Finite vs. infinite problems

Nikolay Brodskiy

University of Tennessee

Calculus is the first course where students consider a new kind of problems: infinite problems. I call a problem infinite if it does not allow a finite solution.

Example: find the area below the graph of a function.

Immediate questions:

What does it mean to solve an infinite problem?

Classroom assessment: how can a student solve an infinite problem in finite time?

Typical solutions in a Calculus course:

1) A list of standard infinite problems is given with the answers to memorize.

Example:  area of a circle of radius R is equal to ℼ (R^2).

Example:  (lim_(x→0))((sin(x))/x)=1.

2) Any other problem of the course reduces to one of the standard problems in a finite sequence of steps.

Infinite problems (as opposed to finite problems) require a different thinking process.

Unfortunately, most of the standard examples and exercises in Calculus do not require a different thinking process because a simple recipe is provided:

(lim_(x→a))f(x)=?

(lim_(n→∞))(∑_(i=1)^n)(Subscript[x, i])=?

In case the limit exists, approximations are easy to get:

the closer x to a is the better f(x) approximates (lim_(x→a))f(x)

the greater n is the better (∑_(i=1)^n)(Subscript[x, i]) approximates(lim_(n→∞))(∑_(i=1)^n)(Subscript[x, i])

The formula for success is simple: any improvement leads to success.

Non-standard problem:

(∑_(i,j∈ℤ))(Subscript[x, i,j])=?

How to arrange summation? Not any improvement leads to success.

Analogy with learning: what does it mean to learn Calculus in one semester?

Making progress in learning Calculus is easy: read, watch, solve problems, ...

Assessment: compute, explain in words, illustrate with a picture, write a program, ...

But how to be successful?