For a better view allow EDIT on this page
Donát Bánki Faculty of Mechanical and Safety Engineering
2014/2015 Spring
MATHEMATICS II
COURSE DESCRIPTION
Lectures:
BOOKS:
Wednesdays 15.20-17 in room 112, and
http://ocw.mit.edu/ans7870/resources/Strang/Edited/Calculus/Calculus.pdf
http://www.whitman.edu/mathematics/multivariable/multivariable.pdf
Thursdays 9.50-12.25 in room 113.
Consultation: (by request)
Wednesday 17-18, in room 112.
ADVERTISEMENT
CALCULATORS and PLOTTERS:
http://home.sandiego.edu/~pruski/MathLinks.html
Multivariable: http://www.math.uri.edu/~bkaskosz/flashmo/graph3d2/
LAST LECTURES: MAY 7
Total probability theorem, Bayes theorem. Standard deviation, variance.
Normal distribution. See hand-outs.
SCROLL DOWN FOR ALL LECTURES
Ápril 29-30
Discrete distributions: binomial, Poisson.
Binomial theorem. Properties of binomial coefficients.
Discrete distribution and cumulativ distribution FUNCTION.
Notion of continuous random variable, continuous disrtibution functions, density function.
Expected value (mean).
Test on Des.
Ápril 22-23
Hand-outs!
Combinatorics. Conditional probability. Random variable. Discrete distributions:
Hypergeometric.
April 15-16
SCROLL DOWN FOR ALL LECTURES
PROBABILITY THEORY-handouts:events algebra, notion of propability, Kolmogorov axioms.
Probability book: (more explanation – you need only those proofs, which are int he handout as well. ) Int he lectures we vovered till page:22:
http://www.math.uiuc.edu/~r-ash/BPT/BPT.pdf
Please read handouts and the book above! Furthermore, try to soolve as many as you could from the problems set in the handouts.
ADVERTISEMENT On 22 of April, lectures are officialy concealed, however we probably will have
lectue in the REGULAR lecture time (with Dean’s permission). ONLY In that fortunate case you will
write a test on Des on Tursday, from 9.50-10.50, room 221.
Otherwise extar time will be set.
NEW PRACTICE EXAMPLES-DIFFERENTIAL EQUATIONS:
http://digitus.itk.ppke.hu/~b_novak/BANKI/second_or_de_nonhomogeneous.doc
http://digitus.itk.ppke.hu/~b_novak/BANK/First_Order_DE_solved.docx
April 08-09 DIFFERENTIAL EQUATIONS:
http://digitus.itk.ppke.hu/~b_novak/BANKI/Diff_e_INTRO.ppt
http://digitus.itk.ppke.hu/~b_novak/BANKI/First_order_SK.ppt
http://digitus.itk.ppke.hu/~b_novak/BANKI/second_order_DEs.ppt
SCROLL DOWN FOR ALL LECTURES
http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf
http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx
HOMEWORK: http://digitus.itk.ppke.hu/~b_novak/BANKI/HW_DE_1.xlsx
April 08-09 DIFF.EQs:
http://digitus.itk.ppke.hu/~b_novak/BANKI/Diff_e_INTRO.ppt
http://digitus.itk.ppke.hu/~b_novak/BANKI/First_order_SK.ppt
http://digitus.itk.ppke.hu/~b_novak/BANKI/second_order_DEs.ppt
http://www.stewartcalculus.com/data/CALCULUS%20Concepts%20and%20Contexts/upfiles/3c3-2ndOrderLinearEqns_Stu.pdf
http://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx
HOMEWORK: http://digitus.itk.ppke.hu/~b_novak/BANKI/HW_DE_1.xlsx
Ads
RESET SERIES:ON APRIL 16, 9.50-10.50.
MIDTERM TEST LINEAR ALGEBRA PART: ON 9th of APR!
RESET MIDTERM TEST ON 8th of APR!
Practice:
http://digitus.itk.ppke.hu/~b_novak/BANKI/PRACTICE_MID_I.docx
SCROLL DOWN FOR ALL LECTURES
MIDTERM PARTS MULTI AND SERIES ON APRIL 1
SPRING BREAK
ADs
MIDTERM TEST ON 1st of APRIL!
