Math 112 Honors: MATLAB Assignment 2

TAYLOR SERIES AND POLYNOMIALS Due Date: April 18, 2011

The sample program below calculates 2nd and 4th degree Taylor polynomials for the function

$$f(x)=1/(1+x^2)$$

about the point

$$a=0.$$

The polynomials are plotted, together with the function, over the interval

$$[-1,1]$$

Contents

Sample program

Lines beginning with a percent sign (%) are comments construct Taylor polynomials.

figure(1);clf; %open figure one and clear any plots
x=[-1: .01:1];
y=(1+x.^2).^(-1);  %the function
P2=1-x.^2;         %the second-order Taylor polynomial (calculated by hand)
P4=1-x.^2+x.^4;    %the 4th order Taylor polynomial
% annotating the graph is always important
figure(1);clf; %open and clear the 1st figure
plot(x,y,x,P2,'--',x,P4,'-.')
legend('1/(1+x^2)','P_2','P_4')
xlabel('x')
title('The function and the 2^{nd} and 4^{th} order Taylor Polynomials')
figure(2);clf; %open and clear the 2nd figure
plot(x,abs(y-P2),x,abs(y-P4),'--') % abs is the absolute value
legend('|1/(1+x^2)-P_2|','|1/(1+x^2)-P_4|')
xlabel('x')
title('Errors in the two Taylor Polynomials')
% Program output:

New Matlab Idea

the Legend command generates a legend for your graph. If there are two curves in your graph, the legend command should consist of two strings (defined using single quotation marks). Note that the underscore ('_') is used to make subscripts and the caret ('^') is used to make superscripts. if you want to create a superscript with more than one character in it, enclose the superscripted text in braces ('{}')

For example, to type

$$e^{i x}$$

you would use the string 'e^{ix}'.

Variables used

Your assignment: Radiation from the Stars

A $blackbody$ is a system that absorbs all the radiation that falls on it. Proposed in the late 19th century, the Rayleigh-Jeans Law expresses the energy density of blackbody radiation of wavelength $\lambda$ as:

$$f(\lambda)=\frac{8\pi k T}{\lambda^4}$$

where $\lambda$ is measured in meters, $T$, temperature, in kelvins, and $k$ is the Boltzmann's constant. The Rayleigh-Jeans Law agrees with experiments for long wavelengths but disagrees drastically for short wavelengths. [The experiments show that $f(\lambda)\rightarrow 0$ as $\lambda \rightarrow 0^+$]. The Planck's Law (found by Max Planck in 1900) models the blackbody radiation better:

$$f(\lambda) = \frac{8\pi h c \lambda^{-5}}{e^{hc/(\lambda k T)} - 1}$$

where $\lambda$ is measured in meters, $T$ is the temperature measured in kelvins, $h =6.6262\times 10^{-34}$ J s is Planck's constant, $c=2.997925 \times 10^8$ m/s is the speed of light, and $k=1.3807 \times 10^{-23}$ J/K is the Boltzmann's constant.

I) Show that the Planck's Law models blackbody radiation better for short wavelengths.

II) Use a Taylor polynomial to show that Planck's Law gives approximately the same values as the Rayleigh-Jeans Law for long wavelengths. [Hint: expand the denominator of Planck's Law using Taylor series of $e^x$.]

III) Plot the Planck's Law using 2nd and 3rd degree Taylor polynomials when approximating the denominator along with the Planck's Law for $0<\lambda<4$ $\mu$m. Use $T=6000$ K (the temperature of the sun).

IV) Plot both laws on a same graph and comment on the similarities and differences. Use $T=6000$ K.

BONUS QUESTION: Read the Matlab help files on how to use the Symbolic Math Toolbox. This is a part of the program that can actually manipuate algebra and calculus expressions. Use the command taylor, to compute the first five terms in the Taylor series and repeat part (III).


Published with MATLAB® 7.8