History & Philosophy of Calculus, Session 5

THE DERIVATIVE

FERMAT ON MAXIMA & MINIMA

Suppose you can cut a line anywhere along its length, and use the two resulting parts to make a rectangle. Where should you cut it to get the rectangle of biggest area? This is a problem of maximizing something (the area) given

a finite resource (the line). Such problems have many applications!

Fermat described a method for solving this problem in a letter to Mersenne that was subsequently forwarded to Descartes (Descartes received the letter in 1638. Source: D, J , Str iuk, A Source Book in Mathematics, 1200-1800 ).

MAXIMIZING AREA

a - b b a--b (a – b) x b

b

a

The area of the rectangle is (a—b)b = ab – b 2.Fermat’s idea is to replace b with b + e, where “e” is an

“error” term. We will suppose there’s a value of b that maximizes the area –

that seems reasonable. We then pick any length, b + e, which is the “right” length plus

an error, and work out an expression for the area we get. We then pretend we picked the right length; that is, we set the

area we just found using our length b + e equal to the maximum area, ab – b2.

Finally, we “eliminate” the error term, which must now be zero, from our equation.

FERMAT’S METHOD

a - b b a--b (a – b) x b

b

a

Here’s the algebra: Replacing b with b + e we get:

(a — (b + e))(b + e) = (a — b — e)(b + e) = ab – b 2 – 2be + ae – e2

This is supposed to be equal to ab – b2, so: ab – b2 = ab – b2 – 2be + ae – e2

Subtracting ab – b2 from both sides gives us 0 = – 2be + ae – e2

Dividing through by e: 0 = – 2b + a – e

2b = a – e Eliminating e:

2b = a so we should always pick the halfway point on the line! What’s happened?

As it stands this would not be considered a legitimate argument by modern mathematical standards.

But the answer is correct!

THE DISAPPEARING ERROR

PHYSICS AND THE DERIVATIVE

DISTANCE-TIME GRAPHS

DISTANCE-TIME GRAPHS

DISTANCE-TIME GRAPHS

DISTANCE-TIME GRAPHS

Last week we saw how to go from speed to distance travelled using the integral.

This week we look at how to go the other way: from distance to speed, using the derivative.

THE BIG PICTURE

Speed

DistanceIntegration Differentiatio

n

TANGENTS

AVERAGE SPEED

SECANT LINES• Imagine the red curve

represents not an abstract function involving two numbers but the height of a falling object as a function of time.

• We’re interested in the rate of change of height between 2 seconds and 8 seconds.• This is the speed at which the

object is falling; but its speed is constantly changing, so we’re only looking at the average speed.

• The average rate of change is the slope of the grey line.• The grey line is called a

secant line.

Barrow was the first Lucasian Professor of Mathematics at Cambridge University; his successor was Newton. Barrow’s work on tangents was crucial for Newton’s radical reformulation of physics. His Lucasian lectures, published in 1670, lay out many of

the principles of calculus as it stood at the time, and deeply influenced both Newton and Leibniz.

Barrow was interested in whether and how we can make sense of speed at an instant in time. Remember, the physics of motion was a key motivating

project for the calculus.

ISAAC BARROW

This is the curve corresponding to the equation y = 10 – t2/10.

We’ll use it as an example to investigate the (purely geometric) problem of finding tangents.

But what is a tangent?

WHAT IS A TANGENT?

We picked a point and added a new line (in grey).

The grey line is straight and touches the red curve at the chosen point.

This is the tangent to the curve at that point.

The tangent is the best straight-line approximation to the curve in the region of that point: if you zoom in close enough, the two look very similar.

WHAT IS A TANGENT?

WHAT IS A TANGENT?

WHAT IS A TANGENT?

TANGENT AND “SPEED AT AN INSTANT”

The tangent to a circle at a point is easy to find; join the point to the centre of the circle, then draw another line at right angles to it.

Finding tangents to other curves isn’t so easy, but it’s an important technique in geometry.

But even more important, historically, was that the tangent to a curve can be thought of as the “rate of change at an instant” that Barrow refers to in his lecture. This, rather than the geometric applications, was

what drove the research that eventually produced the differential calculus.

FINDING TANGENTS

It’s not surprising that speed and distance can be derived from each other.

What is surprising is that tangents and areas have this intimate relationship with each other and with the physics.

PHYSICS THROUGH GEOMETRY

Rate of Change

Thing Changing

Area Tangent

NEWTON: THE METHOD OF

FLUXIONS

NEWTON’S CALCULUS

The 1671 Method of Fluxions was written in Latin and never published until after his death (when it was published in English). It speaks freely of “indefinitely little” quantities

of both space and time and uses them in calculations.

By the Principia (1687) he already has something like the idea of limit in the form of the “ultimate ratio”. But the Principia wasn’t written in the new

language of the calculus, which his contemporaries wouldn’t have understood.

The infinitesimals persisted in later work, and drew criticism.

FLUENTS & FLUXIONS

In Newton’s terminology a fluent is a thing that changes over time, and its fluxion is the way in which it changes. Example: the height of a thrown ball over time is a

fluent; its velocity (how the height changes over time) is its fluxion.

These terms haven’t passed into mainstream mathematical usage

In the Method of Fluxions he is clear that there are two key and complementary problems: Given a fluent, find its fluxion: this is the problem

solved by differentiation. Given a fluxion, find its fluent, which will be solved by integration.

Fluxion

Fluent

Integral Derivative

CALCULATING THE DERIVATIVE

• Here x is horizontal and f(x) gives us the height at each point x.

• The horizontal change is h. The vertical change is f(x + h) – f(x).

• The gradient, therefore, is

The only way to “see” that this works is by calculating a derivative, so with apologies…Suppose f(t) = t2:

The x2 and –x2 cancel each other out.

Replace f with what it actually does.

Definition of the derivative.

Multiply out the bracket.

The dt on top and bottom cancel out. Now we’re not dividing by dt any more, we can let it go to 0 and see the result.