Polynomial Interpolation from Data

7 Sam decided to go on a high-protein diet. Sam's mass in \(\mathrm { kg } , M\), after \(t\) days of following the diet is recorded in Fig. 7.1. \begin{table}[h] \end{table} A difference table for the data is shown in Fig. 7.2. \begin{table}[h] \end{table}

1. Complete the copy of the difference table in the Printed Answer Booklet. Sam's doctor uses these data to construct a cubic interpolating polynomial to model Sam's mass at time \(t\) days after starting the diet.
2. Find the model in the form \(\mathrm { M } = \mathrm { at } ^ { 3 } + \mathrm { bt } ^ { 2 } + \mathrm { ct } + \mathrm { d }\), where \(a , b , c\) and \(d\) are constants to be determined. Subsequently it is found that when \(\mathrm { t } = 40 , \mathrm { M } = 78.7\) and when \(\mathrm { t } = 50 , \mathrm { M } = 80.05\).
3. Determine whether the model is a good fit for these data.
4. By completing the extended copy of Fig. 7.2 in the Printed Answer Booklet, explain why a quartic model may be more appropriate for the data.
5. Refine the doctor's model to include a quartic term.
6. Explain whether the new model for Sam's mass is likely to be appropriate over a longer period of time.

2 A car tyre has a slow puncture. Initially the tyre is inflated to a pressure of 34.5 psi . The pressure is checked after 3 days and then again after 5 days. The time \(t\) in days and the pressure, \(P\) psi, are shown in the table below. You are given that the pressure in a car tyre is measured in pounds per square inch (psi). The owner of the car believes the relationship between \(P\) and \(t\) may be modelled by a polynomial.

1. Explain why it is not possible to use Newton's forward difference interpolation method for these data.
2. Use Lagrange's form of the interpolating polynomial to find an interpolating polynomial of degree 2 for these data. The car owner uses the polynomial found in part (b) to model the relationship between \(P\) and \(t\).
   Subsequently it is found that when \(t = 6 , P = 26.0\) and when \(t = 10 , P = 24.4\).
3. Determine whether the owner's model is a good fit for these data.
4. Explain why the model would not be suitable in the long term.

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.

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.

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.

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?