So You Think You’re Educated, But You Don’t Know Calculus

A brief introduction to one of humanity’s greatest inventions

Michael Z. SpiveyDepartment of Mathematics and Computer Science

Samford UniversityDecember 1, 2004

What is Calculus?

• Calculus is the mathematics of change.

• It has two main branches:– Differential calculus

• Involves calculating rates of change from functions– Integral calculus

• Involves determining a function given information about its rate of change

Outline of Talk• The intellectual progression from arithmetic to algebra to calculus• Ideas that led to the development of calculus

– The tangent line problem– The area problem– The Cartesian coordinate system

• Historical side note• What, exactly, is calculus?

– Differentiation– Integration– The Fundamental Theorem of Calculus

• The impact of calculus– Immediate applications– Impact on Western thought and contribution to the Enlightenment

• Calculus today

Arithmetic

• With arithmetic, the unknown always occurs at the end of the problem.

• Example: 357 + 982 = ?

Algebra

• With algebra, the unknown can be incorporated at the beginning of the problem.

• Example. We know that x satisfies the following relationship: x2 – 3x + 4 = 0. Find x.

Calculus

• With calculus, the unknown can be incorporated at the beginning of the problem, and it can be allowed to change.

• Example. We know that x changes according to the following rule:

5dx xdt

Find a formula giving x at any time t.

Ideas That Led to the Development of Calculus, Part I: The Tangent Line Problem

• It takes two points to determine a line.

• So, if we have two points on a curve, then we can determine the line through those points.

• In particular, we can determine the slope of the line.

Ideas That Led to the Development of Calculus, Part I: The Tangent Line Problem

• But how do we determine the slope of the tangent line?

• The problem is that there’s only one point.

Ideas That Led to the Development of Calculus, Part I: The Tangent Line Problem

• The Greeks had solved the tangent line problem for a whole host of geometrical shapes, including the circle, the ellipse, and various spirals.

• But the problem remained: Is there a general method for finding the slope of a tangent line; i.e., a method that will work on any curve?

Ideas That Led to the Development of Calculus, Part II: The Area Problem

• How do you find the area enclosed by a curve?

• Again, the Greeks had solved the area problem for a whole host of geometrical shapes, including the circle and the ellipse.

• Example: The area of a circle is given by A = πr2, where r is the circle’s radius.

r

Ideas That Led to the Development of Calculus, Part II: The Area Problem

• Is there a general method for finding the area enclosed by curves?

Ideas That Led to the Development of Calculus, Part III: The Invention of the

Cartesian Coordinate System• Until the 17th century, algebra

and geometry were considered two separate branches of mathematics.

• The use of a coordinate system shows how the two are related, though:– The set of all points that

satisfy an algebraic equation determines a curve.

– And any curve determines an algebraic expression.

• The invention of the Cartesian coordinate system is credited to Descartes and Fermat and is named after Descartes.

2 2 1x y

Ideas That Led to the Development of Calculus, Part III: The Invention of the

Cartesian Coordinate System• The Cartesian coordinate

system also helps show why the tangent line problem is so important.

• The slope of the tangent line measures the rate of change of the curve.

• In other words, the slope of the tangent line measures how much y changes as x changes.

• In particular, if x represents time, then the slope of the tangent line tells us how fast y is moving.

3 3y x

The Invention (Discovery?) of Calculus

• Using the Cartesian coordinate system as a tool, and building on the work of others, Newton and Leibniz separately solved both the tangent line problem and the area problem.– The tools to solve the tangent line problem and

related problems are the differential calculus.– The tools to solve the area problem and related

problems are the integral calculus.

• Their great accomplishment, though, was to show that the tangent line problem and the area problem are, in some sense, actually inverses of each other!

Historical Side Note – The Controversy• There was a lot of fighting among the scientific community in the late

1600s and early 1700s over who invented calculus first – Newton or Leibniz.– We now know that Newton invented calculus in 1665 but didn’t publish

his results until 1704, in the appendix to his Optiks.• Interestingly enough, calculus is not in the Principia Mathematica, published

in 1687.• Newton used calculus to achieve his scientific results published in the

Principia, but in the book itself he used geometrical arguments, not calculus, to justify his mathematical claims.

– Leibniz invented calculus in 1673 but didn’t publish his results until 1684.

– So Newton invented it first, but Leibniz published first.• Newton’s supporters won the fight, and so today we give Newton

the credit for inventing calculus.• Historical ironies

– We use Leibniz’s notation today when we teach calculus, not Newton’s.– Leibniz’s superior notation and treatment of the subject led to a

flourishing of scientific applications of calculus on the European continent, while British science after Newton languished.

Differentiation• Remember that the slope

of the tangent line to a graph of y versus x is just a way of expressing how much y changes as x changes.

• So the tangent line problem is simply a prototype for the more general problem of finding rates of change.

• The process of finding rates of change is called differentiation.

Differentiation Notation

• The notation for the rate of change of a quantity as x changes is

• So, to express how the quantity y changes as x changes we write

• This is actually Leibniz’s notation.

ddx

( ), or d dyydx dx

Differentiation• There is a very nice technique for finding the

rates of change of all of the most commonly-encountered functions, such as:– Polynomial functions– Rational functions– Trigonometric functions– Exponential functions– Logarithmic functions

• Most of Calculus I involves learning this technique and how to apply it to different kinds of problems.

