Příroda se směje našim potížím s integrací. (Pierre Simon Laplace, francouzský matematik a fyzik)

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mathematics [2020/03/05 07:36] (aktuální)
Řádek 1: Řádek 1:
 +<markdown>
 +Mathematics
 +===========
 +
 +* This is a webpage for Erasmus students at Faculty of Forestry and Wood
 +technology and for students of Faculty of Regional Developement and
 +International Studies.
 +* The lectures are scheduled on Monday 15:00-19:00
 +and Tuesday 15:00-19:00, room Z7 in the building of the [Faculty of
 +Regional Develpement.](http://frrms.mendelu.cz/25929-kontakt)
 +* Lectures start on September 25. There are 5 weeks of lectures. The
 +exam follows.
 +
 +
 +EXAMS, common informations
 +--------------------------
 +
 +* The exam is written, includes problem solving and theoretical
 +    questions, passing is 40%. Exam covers the topics from lectures.
 +* Example of one old exam is [here](angl/exam.pdf).
 +<!--
 +* You get your graduation in the university information system. Anyway,
 +  it is always a good idea (especially if you fail) to come to us
 +  (Building B, campus Cerna Pole, Zemedelska 3) and look at your
 +  answer sheet with marked corrections.
 +* Full credit is for correct answers, partial credit for partially
 +  correct answers, **no credit if you answer another question**.
 +* Always answer the questions. If the question is "*write the
 +  definition* of increasing function" then you are supposed to write
 +  the *definition* of increasing function. No partial credit for
 +  other things (such as the relation of the derivative and
 +  monotonicity). If the question is "*explain the relationship* between
 +  the monotonicity and derivative", you are supposed to *explain the
 +  relationship* between the monotonicity and derivative. No partial
 +  credit for other things (such as definition of monotonicity,
 +  definition of increasing functions, ...)
 +-->
 +
 +
 +EXAMS, Erasmus students 
 +----------------------
 +
 +* Contact the teachers, all exams have to be finished before February 2018.
 +* As soon as the terms of exams will be known, you have to register to the exam in the [University Information
 +    System](http://is.mendelu.cz/?lang=en) (UIS).
 +
 +EXAMS, Faculty of regional development 
 +--------------------------------------
 +
 +* The exams are organized by the faculty staff, as usual on this
 +  faculty. The lecturers just provide the problems for each term,
 +  correct solutions and graduation scale.
 +* It is supposed that one term will be in the week after the lectures, one term in November or December and the other terms
 +  in January 2018.
 +* As soon as the terms of exams will be known, you have to register to the exam in the [University Information
 +    System](http://is.mendelu.cz/?lang=en) (UIS).
 +
 +
 +
 +Materials for year 2017 (September, October) - Monday's lecture
 +-------------------------------
 +
 +1. (Sep. 25. 15-19, Z7) **Precalculus** (one-to-one functions, inverse functions, monotonicity), equations via inverse functions, continuity, theorem of Bolzano, solving inequalities using theorem of Bolzano. **Calculus** (evaluating derivatives)
 +   * [Homework 1](angl/hw1.pdf)
 +   * Derivatives, [step by step solved problems](frvs/derivace.pdf)
 +   * [Online calculator with step by step solution](http://um.mendelu.cz/maw-html/index.php?lang=en&form=derivace), 
 +   * Derivatives, [exercises for exam](angl/derivatives_exam.pdf)
 +   * [Refcard](frvs/ref-cards/formulas.pdf), 
 +1. (Oct. 2. 15-19, Z7) **Derivatives** (linear approximation, derivative and continuity, related rates, derivative and monotonicity, derivative and local extrema)
 +   * [Homework 2](angl/hw2.pdf)
 +   * [Problems on related rates](angl/related_rates.pdf)
 +1. (Oct 9. 15-19, Z7) **Derivatives** (min-max problems,  Newton-Raphson method for nonlinear equations)
 +   * [Homework 3](angl/hw3.pdf)
 +   * [Problems on minima and maxima](angl/aplikace.pdf)
 +   * [Related rates problems](http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx)
 +   * Curve sketching, also exercises for exam, ([PDF, exercises from 15 to 34, skip 30](pdf/mm2.pdf))
 +1. (Oct 16. 15-19, Z7) **Derivatives** (derivative and convexity, function investigation and curve sketching)
 +   * [Homework 4](angl/hw4.pdf)
 +   * Curve sketching, also exercises for exam, ([PDF, exercises from 15 to 34, skip 30](pdf/mm2.pdf))
 +   * Derivatives using [Sagecell](https://sagecell.sagemath.org/):
