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HW5: Least-squares fits to models

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Math 753/853 HW5: Least-squares fits to models
Problem 1. Polynomial least-squares fit
(a) Fit a cubic polynomial to the following data, and make a plot that shows the data as dots and the cubic fit as a smooth curve.

xdata = [-1.0,  -0.8,  -0.6,  -0.4,  -0.2,  0.0,  0.2,  0.4,  0.6,  0.8,  1.0];
ydata = [3.41,  3.19,  2.57,  2.44,  1.90, 1.66, 1.17, 1.46, 1.07, 1.44, 2.28];
(b) Write out the polynomial in the form  P(x)=c0+c1x+c2x2+c3x3  with the coefficients specified as numeric values with three digits.

Problem 2. Exponential fit
(a) Given the following experimental measurements of alpha-particle emission of a radioactive substance, fit an exponential function  y=cexp(at)  to the data using least squares. Make a plot that shows the data as dots and the exponential fit as a smooth curve. What are the values of  c  and  a ?

tdata = [0,   4,  8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48]; # time in hours
ydata = [69, 64, 54, 41, 44, 34, 26, 29, 18, 22, 18, 19, 11]; # alpha particle emission rate;
(b) What is the substance’s half-life? (i.e. the time  t  for which  y(t)=0.5y(0) .

Problem 3. Power-law fit
(a) Fit a power-law curve  y=cta  to the following data, and make a plot showing the datapoints as dots and the fit as a smooth curve.

tdata =  [   2,    3,    4,    5,    6,    7,    8,    9,   10];
ydata =  [12.7, 11.2, 8.99, 8.62, 8.12, 8.47, 7.39, 7.24, 6.99];
(b) Write out the least-squares power-law fit  y=cta  as an explicit function in Julia with  c  and  a  specified as numeric values with three digits. E.g. P(t) = 4.32 t^5.19.

Problem 4. c t exp(at) fit
(a) The following data represent measurements of blood concentration of a drug after intravenous injection as a function of time. Fit a function of the form  y=cteat  to the data using least squares. Make a plot that shows the data as dots and the exponential fit as a smooth curve.

tdata = [4,   8,  12,  16,  20,  24]; # time in hours
ydata = [21, 31,  25,  21,  15,  16]; # concentration in ng/ml;
(b) Write out the model  y=cteat  as an explicit function in Julia with numeric values specified to three digits.

(c) Based on the model, at what time do you expect the concentration to reach 5 ng/ml?