spreadsheet output with additional values of \(x\) and the associated values of \(\mathrm { f } ( x )\). \begin{table}[h]

   \(x\)	3	3.00001	3.0001	3.001	3.01	3.1
   \(\mathrm { f } ( x )\)	6.082763	6.08274	6.082541	6.08054	6.060454	5.848846

   \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{table} These values have been used to produce a sequence of estimates of the gradient of \(\mathrm { f } ( x )\) at \(x = 3\), together with some further analysis. This is shown in the spreadsheet output in Fig. 7.3. \begin{table}[h]

   \(h\)	0.1	0.01	0.001	0.0001	0.00001
   estimate	-2.339165	-2.230883	-2.220532	-2.219501	-2.219398
   difference	0.1082815	0.010352	0.0010307	0.000103
   ratio	0.095602	0.099567	0.0999568

   \captionsetup{labelformat=empty} \caption{Fig. 7.3}
   \end{table} Tommy states that the differences between successive estimates is decreasing so rapidly that the order of convergence of this sequence of estimates is much faster than first order.
2. Explain whether or not Tommy is correct.
3. Use extrapolation to determine the value of the gradient of \(\mathrm { f } ( x )\) at \(x = 3\) as accurately as possible, justifying the precision quoted.
4. Calculate an estimate of the absolute error when \(\mathrm { f } ( 3 )\) is used as an approximation to \(\mathrm { f } ( 3.02 )\).

1

1. 1. Determine the relative error when
      \begin{table}[h]

      	A	B	C
      1	2	1.414214	2

      \captionsetup{labelformat=empty} \caption{Fig. 1}
      \end{table} The formula in cell B1 is = SQRT (A1)
      and the formula in cell C 1 is \(\quad = \mathrm { B } 1 \wedge 2\).
      Ben evaluates \(1.414214 ^ { 2 }\) on his calculator and obtains 2.000001238 . He states that this shows that the value displayed in cell C1 is wrong. Explain whether Ben is correct.

2 The table shows some values of \(x\) and the associated values of \(\mathrm { f } ( x )\).

1. Complete the difference table in the Printed Answer Booklet.
2. Explain why the data may be interpolated by a polynomial of degree 2.
3. Use Newton's forward difference interpolation formula to obtain a polynomial of degree 2 for the data.

4 The table shows some values of \(x\) and the associated values of \(\mathrm { f } ( x )\).

1. Calculate four estimates of the derivative of \(\mathrm { f } ( x )\) at \(x = 4\).
2. Without doing any further calculation, state the value of \(f ^ { \prime } ( 4 )\) as accurately as you can, justifying the precision quoted.

5 When Nina does the weekly grocery shopping she models the total cost by adding up the cost of each item in her head as she goes along. To simplify matters she rounds the cost of each item to the nearest pound. One week Nina buys 48 items.

1. Calculate the maximum possible error in Nina's model in this case. Nina estimated the total cost of her shopping to be \(\pounds 92\). The actual cost is \(\pounds 90.23\).
2. Explain whether this is consistent with Nina's model. The next week her husband, Kareem, does the weekly shopping. He models the total cost by chopping the cost of each item to the nearest pound as he goes along. On this occasion Kareem buys 52 items.
3. Calculate the expected error in Kareem's model in this case. Using his model Kareem estimates the total cost as \(\pounds 76\). The total cost of the shopping is \(\pounds 103.24\).
4. Explain how such a large error could arise. The next week Kareem buys \(n\) items.
5. Write down a formula for the maximum possible error when Kareem uses his model to estimate the total cost of his shopping.
6. Explain how Kareem's model could be adapted so that his formula gives the same expected error as Nina's model when they are both used to estimate the total cost of the shopping.

1

1. Solve the following simultaneous equations. $$\begin{aligned} & x + \quad y = 1 \\ & x + 0.99 y = 2 \end{aligned}$$
2. The coefficient 0.99 is correct to two decimal places. All other coefficients in the equations are exact. With the aid of suitable calculations, explain why your answer to part (i) is unreliable.

4 The table below gives values of a function \(y = \mathrm { f } ( x )\).

1. Calculate three estimates of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) using the central difference method.
2. State the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) to an appropriate degree of accuracy. Justify your answer.

5 A vehicle is moving in a straight line. Its velocity at different times is recorded and shown below. The velocities are recorded to 5 significant figures and the times may be assumed to be exact. It is suggested initially that a quadratic model may be appropriate for this situation.

1. Given that the vehicle is modelled as a particle with constant mass, what assumption about the net force acting on the vehicle leads to a quadratic model?
2. Find Newton's interpolating polynomial of degree 2 to model this situation. Write your answer in the form \(v = a t ^ { 2 } + b t + c\).
3. Comment on whether this model appears to be appropriate.
4. Use this model to find an approximation to the distance travelled over the interval \(5 \leq t \leq 15\). Further investigation suggests that a cubic model may be more appropriate.
5. What technique would you use to fit a cubic model to the data in the table?

Figure 2 below shows a 1.5 metre length of pipe. \includegraphics{figure_16} The symmetrical cross-section of the pipe is shown below, in Figure 3, where \(x\) and \(y\) are measured in centimetres. \includegraphics{figure_16_cross_section} Use the trapezium rule, with the values shown in the table below, to find the best estimate for the volume of the pipe. \begin{array}{|c|c|c|c|c|c|c|} \hline x & 0 & 0.4 & 0.8 & 1.2 & 1.6 & 2
\hline y & -3 & -2.943 & -2.752 & -2.353 & -1.572 & 0
\hline \end{array} [5 marks]

A particle is moving in a straight line with velocity \(v\text{ ms}^{-1}\) at time \(t\) seconds as shown by the graph below. \includegraphics{figure_15}

1. Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period \(20 \leq t \leq 100\) [4 marks]
2. Over the same time period, the curve can be very closely modelled by a particular quadratic. Explain how you could find an alternative estimate using this quadratic. [1 mark]