Foundations of Analysis: Sequences of Functions

A web page to accompany Section 4.5 of
Foundations of Analysis by David Belding and Kevin Mitchell
(For instructions, see below.)

Quick Instructions

The program, above, shows the graph of a function f(x) that can depend on an iteger parameter, n. The program can animate the graph by displaying it for a sequence of parameter values. The user controls the animation using a strip of controls on the bottom of the program. There are input boxes where the user can specify:

• the range of values for x and y
• the range of values for the sequence parameter n and the current value of n
• a new function expression (see below)
• buttons to control the animation
• and a pull-down menu of pre-load examples at the top of the program.

Hover your mouse over an interface element for some information about it!

Mathematical Expressions in the Program

The program works with "expressions" such as "x+3" or "sqrt(n*x)". Expressions can use the operators +, -, *, /, ^, and ** (where both ^ and ** indicate exponentiation), and they can use the mathematical constants pi and e. Multiplication must always be indicated explicitly with a * operator. Expressions can include various mathematical functions: sin, cos, tan, sec, cot, csc, arcsin, arccos, arctan, exp, ln, log2, log10, abs, sqrt, trunc, round, floor, ceiling, and cubert. (Here, "abs" is the absolute value function and "cubert" is the cube root function.) The argument of a function myst always be enclosed in parenthese. A particularly useful feature is the use of C-style conditional expressions such as "x<y? x : y", which means "if x<y then x otherwise y". This option can be used to define piecewise or "split" functions.

The Pre-loaded Examples

The pre-loaded examples in the menu at the top of the program come from Section 4.5 of Foundations of Analysis by David Belding and Kevin Mitchell. The first three examples illustrate examples 1, 2, and 3 in this section. The next eight examples correspond to some of the sequences of functions in the problem section. The last three examples are just interesting!

By looking at the syntax of the examples in the program and comparing it to the definition of the sequence in the text, it should be possible for you to create more examples of your own or to illustrate the other problems in the text.

Now, just go ahead and play!

Hobart and William Smith Colleges: Department of Mathematics and Computer Science