Practice:
http://digitus.itk.ppke.hu/~b_novak/BANKI/PRACTICE_MID_I.docx
SCROLL DOWN FOR ALL LECTURES
Lectures on 2 of APRIL(Thursday) are partly already held (4 of March), so there will be NO LECTURE! The remainder 1 lecture will be held on any WEDNESDAY from 17-18 by your wish!
March 18-19
Taylor and MacLarurin series.
http://digitus.itk.ppke.hu/~b_novak/BANKI/SERIES_PRACTICE.doc
Error approximation
HOMEWORK:
http://digitus.itk.ppke.hu/~b_novak/BANKI/SERIES_PRACTICE.doc
Separable Des:
Theory:
http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-i-first-order-differential-equations/conventionsand-preliminary-material/MIT18_03SCF11_s0_4text.pdf
Examples:
http://tutorial.math.lamar.edu/Classes/DE/Separable.aspx
http://www.cliffsnotes.com/math/differential-equations/first-order-equations/separable-equations
SCROLL DOWN FOR ALL LECTURES
March 11-12
Infinite number series, power series
http://digitus.itk.ppke.hu/~b_novak/BANKI/Series.pptx
Homework: Expand sinx and cosx into Maclaurin series (Taylos series centered at the origin)
March 4-5
Multivariable functions: Finding extrema, directional derivative, gradient vector. Linear mapping. Complex eigenvalues,
eigenvectors. Numerical and functional series.
http://digitus.itk.ppke.hu/~b_novak/BANKI/Multivariable_full.ppt
http://archives.math.utk.edu/ICTCM/VOL10/C009/paper.html
HOMEWORK: finding extrema, total derivatives, directional
Febr. 25-26
Multivariable functions: total differential,linear approximation tangent plane, further partial derivatives
http://digitus.itk.ppke.hu/~b_novak/BANKI/Multivariable_full.ppt
http://archives.math.utk.edu/ICTCM/VOL10/C009/paper.html
HOMEWORK:for problems 1-4 in lecture check = of mixed partial derivatives and write total differential
SCROLL DOWN FOR ALL LECTURES
ADVERTISEMENT
THERE WILL BE EXTRA (mandatory) LECTURES NEXT WEEK, on MARCH 4 WEDNESDAY 11.40-12.25 in room 106.
QUIZ on next Wednesday! Topics:
Defintions: partial derivative, eigenvector, eigenvalue
Problems: matching multivariable function graphs with level curves or cross sections and formuli, partial differentitation, total differential.
Febr. 18-19
1.Determinants:
http://digitus.itk.ppke.hu/~b_novak/BANKI/determinants_emg.pptx
Eigenvalues, eigenvectors: Lecture notes: http://digitus.itk.ppke.hu/~b_novak/BANKI/gauss_ang.pdf
Calculating eigenvectors and eigenvalues.
Notion of a multivariable function. Cross sections (level curves).
On-line book: http://www.whitman.edu/mathematics/multivariable/multivariable.pdf
2. Multivariable functions.
Hand-in homework: 1. eigenvalue calculations: Definiton, matrix equation, one vector mapping, characteristic polinom and equation, solution,
graph of solution. Matching graphs, formulas and cross section/level curves
SCROLL DOWN FOR ALL LECTURES
Febr. 11-12
Lecture note on improper integrals: http://digitus.itk.ppke.hu/~b_novak/BANKI/Improper_integral.pdf
Solved practice exercises: http://archives.math.utk.edu/visual.calculus/4/improper.1/
Solving system of linear equations
Lecture notes on Gauss, Gauss-Jordan eliminations and its application to inverse matrix:
http://digitus.itk.ppke.hu/~b_novak/BANKI/gauss_ang.pdf
HAND –IN HOME-quiz for next WEDNESDAY: expand a given determinant in 3 different ways!
SCROLL DOWN FOR ALL LECTURES
SCROLL DOWN FOR ALL LECTURES
SCROLL DOWN FOR ALL LECTURES
PROBABILITY THEORY
1.
Combinatorics: variation and permutations
Theory and practice problems with solutions:
http://www.mathsisfun.com/combinatorics/combinations-permutations.html
http://www.zweigmedia.com/RealWorld/tutorialsf5/framesPC.html
Permutation with repetition. Combination.
http://www.mathsisfun.com/data/probability-events-types.html
Binomial coefficients ("n choose r" ) and their properties. Pascal triangle.
