Mathematics 162                                  Laboratory 1                           Week of Jan. 25, 1993
	  
Name: _____________________________   Lab Partner: ___________________________
	  
Consulted with:  ____________________________________________________________

© Lafayette College, 1994.

The Natural Logarithm Function

The purpose of this laboratory is to explore properties of the natural logarithm function while reviewing many of the Mathematica commands you used last semester.

Recall that the natural logarithm function is defined, for x > 0, by

Mathematica knows this function as Log[x].

Part 1:  Graphical and Numerical Views of ln x

Recall, the command below will calculate the first derivative of ln x.  To execute it, use the mouse to position the cursor over the command.  Click the left mouse button once (firmly) to make the blinking insertion point appear in the line, then strike the Insert key or Shift-Enter.

D[Log[x], x]

Now, modify the command so that it will compute the second derivative of ln x.  You may want to look at the summary of Mathematica commands handed out at the start of this lab.

For what values of x is ln x increasing?  decreasing?

For what values of x will the graph be concave up?  concave down?

Use Plot[Log[x], {x, 0, 10}] to get a first picture of the graph of ln x.  Does this graph support the conclusions that you have drawn above?  Print out a copy of the graph, and indicate on it what features corroborate what you have found, or indicate what features differ with your conclusions.  Explain any differences.

Use

Solve[Log[x] == 1, x]

to determine the value of x for which ln x = 1.

The letter e (E in Mathematica output) represents an irrational number (like π) that is the base of the natural logarithm function.  Use

N[E]

to get a decimal approximation for this number. Mathematica says e @

Locate e graphically by using

Plot[{Log[x], 1}, {x, 0.5, 4}, PlotStyle->{Red,Blue}]

to graph y = ln x and y = 1 on the same set of axes.  Print a copy of this graph and show the location of e on the horizontal axis.  Now shade the region bounded by the graph of ln x, the x-axis, and the vertical line x = e.  Use Integrate to compute the area of the region you have shaded, and write it below.

Use the following steps to investigate the behavior of ln x for large values of x.

Plot[Log[x], {x, 0, 100}]

What does this graph suggest about the behavior of ln x as x gets large?  Can you guess the limiting value of ln x as x gets large from this graph?  What value do you guess?

Recall that you can use the Table command to create a table of values.  Try

Table[{x, Log[x]}, {x, 10., 100., 10.}]

to get a feeling for how rapidly ln x is growing.  Would you describe the growth as rapid or slow?

To check the values of ln x over a wider range of x values, try

Table[{10.^n, Log[10.^n]}, {n, 1, 10}]

What are the smallest and largest inputs you use in this table?  What is the difference of any two successive outputs?  What do you conclude about the limit of ln x as x gets large?

Use a similar approach to investigate the behavior of ln x for values of x near 0.  What do you conclude about the limit of ln x as x approaches zero?

Part 2:  The Function ln x/x

Define the function f, for x > 0, by

f[x_] := Log[x]/x

Describe the graph of f.  Mention in your description of the graph:  the behavior of 
f (x ) when x is very large or very small, the intervals on which f is increasing (decreasing), the intervals on which the graph is concave up (concave down), and the extreme values of f.  You are welcome to include a hand sketch or printout of the graph of f in your discussion, but address all of the other points in a paragraph, using full sentences.

Use the space provided here to take notes on your investigation; write your solution on a separate sheet of paper.