 Department of Mathematics

## Course number 120131

### Syllabus:

• Sequences of real numbers: monotonous sequences, bounded sequences, periodicity. Convergence and divergence. Algebra of convergent sequences; convergence theorems.
• Functions of a real variable. Domain, range, graph of a function. Odd functions, even functions, periodic functions.
• Functions of a real variable; limits, asymptotes to the graph of a function. Comparison of functions on a neighborhood of a real point or on a neighborhood of infinity.
• Continuity at a point, on an interval. Extrema of a function on a closed interval.
• The main properties of a continuous function on a closed interval. The inverse function of a continuous function. Roots and inverse trigonometric functioms.
• Intermediate Value Theorem. Solution of equations: the method of dichotomy.
• Differentiable functions. The algebra of differentiable functions. Derivative of logarithmic function and of exponentials. Logarithmic derivative.
• The main theorems of infinitesimal calculus. The theorems of Rolle, Lagrange, etc; L'Hopital's rule.
• Applications of the previous topics: extrema, asymptotes, convexity/concavity of a function, inflexion points. The study of a function of one real variable (curve discussion and applications). Implicit differentiation. Solution of equations: the method of Newton-Raphson.
• The indefinite integral of a function; primitives. Primitives of usual functions. Techniques of integration: sum, product by a constant, integration by parts, substitution, primitive of a rational function (decomposition into simple fractions).
• The definite integral of a function. Techniques of integration. Integration of odd functions and of even functions. Integration of a parametric integral (by induction).
• Numerical integration: rectangles, trapezes, Simpson formula.
• Applications of the definite integration: length of an arc, areas, volumes,center of mass, envelope area, etc.
• Functions defined by an integral; the main theorem of integral calculus.
• Series of real numbers. Convergence, divergence. Absolutely convergent series. Series with non negative terms.
• Power series. Taylor series. Applications.

### Pace of the course:

• Lectures: Sunday 16:15-17:45 and Wednesday 16:15-17:45 - Lau 540.
• Exercise sessions: 2 hours, once a week.
• Homework every week.

• Three partial exams: 10%. Here is a sample ("bochan ledugma"): pdf, rtf.
• There will be one Moed bet; the above decomposition of the final grade is valid for all the terms.

### Sample of exams:

• A sample of an exam for 5765 's final exam:pdf (sorry for the upside down format) (mivchan leDugma).
• A list of formulas for the exam: pdf.

### Homework:

The weekly homework assignments are posted on the e-learn site of the course. Please download the documents from there.

### Solution sheets:

Solutions for homework assignements are posted on the elearn site. Registered students are invited to use them.

Here are short solutions of the first midterm. We apologize for the uncomfortable format.

And here are solutions of the second midterm.

Next partial exam (second midterm...) will be on Wednesday, January 28. Good luck.

### Booklets and help material:

I wrote some booklets (in Hebrew) containing a lot of material.
Here are pdf files of the classroom presentations:

A few animated presentations (shows):
• Second derivative.
• Riemann Sums.
• Improper Integrals.

• PostScript files are posted here. If you work with Unix/Linux and have Ghostview installed, the solution file will be opened automatically in a separate window.

If you need a previewer under Windows, you can download the executable file , run it and it will install GSview (think of it as Ghostview for Windows). If needed, you can get more information on GSview and more information on how to configure it as a viewer for Netscape Navigator.
Another possibility is to download the Postscript files, then to convert them to pdf files using ps2pdf.
A webbook for this course is available; it is (always) under construction. Enjoy it.
Here a PostScript version of a table of integrals

### Software:

Students who are already acquainted with some mathematical software are invited to use it and to share work with their peers. The MatLab software has been introduced in Ellul and can be used.
The Derive software has been installed in all labs.
We have a few licenses for the Maple software. From time to time Maple worksheets will be added here.
We suggest that you have a look at some graphical examples .

Recommended Texts:
• Glyn James and al.: Modern Engineering Mathematics, Addison-Wesley.
• B. Kohn: Chedva (in Hebrew), Bak Editions.
• Finney - Weir - Giordano: Thomas' Calculus, Addison-Wesley, 2000.
• S. Lipschutz: Infinitesimal Calculus, Schaum Series (exists in Hebrew, edited by Steimatzky).
• H. Anton (translated in Hebrew by the Open University): Chedva 1 and 2.
All these books are in the library. We recommend a visit to the library; there are many recent acquisitions. Look for "Calculus".