Plotting problem 5.mws

1. Shapes of Curves.

The population of honey bee's raised in an apiary started with 50 bees at time t=0, and is modeled by the function P(t) = 75200/(1+1503*exp(-.5932*t)) , where t is time in weeks, with t <25 .

Submission:

Graph this population function, being sure to label axes. Next, graph the derivative, and label the axes. Now think about this question: When was the population growing the fastest? Discuss the answer to the question in the context of both the first and second graphs. Use vocabulary like "point of inflection" and "critical point".

Submission worksheet:

 

2. More on shapes of Curves.

The family of bell-shaped curves y = 1/(sigma*sqrt(2*Pi)) exp(-(x-mu)^2/(2*sigma^2)) occurs in probability and statistics, where it is called the "Normal Density Function". The constant mu is called the mean and the positive constant sigma is called the standard deviation .

Submission:

Investigate this family of curves by doing the following:

(a) Fix mu = 0 and plot, on the same set of coordinate axes, four curves corresponding to different values of sigma .

(b) Fix mu = 50 and plot, on a new set of coordinate axes, four curves corresponding to same values of sigma you used above.

(c) Repeat, with mu = -50 .

(d) Now for the general bell-shaped curve, explain where the maximum value occurs, and where the points of

inflection occur. Also discuss the concavity of the curve.

Submission worksheet:

 

3. Even More on Shapes of Curves.

The hydrogen atom is composed of one proton in the nucleus and one electron, which moves around the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit, but that it occupies a state known as an orbital , which may be thought of as a cloud of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state , or 1s-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function p(r) = 4/(a[0]^3) r^2 exp(-2*r/a[0]) , 0 < r , and where a[0] is the Bohr radius , and has a numerical value of about 5.59x 10^(-11) meters.

Submission:

(a) Plot the function p , over a physically reasonable domain, and label your axes.

(b) What is the maximum value of p ? Where does it occur? Where are the points of inflection of p ?

(Answer these three questions using meters as a unit, and also using a[0] as a unit.)

(c) Discuss the concavity of the curve.

Submission worksheet: