Mathematical Analysis 1¶
Teaching material conceived and developed by Fabio Furini, associate professor at DIAG, Sapienza University of Rome.
The course lecture notes, online: definitions, theorems with their proofs, examples worked out step by step, exercises and interactive graphs to watch mathematics move.
Start with numbers Try an interactive graph All lecture notes as PDF
How each chapter is organized¶
Each chapter is one of the course notes, with the same numbers and the same colors: on the website and in the PDF you find everything in the same place.
Definition
The precise meaning of a new concept.
Theorem
A result to remember, with its hypotheses.
Example
A computation worked out in full, one step at a time.
In green the key point to take away.
Proof — opens with a click
All proofs are included, but folded: first read the statement, then open the proof when you want to study it.
The parts of the course¶
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1 · Numbers and logic
Sets, logic and proof techniques, real numbers and supremum, summations, induction, binomial coefficients, complex numbers.
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2 · Functions
Elementary functions and their graphs: powers, exponentials and logarithms, trigonometric and hyperbolic functions, composite and inverse functions.
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3 · Limits of sequences
Sequences and limits, Euler's number, comparisons and asymptotic estimates, hierarchies of infinities, recursive sequences.
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3 · Limits of functions and continuity
Limits and asymptotes, standard limits, continuous functions, zeros theorem and bisection, Weierstrass and intermediate values.
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4 · Derivatives
The derivative and the rules of differentiation, Fermat and Lagrange, L'Hôpital, second derivative, Taylor, curve sketching, Newton's method.
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5 · Series
Numerical series, convergence tests for series with non-negative terms, series with terms of varying sign, series of functions.
Don't miss¶
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Interactive lab
Move the coefficients with the sliders and watch the graph change; then test yourself with the challenges.
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Exercises with solutions
About 300 exercises by topic, each with a worked solution that opens with a click.
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Exam theorems
The list of results you must be able to prove, each linked to the chapter where it is proved.
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The course
How to use the website, reference books, the full index of the chapters.
Teaching material by Fabio Furini — DIAG, Sapienza University of Rome. Part of the same series as the Operations Research Lab and MIP Modelling.
Questo sito è disponibile anche in italiano.