Longer: http://www.mathsisfun.com/pascals-triangle.html
Shorter: http://www.bymath.com/studyguide/alg/sec/alg31.html
Pattern problems for midterm:
Combinatorics
SCROLL DOWN FOR ALL LECTURES
http://digitus.itk.ppke.hu/~b_novak/BANKI/2_nd_sem_midterm/Problem_set_1.docx
More help on combinatorics:
http://www.math.uiuc.edu/~hildebr/408/combinatorialproblemssol.pdf
http://web.eecs.utk.edu/~booth/311-04/notes/combinatorics.html
2.
Event algebra. Probability: Classical notion. Kolmogorov axioms (introduced on
relative frequency basis).
http://www.zweigmedia.com/RealWorld/tutorialsf15e/frames6_1.html
http://www.zweigmedia.com/RealWorld/tutorialsf15e/frames6_2.html
http://www.zweigmedia.com/RealWorld/tutorialsf15e/frames7_3.html
http://www.mathsisfun.com/data/probability.html
Do you have time? Carry out these below:
http://www.mathsisfun.com/activity/dice-experiment-1.html
http://www.mathsisfun.com/activity/dice-experiment-2.html
Sampling without replacement: hypergeometric distribution:
SCROLL DOWN FOR ALL LECTURES
http://www.youtube.com/watch?v=BCeFgnh6A1U
3.
Notion of a random variable. Sampling with replacement: binomial distribution.
http://www.mathsisfun.com/data/quincunx-explained.html
http://www.zweigmedia.com/RealWorld/tutstats/frames8_2.html
VIDEOS:
http://www.youtube.com/watch?v=eSJ6ufTSJNk
http://www.youtube.com/watch?v=IYdiKeQ9xEI
Summary through examples: http://www.youtube.com/watch?v=Iu25wy7icok
Consolidation: random variable, distribution. Properties of a distribution function.
http://www.zweigmedia.com/RealWorld/tutstats/frames8_1.html
SCROLL DOWN FOR ALL LECTURES
4. : Mean, deviation, expected value, standard deviation.
Poisson distribution.
Continuous distributions: Normal and exponential.
Statistics
PRACTICE FOR FINAL:
Try to solve the problems on your own- do not peep the solutions before that! 
Poisson distribution
http://www.intmath.com/counting-probability/13-poisson-probability-distribution.php
Normal distribution:
http://www.intmath.com/counting-probability/14-normal-probability-distribution.php
SCROLL DOWN FOR ALL LECTURES
SCROLL DOWN FOR ALL LECTURES
SCROLL DOWN FOR ALL LECTURES
Old links from last year:
ATERIAL FOR NEXT WEEK QUICK QUIZ
Practicing exercises on the expecting value:
http://cnx.org/content/m16828/latest/
Practicing exercises on Poisson distribution:
http://www.intmath.com/counting-probability/13-poisson-probability-distribution.php
Week 10 (18 of April): Independent events. Conditional probability. Total probability and Bayes theorem:
http://dtc.pima.edu/~hacker/busmath/problem-sets/problem-set7-ltp-bayes-theorem-sols.pdf
SCROLL DOWN FOR ALL LECTURES
Week 11(18 of April): Conditional probability. Total probability Theorem, Bayes Thorem
Proofs: http://digitus.itk.ppke.hu/~b_novak/BANKI/Total_Bayes.pdf
http://www.zweigmedia.com/RealWorld/tutorialsf15e/frames7_5.html
Hand-out: homework on conditional, total and Bayes, help to the solution:
http://www.zweigmedia.com/RealWorld/tutorialsf3/frames6_6.html
Week 12 (25 of April): Distribution
FUNCTION and its properties. Well known (continuous) distribution
function:
exponential, normal. Normal distribution and its properties
General:
http://digitus.itk.ppke.hu/~b_novak/BANKI/cont_distr_general.doc
Exponential:
http://digitus.itk.ppke.hu/~b_novak/BANKI/exponential.pdf
Normal distirbution:
http://digitus.itk.ppke.hu/~b_novak/BANKI/Normal.ppt
SCROLL DOWN FOR ALL LECTURES
original:
http://www.stanford.edu/~kcobb/hrp259/lecture6.ppt
http://digitus.itk.ppke.hu/~b_novak/BANKI/Normal_distrib_PRENHALL_public.pdf
http://digitus.itk.ppke.hu/~b_novak/BANKI/exp_and_norm.pdf
SCROLL DOWN FOR ALL LECTURES
Week 13 ( 2 of May): Basic Statistics. Trials. Confidence intervals
Week 14 (9 of May): Consolidation
Week 15 (16 of May): Final
SCROLL DOWN FOR ALL LECTURES
OLD PAGES
MATHEMATICS: CALCULUS I.
Lecturers: Ágnes Bércesné Novák PhD, Pál Pentelényi PhD
Curricula
RELIABLE FUNCTION PLOTTER
RELIABLE Calculus pages: http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx
http://archives.math.utk.edu/visual.calculus/
Week 1September 12-13
Wednesday: Polynomials - Long division (Seminar)
Solved examples (in Hungarian):
http://digitus.itk.ppke.hu/~b_novak/BANKI/polinomosztas.pdf
Thursday:
Functions, Elementary functions, intuitive knowledge about functional limit, limits of elementary
functions.
READING (INTUITIVE MEANING OF THE LIMIT) : http://www.mathsisfun.com/calculus/limits.html
SCROLL DOWN FOR ALL LECTURES
VIDEO HELP:
http://www.calculus-help.com/tutorials/
Lesson 1: What Is a Limit?
Lesson 2: When Does a Limit Exist?
Lesson 3: How do you evaluate limits?
SCROLL DOWN FOR ALL LECTURES
Week 2 September 19-20:
Wednesday: Calculating limiting values of rational functions.
Strategy:
1. Substitute – if fails:
2. Factorize (use root factors: (x-root) – if fails:
3. Use conjugate
Thursday:
Quick quiz: long division of polynomials
Limits in positive and negative infinity, and infinite „limits”
READING (limits in infinity):
QUICK QUIZ next week: http://www.sosmath.com/calculus/limcon/limcon04/limcon04.html#answer4
ANOTHER READING (limits in infinity): http://www.mathsisfun.com/calculus/limits-infinity.html
READING (how to calculate limit): http://www.mathsisfun.com/calculus/limits-evaluating.html
VIDEO HELP:
http://www.calculus-help.com/tutorials/
Lesson 4: Limits and Infinity
VISIT ONE OF THE LEADING UNIVERSITIES IN THE WORLD: THE MIT
MIT FREE VIDEOS: Only till 50th
minutes : Lecture 02: Limits, continuity;
Trigonometric limits
SCROLL DOWN FOR ALL LECTURES
114.9 MB
Week 3 September 25-26:-27:
Wednesday:
Well known limits.Pinching (squeeze, sandwich ) theorem
VIDEO HELP: http://www.youtube.com/watch?v=Ve997biD1KtA (little bit different from my proof)
Special functions: Sequences. Limit of a sequence.
Inverse functions. The inverse to the trigonometrical functions.
http://www.sosmath.com/algebra/invfunc/fnc1.html
Thursday:
QUICK QUIZ next week: http://www.sosmath.com/calculus/limcon/limcon04/limcon04.html#answer4
Formal definition of limit and continuity
Hand-out: Functional limit, continuous functions
READING: http://www.mathsisfun.com/calculus/limits-formal.html
VIDEO HELP: http://www.calculus-help.com/tutorials/
Lesson 5: Continuity
Lesson 6: The Intermediate Value Theorem
VISIT ONE OF THE LEADING UNIVERSITIES IN THE WORLD: THE MIT
SCROLL DOWN FOR ALL LECTURES
MIT FREE VIDEOS: Only till 50th minutes : Lecture 02: Limits,
114.9 MB
continuity; Trigonometric limits
Week 4 October 3-4:
Wednesday: Sequences, limit of a sequence, treshold number and its calculation.
LEARN: http://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx
Thursday:
Quick quiz: THIS WILL BE FOR 2 POINTS!
-
Intermediate value theorem or limit of a function (formal one) ONLY PRECIZE STATEMENTS WILL BE MARKED!
-
Limit calculations using well-known limits – unfortunately this should be write again!
RECALL: Trigonometrical functions:
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/trig.html
LEARN: Inverse trigonometrical functions: http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/invtrig.html
Equation of a line. Review: http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/zoo/line.html
Velocity: http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/ball1.html
Difference quotient, differential quotient and their geometrical (slope of secant, tangent) and physical interprtations
(average velocity, instantenous velocity).
