Mathematical Modeling and Mathematica

Lab #1 Exercises: Introduction to Mathematica

(Your Name Here)

Exercises

A notebook containing solutions to the following exercises should be "turned in" as your lab report (as described in the first day handout).  Each student is responsible for writing a lab report independently.

Exercise 1

Find a 10-digit number which is prime.

Exercise 2

Plot a graph of the function f(x,y) = Cos[Sqrt[x2 + y2]] over some suitable region in the xy-plane containing the origin.  Experiment with the region and possibly with the PlotPoints option of Plot3D to obtain a pleasing and informative rendering of the graph.

Exercise 3

Modify the Maclaurin polynomial graphing code found in the Programming in Mathematica  section of Lab 1 to compare the graphs of Cos[x] with those of its first six (not eight) Maclaurin polynomials.

Exercise 4

Let  f(x) = 2x3 - 6x + 3  and  g(x) = -4x3 + 3x2 + 2 .  
(a)  Use the graphing capabilities of Mathematica  to graph both functions on a single set of axes.  Examine the graphs and visually estimate the number of points of intersection of the two curves as well as the coordinates of the points of intersection.

(b)  Use the symbolic capabilities of Mathematica  to find the exact values of the x and y coordinates of the points of intersection of the two curves.

(c)  Use the numerical capabilities of Mathematica  to find approximate values for the x and y coordinates of intersection of the two curves.

Exercise 5

Consider the function f(x) = (x3+x+1)/(x4+1). 

(a)  Using the symbolic capabilities of Mathematica , find simplified formulas for f'(x) and for f''(x).

(b)Find all real roots of f'(x) and of f''(x).  These numbers are then candidates for relative extrema and for inflection points.

(c)  Rather than applying the first or second derivative tests to classify the critical points, simply graph f(x), f'(x), and f''(x) on the same set of axes over some appropriate interval and interpret what you see.  Give a full explanation of which points found in (b) above are relative extrema and which are inflection points.  To keep the three graphs straight, plot them in blue, red, and green, respectively.