Introduction: A Parable of Precalculus

Studying precalculus is like crossing a bridge. You're between two worlds; one old, one new.

On one side of the bridge is an ancient land that was inhabited long ago, long before the bridge itself could even be imagined. Over many centuries, mathematical settlements formed there. They grew into kingdoms with names like Arithmetic, Algebra, Geometry, and Trigonometry...

Each kingdom developed its own tools, its own laws, and its own methods for doing business. Mathematical guilds evolved, and through slow refinement of their techniques they were eventually able to craft elegant solutions to many mathematical problems, both real and imagined.

Still, attempts to solve other, practical problems only snapped, crumbled, and broke. The tools and the methods of the individual guilds were too crude.

It took time for a network of communication to develop among the separate guilds. Slowly, however, the network turned into trade roads, and the kingdoms coalesced into the multi-cultural city-state of Modern Mathematics.

In the new metropolis, tracts by the likes of Leibniz, Newton, and Seki were circulated. These tracts contained intricate diagrams and detailed calculations, parts of a revolutionary blueprint for what would become "The Bridge".

But there were many obstacles to these visionary plans. The bridge took several centuries to construct...

Where it finally led, however, was a wonderous place, a fertile land spreading to the horizon in every direction. Settlers crossing the bridge called this new place: Calculus.

In Calculus, ancient mathematical guilds found new life, working together to solve problems from virtually every area of inquiry. Old tools were forged into instruments of tremendous power, and massive projects of engineering were undertaken.

The bridge itself came to be called Precalculus. Today, anyone wishing to cross the bridge must still serve an apprenticeship among the well-worn shop benches in the guilds of Arithmetic, Algebra, Geometry, and Trigonometry. This is how one pays the toll. This is the only way to reach the opportunities on the other side.

Of course you are eager to cross, and see for yourself this wonderous place. But you must be patient...

When you do make your passage, take time to stop and look from the bridge's highest heights, rising between the two lands, between the old and the new, between your own mathematical apprenticeship and your mathematical maturity. The view is quite incredible, and getting across is part of the fun.

***

Functions

You may be unable to name the highest peaks of the Moluccas � remote islands in the Malay Archipelago � and yet still be quite familiar with the many spices that grow there. You can stir nutmeg into holiday eggnog, and appreciate the flavor, without the slightest knowledge that the nutmeg's 400 species of tropical evergreens cover the Moluccas' spectacular mountains and volcanoes.

Your experience with functions is perhaps somewhat similar. You've probably heard of functions. You've probably tasted their distinctive mathematical "flavor". You may even recognize some of their many varieties. And yet, faced with the information that there are far more than 400 types of function, covering the mountainsides in the foreign land of Calculus, you may begin to doubt if you know them at all.

Precalculus will talk about functions in a way that may, at first, be unfamiliar to you. It would be a mistake, however, to tell yourself that you really don't know functions at all. You probably know a lot about functions!

For example, you've probably worked with equations like this:

drawn graphs in an x,y-coordinate system like this:

These are all functions! They may look like quite different things, but they can all be given a very general, unifying description.

Here's how.

Each of the examples above involves a pair of values: x� and �y . Lots of things come in matched pairs like this.

Functions are just the rules for finding the one thing that uniquely corresponds with another. In the examples, there is always a unique �y� corresponding to each �x . The rule tells you how to find the proper mate: plug into the equation, read the graph, read the table.

If you can appreciate how each of these examples fits this very general notion of "function", then you've already made your first steps on the bridge towards Calculus.

***

Models

Functions cover the landscape of Calculus. Once you arrive there, their ubiquitous presence will provide a pleasant reminder that you really haven't travelled too far from what you already know.

In fact, you will recognize many things in Calculus. The traditions of Arithmetic, Algebra, Geometry, and Trigonometry flourish there.

It is those two large peaks � the Derivative and the Integral � that may seem cold and intimidating. You may wonder if you'll ever be able to scale them and cross that bridge in the sky, the Fundamental Theorem of Calculus.

Don't worry. You'll do it.

The peaks only look big from where you are now. Their size is actually something of a mirage. As you get closer, you'll see them begin to diminish.

Eventually, you will discover one of Calculus' great secrets: the Derivative and the Integral are actually only mathematical models. Models � like in a toy store! They may resemble the Alps from a distance, but, like the model mountains of a toy train set, they're not as big as we imagine.

