Assignment 4 - 2010

Due date(s):	08 November 2010
(PDF)	Assignment questions
Other instructions	Hand-in at class.

<rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/> .. |m3| replace:: m\ :sup:`3`

.. rubric:: Questions 1, 2 and 3 will be posted at the end of the semester in assignment 4(a).

Question 4 [2]

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.. Similar to Tutorial 8, 2009

The viscosity of sulphuric acid, :math:`\nu`, varies with purity, :math:`p` in the following manner:

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math:`p` [%] 20 60 80

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math:`\nu` [millipascal] 1.40 5.37 17.4

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1. . Express :math:`\nu(p)` as a quadratic function using Lagrange interpolating polynomials. Do not simplify the polynomial.
2. . Express :math:`\nu(p)` as a quadratic function using Newton interpolating polynomials. Do not simplify the polynomial.
3. . Fit a cubic spline through the data points: clearly show your :math:`{\bf Xa = y}` linear system of equations, then solve them using computer software; finally report your spline coefficients.
4. . Use computer software to plot:

	*	the Newton interpolating polynomial 

* the cubic spline, * and the 3 data points on the same graph.

1. . What is the estimated viscosity at :math:`p` = 40% purity using linear interpolation?
2. . Which of the estimation procedures that you used above has the closest estimate to the true value of 2.51 millipascal?

Question 5 [2]

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The following data are collected from a bioreactor experiment, during the growth phase.

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Time [hours] 0 1.0 2.0 4.0 6.0

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math:`C` [g/L] 0.1 0.341 1.102 4.95 11.24

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Fit a natural cubic spline for these data and use it to estimate the number of cells at time 3, 5, and 7 hours.

Show your matrix derivation for the linear system of equations, and solve it using computer software. Plot the cubic spline over the time range 0 to 8 hours.

Bonus question [0.5]

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Use the cubic spline from the previous question and find the time where the cell count was approximately 10.0 g/L. Do not solve the equation by hand, but investigate `MATLAB's <http://www.mathworks.com/help/techdoc/ref/roots.html>`_ or `Python's <http://docs.scipy.org/doc/numpy/reference/generated/numpy.roots.html>`_ polynomial root finding function: ``roots(...)``

.. raw:: latex

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