Homework # 3a:  Functions and Expressions; Continuity and Derivatives

Below are a few simple exercises that will help you understand some of the problems you may have encountered while working on Module 3a.  Be sure you understand the solutions to these exercises because these problems will come up again later in this course.

You may use the Documentation Center to assist you with these problems, but you should not copy your answers verbatim from the Documentation Center.  After you understand the solution in the Documentation Center, you should present the solution in your own words.  Try to avoid jargon and special notation;  use standard English and classical Mathematical notation.  Also you should not copy examples from the Documentation Center;  rather you should provide your own examples to ensure that you understand the concepts.

Please format your solutions appropriately.  Use text format, not input format, when you are typing text.  Write coherent sentences and paragraphs;  part of your grade will depend on how clearly you present your ideas.

You must work on these exercises by yourself, not in pairs;  if you get stuck and cannot find the solution in the Documentation Center, you may consult the labbie or the instructor during their office hours.

Functions and Expressions

Mathematica typically distinguishes between functions and expressions.  This distinction is basic to many functions that you will use later in Mathematica, so it is important for you to be clear now about this distinction.

Problems

1.  Explain in your own words the difference in Mathematica between a function and an expression.

2.  Give 3 examples to illustrate the difference between a function and an expression.

3.  Give 3 examples of procedures in Mathematica that will work only if you pass functions, but will not work correctly if you pass expressions.  (Hint:  Use the Documentation Center.)

Continuity

Here are a few problems to make sure that you understand the notion of continuity.

Problems

1.  Explain in your own words the what it means for a function to be continuous at

a.  a single point,

b.  in an interval.

2.  Is the function y = |x| continuous at x = 0?  Give a reason for your answer.

3.  Consider the function f[x] given by

f[x] = x  if x is rational,
	f[x] = 0  if x is irrational.

Where is this function continuous?

Derivatives

Here are a few problems to make sure that you understand the notion of a derivative.

Problems

1.  What is the definition of the derivative?

2.  Is the function y = |x| differentiable at x = 0?  Give a reason for your answer.

3.  Consider the function f[x] given by

f[x] = (x^2)  if x is rational,
	f[x] = 0    if x is irrational.

Where is this function differentiable?