Office hours: Wednesday 14:00-15:00 or by appointment, e-mail: hoyt at_mark mat.uniroma2.it
2026/2027 First semester (with G. Cipolloni)
Course description, time table: Mon-Thu 9:00-11:00, Rooms
Lecture notes (of the part taught by Tanimoto).
Lecture diary:
- 09/14 Introduction of the course. Application of mathematics (tax rate, minimum wage). Functions. Graph a function. x-intercept, y-intercept. Examples: linear functions, quadratic functions, square root.
- 09/15 Examples of functions, domain. Linear functions, quadratic functions. Intervals of real numbers, piecewise linear functions (Italian tax rate). Absolute value, signature, rational functions, roots.
- 09/17 Definition of limit and non-existence of limit. Quadratic functions, signature, linear functions, constant functions. Rules of limits for sum, difference, product and quotient. Examples.
- 09/21 Infinite limits, examples, graph of diverging functions. Limits to infinity, one-sided limit. Rules of limits to infinity, crop yield model, tuition discount example.
- 09/22 Continuity, definition and examples. Rules of continity. Continuity of functions defined in two adjacent
intervals. Continuity on a closed interval. The intermediate value theorem, idea of the proof, application to the existence of square root.
- 09/23 Excercises on functions, domains, limits and continuity.
- 09/24 Definition of natural, integer and rational powers, power functions. Definition of exponential functions,
examples, basic properties and graphs. Popolation growth, radioactive decay and interest as examples of exponential functions. Napier's number. Polynomial vs exponential.
- 09/28 Definition of logarithm, numerical examples, properties of logarithm, graphs of the logarithmic functions, inversion and conversion formulas, examples from investiment, carbon dating and population growth.
- 09/29 Definition of generalized angles, definition of trigonometric functions, the graphs of sin x and cos x and their properties. Various limits including composition, exp x, log x and sin x, exercises.
- 09/30 The slope of a segment, average velocity, velocity at a moment, definition of derivative, rules of the derivative, examples (polynomials and rational functions). Functions that are not differentiable everywhere.
- 10/01 The derivatives of exp x, log x, sin x, cos x. The chain rule, examples. The derivative and the behaviour of the function.
- 10/05 Increasing and decreasing parts of a function, critical points, the first derivative test for relative minima and relative maxima, saddle points. Various asymptotes, examples.
- 10/06 Higher order derivatives, the second derivative test for relative extrema, convexity, concavity and points of inflection. Graph sketching. Examples.
- 10/07 Exercises on graph sketching. Global minima and maxima, examples. Marginal analysis in economics.