Mere models probably deserve names that aren't quite so intimidating. Let's call the peaks "derivative" and "integral", just as they're called in the land of calculus. (That's right, "calculus", not "Calculus".) And that bridge? You may have already guessed: On the other side, it's called only "precalculus".

How curious a place, this calculus! Are the mountains there big, or are they small?

It's important to have proper perspective about the human scale of things in calculus. We wouldn't want to be frightened away before our trip has even begun.

Just as importantly, however, we wouldn't want to confuse calculus' accessibility with any sort of diminished significance. It took tremendous craftsmanship to build the derivative and the integral, and they are two of the most ingeniously useful mathematical models ever devised. They do indeed tower above all of mathematics.

The usefulness of these models comes from their ability to reflect, like great works of art, universal aspects of our everyday experience. We don't have to fully appreciate this ponderous achievement right now, but understanding what goes into the construction of these models is an important part of our preparation for calculus.

***

Rates of Change

What, then, might the derivative be a model of?

It's not the model of a thing ...but rather of an idea.

It's not a toy store model, constructed from plastic and glue ...but rather a mathematical model, constructed from functions and their changes.

Let's squint a little harder...

Functions tell us how one thing depends on another: The unique �y� that corresponds to a particular �x . Functions are themselves models. They provide a mathematical description of things as various as:

* The amount of water in your bathtub

* The speed of your car as you brake

* The effects of calorie consumption on your weight

Or:

* World population

* The effect of pollution on marine organisms

* The speed of rotation of a galaxy so far from its center

Lots of things! Functions let us describe many cause-and-effect relationships in the world.

Of course, things change. The �y� that corresponds to a particular �x� today may not be the �y� that corresponds to that �x� tomorrow. Your bathtub fills up. Your car slows down. The more you eat, the heavier you become.

Often, we can't keep track of all the world's changing values. Instead, we summarize what we see by talking about the rate at which something is changing:

* Your bathtub is filling at a rate of one gallon every two seconds

* Your car slows down at a rate of 10 mph every second

* An extra 500 calories each day translates into an additional pound

It's more likely that we will know the rate at which something is changing than have a complete knowledge of the thing itself.

The derivative gives us a mathematical way to talk about rates: Anything described by "So much change in �y� per so much change in �x".

Don't get it? Don't worry. Derivatives are given a careful study in calculus, not precalculus. Precalculus will talk a lot about functions, how they change, and, indeed, their rates of change. However, if you understand how "rate" is used in everyday language, then you already understand how it is used in precalculus.

Just keep listening for the word: "rate", "rate", "rate". It's like listening to foreign speakers on an audio tape. By the time you get to calculus, the idea will be so second-nature that the careful definition of "derivative" will seem like only a clever turn of phrase.

***

accumulation

The derivative gives us a mathematical way to talk about something we encounter every day in a hundred different places: the rates at which things change.

The integral lets us describe something equally ubiquitous: accumulation.

You know about accumulation:

* Dust under your bed

* Cereal poured into your bowl

* Money in your bank account

Or:

* People on earth

* Zebra mussels in the Great Lakes

* Charge on a capacitor in an electronic circuit

All of these things involve changing quantities, and all of them result in accumulations.

If you can provide a clear description of a changing quantity � that is, if you can provide a function � the integral will describe the amount that is accumulating. The integral models accumulations in the same way that the derivative models rates of change.

Calculus wouldn't be so powerful if it didn't describe, with just these two basic models, so much of the changing world.

Like the derivative, the integral is given a careful study in calculus, not precalculus. However, just as precalculus will talk repeatedly about the intuitive notion of "rate", it will also talk repeatedly about the intuitive notion of "accumulation".

Trust your intuition about the meaning of these words, and use your study of precalculus to gain experience with the many places where they are used. When you get to calculus, you'll be pleasantly surprised to find how much you've already mastered.

Before you venture off across the bridge, take a moment to ponder, one last time, that engineering marvel stretching between the peaks of the derivative and the integral: The Fundamental Theorem of Calculus. The derivative is a model of rates of change. The integral is a model of accumulation. The more rapidly something changes, the more quickly it accumulates. The more slowly something changes, the more gradually it accumulates. Is it really so difficult to imagine a connection?

Maybe you can climb the peaks of Calculus, and cross that bridge in the sky, after all...

