Why Calculus?

Sir Isaac Newton, 1643-1727 Gottfried Wilhelm von
Leibniz, 1646-1716
Sir Isaac Newton, 1643-1727 Gottfried Wilhelm von Leibniz, 1646-1716

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Objectives of the Module

The goal of the course is to show why calculus has served as the principal quantitative language of science for more than three hundred years. How did Newton and Leibniz transform a bag of tricks into a powerful tool for both mathematics and science? Why is calculus so useful in geometry, physics, probability and economics? Why are mathematicians so concerned with rigor in calculus? Since calculus is about calculating, what is the relationship between calculus and computers? What is the relationship between calculus and new topics like chaos and nonlinearity? If you want to understand what calculus is really about, then this is the course for you.

Topics to be Covered

Ancient peoples, driven by natural curiosity and the demands of applications, confronted the problems of finding areas and volumes of various shapes. Their methods of solving these problems may be regarded as precursors to integration. Outstanding in this regard was the work of the Greeks, exemplified by Archimedes' solutions to numerous problems of quadrature, and the works of the Chinese mathematicians Liu Hui and Zu Chongzhi. Concepts resembling differentiation did not arise until much later. Precursors to differentiation can be recognized in the work of Fermat and Descartes on tangents, and finding maxima and minima.

Greek mathematics separated algebra and geometry. The invention of Cartesian coordinates and modern symbolism allowed Newton and Leibniz to create calculus. Our focus is on how they used the Fundamental Theorem of Calculus to transform a bag of tricks into a powerful tool.

The power of calculus was clear from the start. It achieved spectacular successes in geometry, physics, probability and economics.

The foundations of calculus were not secure at the time of invention, and the limitations of calculus were obvious to many critics. However, mathematicians gradually succeeded in putting calculus on a firm foundation. This required developing a clear understanding of infinity.

Calculus is fundamentally a theory of continuous objects. However, in many applications the theory is extended through the interplay between discreteness and continuity. Discrete problems may be described by continuous models and discrete methods may be applied to solve continuous problems.

Part of the success of calculus comes from the fact that we can simplify problems by linear approximations. For example, we look at the tangent line rather than the whole curve. However, calculus is about calculating and with the advent of computers it has become possible to attack non-linear problems. The contrast between linearity and non-linearity shows up in the new and unexpected world of chaos.

Recommended Background

It is recommended that you have seen some calculus already. Like knowing how to differentiate and integrate polynomials. But there are no formal requirements.

Mode of Teaching

There will be two hours of lectures and two hours of tutorials a week.


The final exam counts 60% of your grade. You have to write a project that counts 40%. A lot of topics will only be touched upon in lectures, and I hope that you will explore them further on your own in the projects. I will provide a list of possible topics, but I also encourage you to propose your own topics and send them to me for approval. The project can be a normal paper project, or it can be a web page with animations and graphics.

The tomb of Archimedes,
287-212 B.C.
The tomb of Archimedes, 287-212 B.C.

Course Content

Precursors to Integration

  1. An Egyptian papyrus exists (the Moscow Papyrus) giving the answer to the question of finding the volume of a pyramidal frustrum with given dimensions. This indicates that around 1800 B.C., the Egyptians possibly knew a formula for finding the volume of a pyramidal frustrum.
    1. How could the Egyptians have discovered the formula?
    2. Consider a step pyramid built up by putting together a number of rectangular blocks. Perform experiments, mental or otherwise, to find the volume of such a step pyramid with prescribed base, height and block size.
    3. How does (b) help to answer (a)?
  2. Approaches to integration in the ancient world - quadrature and cubature.
    1. Area of the circle - Chinese approach by Liu Hui, Eudoxus' method of exhaustion.
    2. Investigation of a quadrature or cubature problem by the method of exhaustion, e.g., volume of a sphere, quadrature of a parabola.
    3. Archimedes' “method”. Investigate the same problem in (b) using Archimedes' method. Areas and volumes are thought of as being made up of line or planar segments, which are then balanced against each other on a lever. Infinitesimals and the atomism of Democritus.
    4. Why did Archimedes think that the arguments in his “method” did not constitute proofs?
  3. Further developments in Renaissance Europe.
    1. Kepler's calculations of areas and volumes (Nova stereometria) - free use of infinitesimal methods.
    2. Cavalieri's Theorem. Cavalieri's approach to the calculation of the volume of a sphere.