See how the secant approaches the tangent: http://www.math.umn.edu/~garrett/qy/Secant.html
Just tangents: http://www.math.umn.edu/~garrett/qy/TraceTangent.html
Examples: differential quotient of x2 and x ½. Similar examples:
http://tutorial.math.lamar.edu/Classes/CalcI/Tangents_Rates.aspx
http://archives.math.utk.edu/visual.calculus/2/definition.8/index.html
(some other nice animations: http://www2.latech.edu/~schroder/animations.htm)
SCROLL DOWN FOR ALL LECTURES
Notion of a derivative (function).
HOMEWORK: arc cot (x), and the limit problems.
SCROLL DOWN FOR ALL LECTURES
Week 5 October 10-11:
Wednesday:
The notion of the derivative, the derivative of elementary functions. Derivation rules.
http://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDerivative.aspx
http://tutorial.math.lamar.edu/Classes/CalcI/DiffFormulas.aspx
Thursday: Quick quiz: Limit calculations using well-known limits
The notion of the derivative, the derivative of elementary functions. Derivation rules. (cont.)
http://tutorial.math.lamar.edu/Classes/CalcI/ProductQuotientRule.aspx
Linear approximation of functions:
http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/approx.html
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/syllabus/
HANDOUT: LEARN: 2.1, 2.2, 2.4, 2.5., except Example 3., 4., 5. on pages 103-104.
READ: 1.6, 2.3.
Homeworks for WEDNESDAY:
Page 111/57, 59., 60., page 120/36., 40., 52., page 130/7., Page 138/51., page 144/1-4., 24., 26., 33.
NEXT QUICK QUIZ will be on very complex differentitation, involving some chain-rule
applications as well.
SCROLL DOWN FOR ALL LECTURES
WEEK 6 October 17-18:
WEDNESDAY: Logarithmic derivation, implicit derivation (See latest handout)
THURSDAY: ROLLE’S THEOREM:
http://oregonstate.edu/instruct/mth251/cq/Stage7/Lesson/rolles.html
Finding turning=critical=stationary points – at which the extrema MAY (or MAY NOT) occur.
THERE WILL BE NO QUICK QUIZ NEXT WEEK!
BUT YOU NEED PRACTICE FOR THE MIDTERM EXAM:
Chain rule: (solved) examples: http://bhageno5.wikispaces.com/Derivatives
Solve these-except 33 and 37 (solutions are at the end of the file)
http://www.math.montana.edu/~michels/MATH%20171%20DeriWksh.pdf
Find critical points:http://www.analyzemath.com/calculus/Problems/tangent_lines.html
Find the equation to a circle at point x=½ given by the following formula: x2+y2=4
MIDTERM PATTERN IS HERE:
http://digitus.itk.ppke.hu/~b_novak/BANKI/english_midterm_pattern.doc
SCROLL DOWN FOR ALL LECTURES
WEEK 7 October 24-25 and 27- MIDTERM EXAM!!:
WEDNESDAY: MEAN VALUE (LAGRANGE’s) THEOREM:
Illustration:
http://www.analyzemath.com/calculus/MeanValueTheorem/MeanValueTheorem.html
Text:
http://oregonstate.edu/instruct/mth251/cq/Stage7/Lesson/MVT.html
PROOF:
http://oregonstate.edu/instruct/mth251/cq/Stage7/Lesson/rolles.ii.html
Function investigation: finding minima and maxima
L’Hospital Rule for calculating limits (not in the midterm exam)
THURSDAY:
-
concavity, finding points of inflexion(=inflection) (not in the midterm exam)
-
training for the midterm exam
SCROLL DOWN FOR ALL LECTURES
SATURDAY: (is defined as THURSDAY): MIDTERM TEST
(+ antiderivatives by Pál Pentelényi)
WEEK 8 October 31:
WEDNESDAY:
- complete function investigation and sketching graphs (hyperbolic functions and their inverses)
- applications: optimization problems
Homework: handout – each student will have a function for investigating and sketching for 2
ponts.
DUE DATE: NOVEMBER 7 (next Wednesday) – late hands-in are NOT EXPECTED!
http://www.softouch.on.ca/kb/data/Calculus%20Workbook%20For%20Dummies.pdf
SCROLL DOWN FOR ALL LECTURES