Studying precalculus is like crossing a bridge. You're between two worlds; one old, one new.

On one side of the bridge is an ancient land that was inhabited long ago, long before the bridge itself could even be imagined. Over many centuries, mathematical settlements formed there. They grew into kingdoms with names like Arithmetic, Algebra, Geometry, and Trigonometry...

Each kingdom developed its own tools, its own laws, and its own methods for doing business. Mathematical guilds evolved, and through slow refinement of their techniques they were eventually able to craft elegant solutions to many mathematical problems, both real and imagined.

Still, attempts to solve other, practical problems only snapped, crumbled, and broke. The tools and the methods of the individual guilds were too crude.

It took time for a network of communication to develop among the separate guilds. Slowly, however, the network turned into trade roads, and the kingdoms coalesced into the multi-cultural city-state of Modern Mathematics.

In the new metropolis, tracts by the likes of Leibniz, Newton, and Seki were circulated. These tracts contained intricate diagrams and detailed calculations, parts of a revolutionary blueprint for what would become "The Bridge".

But there were many obstacles to these visionary plans. The bridge took several centuries to construct...

Where it finally led, however, was a wonderous place, a fertile land spreading to the horizon in every direction. Settlers crossing the bridge called this new place: Calculus.

In Calculus, ancient mathematical guilds found new life, working together to solve problems from virtually every area of inquiry. Old tools were forged into instruments of tremendous power, and massive projects of engineering were undertaken.

The bridge itself came to be called Precalculus. Today, anyone wishing to cross the bridge must still serve an apprenticeship among the well-worn shop benches in the guilds of Arithmetic, Algebra, Geometry, and Trigonometry. This is how one pays the toll. This is the only way to reach the opportunities on the other side.

Of course you are eager to cross, and see for yourself this wonderous place. But you must be patient...

When you do make your passage, take time to stop and look from the bridge's highest heights, rising between the two lands, between the old and the new, between your own mathematical apprenticeship and your mathematical maturity. The view is quite incredible, and getting across is part of the fun.

***

Functions

You may be unable to name the highest peaks of the Moluccas � remote islands in the Malay Archipelago � and yet still be quite familiar with the many spices that grow there. You can stir nutmeg into holiday eggnog, and appreciate the flavor, without the slightest knowledge that the nutmeg's 400 species of tropical evergreens cover the Moluccas' spectacular mountains and volcanoes.

Your experience with functions is perhaps somewhat similar. You've probably heard of functions. You've probably tasted their distinctive mathematical "flavor". You may even recognize some of their many varieties. And yet, faced with the information that there are far more than 400 types of function, covering the mountainsides in the foreign land of Calculus, you may begin to doubt if you know them at all.

Precalculus will talk about functions in a way that may, at first, be unfamiliar to you. It would be a mistake, however, to tell yourself that you really don't know functions at all. You probably know a lot about functions!

For example, you've probably worked with equations like this:

**y = x2 + 2x + 3 ,**drawn graphs in an x,y-coordinate system like this:

These are all functions! They may look like quite different things, but they can all be given a very general, unifying description.

Here's how.

Each of the examples above involves a pair of values: x� and �y . Lots of things come in matched pairs like this.

Functions are just the rules for finding the one thing that uniquely corresponds with another. In the examples, there is always a unique �y� corresponding to each �x . The rule tells you how to find the proper mate: plug into the equation, read the graph, read the table.

If you can appreciate how each of these examples fits this very general notion of "function", then you've already made your first steps on the bridge towards Calculus.

***

Models

Functions cover the landscape of Calculus. Once you arrive there, their ubiquitous presence will provide a pleasant reminder that you really haven't travelled too far from what you already know.

In fact, you will recognize many things in Calculus. The traditions of Arithmetic, Algebra, Geometry, and Trigonometry flourish there.

It is those two large peaks � the Derivative and the Integral � that may seem cold and intimidating. You may wonder if you'll ever be able to scale them and cross that bridge in the sky, the Fundamental Theorem of Calculus.

Don't worry. You'll do it.

The peaks only look big from where you are now. Their size is actually something of a mirage. As you get closer, you'll see them begin to diminish.

Eventually, you will discover one of Calculus' great secrets: the Derivative and the Integral are actually only mathematical models. Models � like in a toy store! They may resemble the Alps from a distance, but, like the model mountains of a toy train set, they're not as big as we imagine.