Precursors to Differentiation

  1. What is a curve?
    1. Greek conception of curves is mostly specific and descriptive, e.g., conic sections. Curves were geometric figures and thus treated synthetically. Occasionally, curves were described dynamically, e.g., Archimedes' spiral.
    2. The “latitude of forms” in the Middle Ages considered variable quantities. These came to be depicted geometrically by two-dimensional pictures.
    3. Descartes' coordinate geometry arithmetized geometry. Curves came to be described algebraically by equations.
  2. Tangents to curves.
    1. Static description of tangents in Greek geometry.
    2. Tangent as instantaneous direction of motion for curves described kinematically, e.g., Archimedes' spiral, cycloid.
    3. Observation on the relationship between tangents and maxima of curves.
    4. Descartes' method for finding tangents.
    5. Fermat's variational method for finding maxima, and a similar method for finding tangents.
    6. Fermat solved problems on finding centers of gravity by variational methods. He also reduced certain rectification problems to problems of quadrature. But he did not have an explicit notion of the relationship between the two types of problems - summation and tangent. Also, he saw these as specific (and geometrical) problems and did not think of them together as forming a whole method of analysis.


Various strands had to come together before “the Calculus” developed into what we recognize it to be today - a coherent method that can be applied algorithmically to solve certain types of problems. These strands include psychological and philosophical shifts no less than the discovery of new mathematical techniques.

  1. Algebra vs. geometry.
  2. Variable quantities.
  3. Arithmetization of geometry - expressing geometric objects by equations, analytic geometry.
  4. Concept of function.
  5. Concept of limit. Newton's theory of “prime and ultimate ratios”.
  6. Notation used in Calculus. Leibniz's notation.

Calculus: Putting it All Together

  1. Differentiation. Velocity. Slope. Tangent lines. Rate of change.
  2. Integration. Distance. Area. Summation.
  3. Fundamental Theorem of Calculus.
  4. Newton's algorithm - FTC and use of infinite series. Binomial series.
The Lorenz Attractor in
3D by Paul Bourke
The Lorenz Attractor in 3D by Paul Bourke

The Power of Calculus

Applications of calculus

  1. Mechanics.
  2. Geometry - tangent, area, curvature.
  3. Optimization.
  4. Differential equations - modeling the real world.
  5. Probability - probability distributions, expected value.
    1. Discrete probability distributions. A short example - binomial distribution.
    2. Continuous probability distributions. An example - a random variable that measures time.
    3. Expected valued - how “weighted average” becomes an integral “in the limit”.
    4. Central Limit Theorem - continuous random variables arise naturally when one considers large sets of discrete data. A visual demonstration (computer plot) of the convergence of binomial distribution to normal distribution.
  6. Economics.
    1. Marginal quantities. Profit is maximized when marginal revenue equals marginal cost. Explain this as a calculus result.
    2. Elasticity.

Infinity and Infinitesimals

  1. Philosophical and foundational issues.
    1. Debate on infinitesimals. What are differentials? What is wrong with using differentials? (Nonstandard analysis.)
    2. What is instantaneous velocity?
    3. What is important in mathematics? Intuition vs. theory. Newton & Leibniz vs modern calculus
  2. Zeno's Paradoxes.
  3. Berkeley's criticism.
  4. D'Alembert - the limit as the central concept.
  5. The concept of function. The derivative as a quantity (function) vs as a ratio (of differentials).
  6. “Modern” formulations - limits and continuity, real numbers examined, theory of infinite series.

Discreteness and Continuity

Problems in economics usually involve discrete variables, but we can treat them as continuous variables and use the tools of calculus. Computers are inherently discrete, but we can still use them to approximate continuous problems. In a similar manner, integration, as in the Greek method of exhaustion, starts out as a discrete approximation before we take a limit.

Linearity and Nonlinearity

Calculus is about calculating, and we show how computers have changed mathematics. One of the basic ideas behind calculus is linearization, but thanks to computers, nonlinearity is now a hot topic. This has lead to chaos theory. Finally, we discuss how computers have brought continuous and discrete mathematics closer together.

Riemann sums


A nowhere
differentiable function by Douglas N. Arnold
A nowhere differentiable function by Douglas N. Arnold

Course Schedule

Week Topic Tutorial Project
1 Precursors to Integration
2 Precursors to Integration
3 Precursors to Differentiation 1
4 Symbolism 2
5 Calculus: Putting it All Together 3 Proposal for Project due
6 Recess
7 The Power of Calculus 4
8 The Power of Calculus 5
9 Infinity and Infinitesimals 6
10 Discreteness and Continuity 7 Project due
11 Linearity and Nonlinearity 8
12 Linearity and Nonlinearity 9
13 Revision


Project Topics

Most of the topics we talk about in lecture can be extended to projects. Here are some suggestions. I also encourage you to propose your own topics and send them to me for approval.

Helmer Aslaksen
Department of Mathematics
National University of Singapore

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