Differential and Integral Calculus 1M 104018

Differential and Integral Calculus 1M 104018
Spring 2016-17
(5 credits)

Dr. Rom Pinchasi, room@math.technion.ac.il
4 weekly hours of lecture
2 weekly hours of TA

1. Introduction (8 hours)
– notation and basic notions: sets of numbers
(natural, integers, real, rationales), intervals, absolute value, basic
rules of inequalities, triangle inequality, the function [x] (integer part of x).
|x|<M <===> -M < x < M.
– functions: domain, range, image, graph, onto (surjection),
one to one (injection), monotone function, bounded function.
– operations on functions including composition and inverse functions.
– notions of: definition, axiom, theorem, negation of statements, disproof,
2. Limit of functions (6 hours)
– definition of a limit, punctured neighborhood, basic theorems, arithmetic of
– one sided neighborhood, one sided limits, basic theorems.
– limit at infinity, infinite limits.
– sandwich theorem, “pizza” theorem
– trigonometric functions, lim sin x /x.
– bounded sets and functions, sup, inf, min, max, axiom of supremum,
monotone bounded functions.
3. sequences (4 hours)
– definition of limit, basic theorems, arithmetic of limits.
– sandwich rule, q^n, n^{1/n}, a^{1/n}.
– monotone sequences, bounded sequences, recursively defined sequences.
– (1+1/n)^n monotone and bounded, e.
– subsequence, partial limits.
– Heine’s theorem
4. continuous functions (3 hours)
– definitions
– arithmetic rules, composition, inverse functions.
– discontinuities.
– mean-value theorem
– Weierstrass theorem
5. the derivative (3 hours)
– definition + geometric meaning + in physics.
– derivatives of elementary functions
– one sided derivatives, differentiability ==> continuity
– arithmetic rules
– chain rule, log x, inverse function
– higher order derivatives
6. more on derivatives (8 hours)
– extremal points, Fermat’s theorem
– theorems of Roll, Lagrange, Cauchy.
– derivatives of monotone functions
– min/max via the second derivative
– Lhopital’s theorem
– convexity, inflexion point, drawing graphs of functions
7. Order of magnitude (2 hours)
– order of magnitude
– linear approximation
– taylor’s polynomial
8. Integral (8 hours)
– indefinite integrals: integration by parts, substitution, partial fractions.
– definite integral, Riemann sum, geometric meaning, arithmetic rules.
– fundamental theorem + applications to definite integrals
– arc length, volume of rotational bodies (about the x-axis and y-axis)
9. generalized integrals (3 hours)
– integral of bounded function on a ray, integral of unbounded function on interval.
– comparison theorems
– absolute convergence
10. series (3 hours)
– definition through partial sums, geometric and telescopic series
– necessary conditions for convergence.
– comparison theorems
– root and ratio test
– integral test
– series with alternating signs. Leibniz theorem
– absolute convergence
11. power series (4 hours)
– definition, radius of convergence, domain of convergence,
– root and ratio method of finding the radius
– integration and derivation of power series
– Taylor series
The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.
In addition, the specific ethics guidelines for this course are:
(1) No mobile phones are allowed during class
(2) No web surfing is allowed during class
Report any violations you witness to the instructor.
Students with Disabilities
Any student with a disability who may need accommodations in this class must obtain an accommodation letter from Technion International’s guidance counselor, at: counselor@int.technion.ac.il