Integration• Solving the area problem

involves the process of integration.

• The integration notation is as follows.– Let f(x) be the height of the

curve forming the top boundary of the enclosed area to the right.

– Then the area under the curve from a to b is denoted:

a b

f(x)

( )b

a

f x dx

The Fundamental Theorem of Calculus

• Unfortunately, there are not any nice, direct integration techniques like the one for differentiation.

• The great accomplishment of Newton and Leibniz was to realize the truth of what we call the Fundamental Theorem of Calculus:

Differentiation and integration are essentially inverse processes.

• This means that the area problem can be solved by using the differentiation technique backwards.

The Fundamental Theorem of Calculus

• Let the upper bound on the region be variable; call it x.

• If I make a tiny increase in x, I’m adding a thin rectangle to the area of the region.

• The area of that rectangle is height of the rectangle, f(x), times the change in x.

• Mathematically, we write:

• What this means, then, is that the rate at which the area of the region changes as x changes is f(x).

• In mathematical notation, this last statement is expressed as:

( ) ( )x

a

d f x dx f xdx

a x

f(x)

( )Area f x x

The Fundamental Theorem of Calculus

• Looking more closely at what we have here, we see that if we integrate a function from a to x and then differentiate it with respect to x, we get back the original function.

• So differentiation and integration are inverse processes!

• And so the area problem can be solved by doing differentiation backwards!

( ) ( )x

a

d f x dx f xdx

Immediate Applications of Calculus

• The tangent line problem is, as we’ve said, a prototype problem for anything involving a rate of change, including:– Velocity– Acceleration

• The area problem is also a prototype problem for a whole host of other problems, including:– Volume– Mass and center of mass– Work

Calculus and Mechanics• With calculus as his tool, Newton was able to use his theory of

gravity to solve and place into one theoretical framework nearly all of the outstanding problems in terrestrial and celestial mechanics.

• Basically, he was able to explain why nearly everything on earth and in space moved the way it did.

• For example, he was able to:– Give a theoretical justification for Kepler’s prediction of the elliptical

orbits of the planets– Explain the movements of the comets– Explain why tides occur– Describe the motion of pendulums

• “Newton singlehandedly completed the scientific revolution.” http://www.phy.hr/~dpaar/fizicari/xnewton.html

Nature and Nature’s lawsLay hid in night;God said, “Let Newton be!”And all was light.

- Alexander Pope

The Impact on Western Thought

• Newton and Leibniz had invented a new mathematical tool, based on a relationship that no one had noticed before.

• Newton then used that tool and his theory of gravity to explain the motion of nearly everything on earth and in the heavens.

• What are the implications of this?

Implication #1: We don’t need God to explain why things move.

• If gravity can explain why the planets move the way that they do, we don’t have to posit a God that keeps them in motion.

• However, gravity doesn’t explain why the planets started moving in the first place – only why they stay in motion.

• So we need to presuppose God in order to explain why the planets first started moving.

• Possible conclusion– Maybe that’s all that God does.– He doesn’t interact with His creation.– He creates, starts things moving, and then steps back to watch.– This is Deism – the idea of a Watchmaker God.

Implication #2: We can use mathematics to model the universe.• While many scientists before Newton had used

mathematics in their work, much of the science up to Newton’s time was descriptive, not quantitative.– For example, Aristotle said that every object has a natural resting

place. Fire naturally wants to be in the heavens, and stones naturally want to be near the center of the earth.

– There’s no mathematics in this.

• Calculus was so successful as a tool for solving physical problems that it revolutionized the way science was done.

• Since Newton, mathematics has been the language of science; since Newton, we have used mathematics to model the universe.

Implication #3: Why should these discoveries stop?

• Calculus was extremely powerful at solving problems in mechanics.

• Scientists who came after Newton were able to use calculus on many other problems as well, such as:– Motion of a spring– Fluid force and fluid flow

• Discoveries were being made in other areas of science, too.

• If we can solve so many problems now, why can’t we continue to solve scientific problems?

Implication #4: Maybe there is a calculus for other fields of knowledge.

• Maybe other areas of human life have their own “calculus,” too.

• Maybe each field has its own fundamental principle that explains everything in it.

• If this is true, and we can figure out what these principles are, then we can solve all of society’s problems in areas such as– Ethics– Economics– Government– History– Philosophy

Calculus Today

• Calculus isn’t starting any more intellectual revolutions today.

• But it’s still being taught.– It’s still the most powerful mathematical tool in

existence.– It’s still the basis for much of modern science.

Some Modern Applications of Calculus

• Population modeling (ecology)– If we can describe the growth rate of a population, we can use

calculus to find a formula for the population at any time.

• Marginal analysis (economics)– The additional cost of making one more item is the rate of

change of the cost with respect to the number of items made.

• Weather forecasting (meteorology)– Historically, this is the origin of the study of chaos.

• Determining spaceship orbits and re-entry (space exploration)