Q1. The goal of this question is to find the date and time of the earliest sunrise and the date and time of the latest sunset. You should achieve this by interpolating the above data points.
(a) Explain how you define the nodes xk that represent the dates above. (Note that there are several options for your answer here! Choose only one.)
(b) Explain how you define the times of sunrise and sunset represented by function values at the nodes xk. (Again, there are several options for your answer here, but you should choose only one.)
(c) Define a global interpolating polynomial Pr that passes through all data points given for the time of sunrise. Also define a global interpolating polynomial Ps that passes through all data points given for the time of sunset.
- Specify the coefficients of Pr in the form Pr(x) = aN xN +.............+ a1x + a0.
- Specify the coefficients of Ps in the form Ps(x) = bNxN + .......+ b1x + b0.
- Use Pr to find an approximate date for the earliest sunrise and state at what time the sun is estimated to rise on that date. Clearly explain how you go about this.
Use Ps to find an approximate date for the latest sunset and state at what time the sun is estimated to set on that date.
(d) Include a plot with your answer that shows the graphs of Pr and Ps in one figure, along with the data points as well as the locations on the graphs of the earliest sunrise and latest sunset. Here, you may use the units as chosen in parts (a) and (b).
(e) How accurate do you think your predictions are? Note that it is easy to find data about sunrise and sunset online; you are welcome to compare your findings with this data.
Q2. Now consider the resulting lengths of daylight for each data point. The goal is to find the date and length of the longest day, that is, the day with the largest length of daylight.
(a) Explain how you can use Pr and Ps from Q1 to define an interpolating polynomial PL that represents the length of daylight.
- Specify the coefficients of PL in the form PL(x) = cNxN +..........+ c1x + c0.
- Use PL to find an approximate date and length for the longest day.
(b) Compare your findings with the plot from Q1(d) and discuss any discrepancies.
Q3. Consider the integral
x0∫x3 PL (x) dx,
where x0 represents 13 November and x3 represents 24 January and PL is the interpolated polynomial from Q2a.
(a) Approximate the integral with the composite trapezoidal rule using PL only at the measurement data itself. Explain how many subdivisions you take and why.
(b) Use the composite trapezoidal rule with N = 100 subdivisions to approximate the integral.
(c) The error of the trapezoidal rule can be bounded using an upper bound for the second derivative of the integrand. Describe how you can use PL to find such a bound and give estimated bounds on the errors of your calculations in parts (a) and (b).
(d) Explain how to interpret the meaning of this integral in terms of daylight and which units you used for your answers.