## Calculus and Linear Algebra, UCSCIMAT11, group 1, semester 1, 2020/ 2021

### General information

Lecturer
Fabian Ziltener (UU)

Assistant
Catalina Blom, c.s.e.blom98 [special symbol] gmail.com

Lectures
Mondays 9:30 - 10:15 and Wednesdays 12:00 - 12:45, room N-D

Tutorials
Mondays 10:30 - 11:15 and Wednesdays 13:00 - 13:45, room N-D

Hand-in assignments
There are hand-in assignments every week. You hand them in before the lecture on Monday of the following week by sending your solution to the assistant by e-mail. The assistant grades your solutions and hands the graded assignments back to you within one week.

When solving hand-in exercises, it is important to justify every step you make in full sentences. You are encouraged to discuss the assignments with one another, but the work you hand in must reflect your own understanding.

Presentations
Each student gives one ungraded presentation of maximally 15 minutes during the tutorials. The topics of the presentations will be assigned during the tutorials. Giving a presentation is a necessary ingredient for passing the course.

Midterm exam
The midterm exam will take place on October 14, from 12:00 to 13:45. It will be about all the material that will have been covered in the lectures and the tutorials up to that point.

Final exam
The final exam will take place on December 16, from 12:00 to 13:45. It will be about all the material that has been covered in the lectures and the tutorials.

The problem sheet of each exam will contain a list with formulae, which you will be allowed to use. You are also allowed to use one sheet of paper with hand-written notes (A4 format, both sides). Other aids (for example calculators) are not allowed.

More details about the exams will be announced later.

Your final grade will be equal to 0.4a+0.3m+0.3f, where a,m,f are the grades for the assignments, midterm exam, and final exam, respectively.

Information about the lectures, tutorials, and work at home
During the lectures I (the lecturer) explain some concepts, ideas, and examples. During the tutorials you work on the (hand-in) problems and give presentations (see below). The teaching assistant and I (if necessary) are available for questions during the tutorials. The tutorials are also used for your presentations.

Due to the corona pandemic only a limited number of students is admitted to the class room. If possible, then the lectures are live-streamed for the other students. You take turns attending the lectures and tutorials on campus.

In addition to the lectures and tutorials, you are expected to work for ten hours a week on average at home, reading the lecture notes, solving (hand-in) assignments, preparing your presentation, and studying for the exams. You can make most efficient use of class time by looking through the problems in advance, so that you can ask questions about the points you find most difficult.

### What are calculus and linear algebra?

Calculus is the theory of derivatives and integrals of functions. The derivative of a function at some point is the slope of the tangent line at the corresponding point of the graph of the function. The integral of a function is the area between the abscissa and the graph of the function.

The notions of a derivative and of an integral are based on the notion of convergence. A sequence of numbers converges to some number x if the members of the sequence approach x more and more, as the sequence progresses. If this happens, then x is called the limit of the sequence.

Derivatives and integrals are fundamental for large parts of mathematics, physics, and engineering. They also have applications in virtually every other scientific field, for example in economics, biology, and medicine.

As an example, in physics the velocity of an object is the derivative of the position of the object, viewed as a function of time. As another example, in economics the profit made with some product can be viewed as a function of the price of the product. In order to maximize this profit, one finds the points at which the derivative of the function vanishes.

A differential equation is an equation for a function, which involves derivatives of the function. (Hence the unknown is a function, rather than a number.) These equations abound in science and engineering, where they model many natural phenomena. As an example, the equation that a function equals its own derivative models unrestricted growth of an animal population and the spread of a disease at an early stage.

Linear algebra is the theory of vectors and linear equations. An equation is linear if the sum of two solutions and any scalar multiple of a solution is again a solution of the equation. Linear differential equations are the simplest types of differential equations. The above equation modelling unrestricted growth is an example.

### Aim

The aim of this course is to provide a glimpse of modern mathematics and its applications in the sciences. Here modern'' means at most 200 years old.

### Contents of this course

Theory:
• Achilles and the turtle, convergence and limits
• derivative of a function
• integral of a function
• techniques for calculating derivatives and integrals, such as the Chain Rule and the Leibnitz Rule (product rule)
• Fundamental Theorem of Calculus, which relates the derivative and the integral
• ordinary differential equations
Applications:
• profit maximization
• models for the growth of an animal population and for the spread of a disease:
• unrestricted, exponential growth
• self-limited, logistic growth
• predator-prey model, Lotka-Volterra equations
We will develop and use linear algebra, as the need arises.

Some scientists whom we will encounter in this course, are the following:
 Isaac Newton, 1643 - 1727 Gottfried Wilhelm Leibniz, 1646 - 1716 Alfred J. Lotka, 1880 - 1949 Vito Volterra, 1860 - 1940

### Literature

James Stewart, Single Variable Calculus, 7th ed.

### Prerequisites

Secondary school knowledge of mathematics is required. This can be acquired in IB Higher Level Math, Dutch VWO Wiskunde B or Wiskunde D, or similar Calculus and Algebra courses from a foreign high school. If in doubt, contact the lecturer before you enroll for this course. If you do not meet the requirements then you should take UCACCMAT01 first.

### The ultimate question (with thanks to Arjen Baarsma for programming this)

And now comes the ultimate question: Is the exponential function periodic?

yes, of course no, of course not What in God's name is meant by “function”?