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#### Calculating e

Write a

separate function that uses the expression:e = lim (1 + 1/n)

^{n}n → ∞

to find approximations for the mathematical constant e. Your function should take as input a long double precision number, and return an approximate value for*e*as a long double precision number. Be careful about the types that you use for numbers in this program. Your main program should ask for the user to input a number that you will pass to your function for it to use as*n*in the formula above for calculating*e*. Your program should compare your approximation for*e*to the value that you get by calculating:e^{1}. You can do so using the exp function. Note to use this function, you must have an#include <cmath>

line near the top of your program. To use this function include a line something like:cout << exp(1.0L) << '\n';

This will give exp a long double version of 1.0, which will give you a fairly precise value for*e*. Also, calculate and print out the percent error in your calculation of*e*. Note that you also need to use the pow(x,y) function which calculates x^{y}.Test your program with the following input: 1., 10., 100., 1000., 10.

^{6}, 10.^{9}and 10.^{12}. Send your results in with your assignment. Comment on your results.Use the

*cout.precision(j)*command to force cout to show*j*digits. Use this function before the places where you are going to print out results to force the program to show you enough digits so that you can tell the accuracy of your program. 8 or 10 digits will probably be sufficient.#### Blackbody radiation

Write a program which find the peak wavelength in the light emitted by a blackbody. A blackbody is an ideal object that is both a perfect emitter and a perfect absorber of light. (See section 38.8 Continous Spectra, p.1334-1337, in

*University Physics*by Young and Freeman, a similar text, or Wikipedia for more details.) The spectrum of light from follows the form of the Plank radiation law:I(λ) = 2πhc

where I(λ) is emittance, λ is wavelength, h is Plank's constant, c is the speed of light, and T is the temperature.^{2}/(λ^{5}(e^{hc/λkT}- 1))Plank's radiation law gives the amount of light emitted as a function of the temperature the object and the wavelength of the light. Plots of Plank's radiation law give curves which peak at wavelength's given by Wien's law:

λ

where λ_{m}= b/T_{m}is the wavelength at which the maximum amount of light is emitted, T is the absolute temperature, and b is a constant which is 2.8977685 x 10^{-3}K*m in SI units. For this problem it will be most convenient if you use temperature in Kelvin (K), wavelengths in nm, and b = 2.8977685 x 10^{6}K*nm.Your program should have a function that will take a temperature in Kelvin and return a wavelength in nm. The main function of your program should call the calculation function with the input list below.

- Your program should calculate the wavelength of the maximum emission for the following temperatures: 2.7 K (cosmic microwave background), 250 K (cold winter day), 295 K (room temperature on Earth), 2000 K (molten lava), and 5800 K (surface of Sun). Your program should print the results for these temperature to the screen. Attach a copy of those results with your assignment.
- Your program should also create a plot of the wavelength versus temperature for the data above. Use the plplot library to create your plot. Save your plot to a file and attach a copy of your plot with your assignment.

### Using PLPlot

Before trying to use plplot, take a look at the example of a simple program using plplot. You may want to download that program and compile it using the instructions in its header. Note that there are several lines from that program that you will need to copy. You will need the include line:#include <plstream.h>

You will also need to define all of the numeric variables that you use with the plot as types PLFLT and PLINT, in place of float and int. You will also need corresponding lines like the rest of the example. To compile your program, you should add:-I/usr/include/plplot -lplplotcxxd -lplplotd

to the end of your usual compiling command. PLPlot allows you to choose from many different forms of output. While developing your program you should probably choose*<11> wxwidget*or*<1> xwin*. For your final postscript plot output you should choose*<3> ps*or hit the save file button from the wxwidget window.-
#### Answer the following Review Questions from your textbook:

3.7, 3.8, and 4.6.