Mere models probably deserve names that aren't quite so intimidating. Let's call the peaks "derivative" and "integral", just as they're called in the land of calculus. (That's right, "calculus", not "Calculus".) And that bridge? You may have already guessed: On the other side, it's called only "precalculus".

How curious a place, this calculus! Are the mountains there big, or are they small?

It's important to have proper perspective about the human scale of things in calculus. We wouldn't want to be frightened away before our trip has even begun.

Just as importantly, however, we wouldn't want to confuse calculus' accessibility with any sort of diminished significance. It took tremendous craftsmanship to build the derivative and the integral, and they are two of the most ingeniously useful mathematical models ever devised. They do indeed tower above all of mathematics.

The usefulness of these models comes from their ability to reflect, like great works of art, universal aspects of our everyday experience. We don't have to fully appreciate this ponderous achievement right now, but understanding what goes into the construction of these models is an important part of our preparation for calculus.

***

Rates of Change

What, then, might the derivative be a model of?

It's not the model of a thing ...but rather of an idea.

It's not a toy store model, constructed from plastic and glue ...but rather a mathematical model, constructed from functions and their changes.

Let's squint a little harder...

Functions tell us how one thing depends on another: The unique �y� that corresponds to a particular �x . Functions are themselves models. They provide a mathematical description of things as various as:

* The amount of water in your bathtub

* The speed of your car as you brake

* The effects of calorie consumption on your weight

Or:

* World population

* The effect of pollution on marine organisms

* The speed of rotation of a galaxy so far from its center

Lots of things! Functions let us describe many cause-and-effect relationships in the world.

Of course, things change. The �y� that corresponds to a particular �x� today may not be the �y� that corresponds to that �x� tomorrow. Your bathtub fills up. Your car slows down. The more you eat, the heavier you become.

Often, we can't keep track of all the world's changing values. Instead, we summarize what we see by talking about the rate at which something is changing:

* Your bathtub is filling at a rate of one gallon every two seconds

* Your car slows down at a rate of 10 mph every second

* An extra 500 calories each day translates into an additional pound

It's more likely that we will know the rate at which something is changing than have a complete knowledge of the thing itself.

The derivative gives us a mathematical way to talk about rates: Anything described by "So much change in �y� per so much change in �x".

Don't get it? Don't worry. Derivatives are given a careful study in calculus, not precalculus. Precalculus will talk a lot about functions, how they change, and, indeed, their rates of change. However, if you understand how "rate" is used in everyday language, then you already understand how it is used in precalculus.

Just keep listening for the word: "rate", "rate", "rate". It's like listening to foreign speakers on an audio tape. By the time you get to calculus, the idea will be so second-nature that the careful definition of "derivative" will seem like only a clever turn of phrase.

***

accumulation

The derivative gives us a mathematical way to talk about something we encounter every day in a hundred different places: the rates at which things change.

The integral lets us describe something equally ubiquitous: accumulation.

You know about accumulation:

* Dust under your bed

* Cereal poured into your bowl

* Money in your bank account

Or:

* People on earth

* Zebra mussels in the Great Lakes

* Charge on a capacitor in an electronic circuit

All of these things involve changing quantities, and all of them result in accumulations.

If you can provide a clear description of a changing quantity � that is, if you can provide a function � the integral will describe the amount that is accumulating. The integral models accumulations in the same way that the derivative models rates of change.

Calculus wouldn't be so powerful if it didn't describe, with just these two basic models, so much of the changing world.

Like the derivative, the integral is given a careful study in calculus, not precalculus. However, just as precalculus will talk repeatedly about the intuitive notion of "rate", it will also talk repeatedly about the intuitive notion of "accumulation".

Trust your intuition about the meaning of these words, and use your study of precalculus to gain experience with the many places where they are used. When you get to calculus, you'll be pleasantly surprised to find how much you've already mastered.

Before you venture off across the bridge, take a moment to ponder, one last time, that engineering marvel stretching between the peaks of the derivative and the integral: The Fundamental Theorem of Calculus. The derivative is a model of rates of change. The integral is a model of accumulation. The more rapidly something changes, the more quickly it accumulates. The more slowly something changes, the more gradually it accumulates. Is it really so difficult to imagine a connection?

Maybe you can climb the peaks of Calculus, and cross that bridge in the sky, after all...