 +       * `((3*x^2+1)/x).factor().show() # factorization`
 +       * `((3*x^2+1)/x).diff(x).factor().show() # f' and factorization`
 +       * `((3*x^2+1)/x).diff(x,2).factor().show() # f'' and factorization`
 +   * Derivatives using [Wolfram Alpha](https://www.wolframalpha.com/):
 +       * `factor (3x^2+1)/x`
 +       * `differentiate (3x^2+1)/x`
 +       * `second derivative of (3x^2+1)/x`
 +1. (Oct 23. 15-19, Z7) **Summary of calculus** (Robert Mařík)
 +
 +
 +
 +Literature 
 +----------
 +- Literature authored by Robert Mařík (precalculus, calculus, polynomials)
 +  * Paper-friendly textbook ([PDF](angl/paper.pdf), approx 90 pages)
 +  * Shortened version of textbook ([PDF](angl/short_math.pdf), approx 20 pages)
 +  * Screen-friendly textbook ([PDF](angl/screen.pdf))
 +  * [Presentations, real world applications, solved problems,
 +quizzes](frvs/index.html)
 +- Literature authored by Simona Fišnarová (integral calculus, linear algebra)
 +  *   [Home page](http://user.mendelu.cz/fisnarov/?page=3)
 +
 +
 +<hr>
 +<hr>
 +
 +
 +*Materials from years 2012--2016*
 +-------------------------------
 +
 +
 +
 +Materials from year 2016 (September, October)
 +-------------------------------
 +
 +1. (Sep. 21, 13-17) **Precalculus** (one-to-one functions, inverse functions, monotonicity), equations via inverse functions, continuity, theorem of Bolzano, solving inequalities using theorem of Bolzano. **Calculus** (evaluating derivatives)
 +1. (Sep. 28) **Canceled** - [public holliday in the Czech Republic](https://en.wikipedia.org/wiki/Public_holidays_in_the_Czech_Republic)
 +1. (Oct. 5, 13-17) **Derivatives** (derivative and continuity, derivative and monotonicity, derivative and convexity, linear approximation, derivative and local extrema)
 +   * [online calculator with step by step solution](http://um.mendelu.cz/maw-html/index.php?lang=en&form=derivace), 
 +   * Derivatives, [exercises for exam](angl/derivatives_exam.pdf)
 +1. (Oct. 12, <span style="color:red">15-17</span>) **Derivatives** (min-max problems, related rates)
 +<span style="color:red">*The lecture starts at 15:00, since from 13:00 to 15:00 there is a meeting of university staff with the rector and the rector canceled the lectures from 13 to 15.*</span>
 +   * [Related rates problems](http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx)
 +   * Curve sketching, also exercises for exam, ([PDF, exercises from 15 to 34, skip 30](pdf/mm2.pdf))
 +1. (Oct. 19, 13-17) **Derivatives** (function investigation and curve sketching, Newton-Raphson method for nonlinear equations)
 +   * [Related rates problems](http://tutorial.math.lamar.edu/Classes/CalcI/RelatedRates.aspx)
 +   * Curve sketching, aslo exercises for exam, ([PDF, exercises from 15 to 34, skip 30](pdf/mm2.pdf))
 +
 +
 +
 +
 +
 +
 +Materials for year 2015 (November, December)
 +-------------------------------
 +
 +1. (Fri, Nov 6, 13:00) <br>**Precalculus** (Radka Smýkalová)
 +1. (Fri, Nov 13, 13:00, room B44)<br>**Calculus, derivatives and applications of derivatives (related rates, linear approximation, nonlinear equations)** (Robert Mařík)
 +   <br>Materials from lecture
 +     * [Handwritten slides from the lecture](angl/2015_11_13.pdf)
 +     * [Refcard](frvs/ref-cards/formulas.pdf), 
 +     * Derivatives, [step by step solved problems](frvs/derivace.pdf)
 +     * Derivatives, [exercises for exam](angl/derivatives_exam.pdf)
 +     * Root finding using Newton-Raphson method  ([wikipedia](http://en.wikipedia.org/wiki/Newton%27s_method))
 +     * [Kinetic energy, Einstein's formula and linear approximation](frvs/einstein.pdf)
 +1. (Fri, Nov 20, 13:00, room B44)<br>**Calculus, derivatives, limits and continuity (Bolzano theorem), applications of derivatives (maxima and minima)** (Robert Mařík)
 +     * [Handwritten slides from the lecture](angl/2015_11_20.pdf)
 +1. (Fri, Nov 27, 13:00, room B44)<br>**Calculus, derivatives, curve sketching** (Robert Mařík)
 +     * Curve sketching -- exercises for exam, ([PDF, exercises from 15 to 34, skip 30](pdf/mm2.pdf))
 +     * [Handwritten slides from the lecture](angl/2015_11_27.pdf)
 +1. (Fri, Dec 4, 13:00, room B44)<br>**Summary of calculus** (Robert Mařík)
 +     * [Handwritten slides from the lecture](angl/2015_12_04.pdf)
 +
 +
 +Materials for year 2014 (January)
 +-------------------------------
 +Exam
 +----
 +-   Written, includes problem solving and theoretical questions, passing
 +    is 40%. Exam covers the topics from lectures. 
 + *  Example of one old exam is [here](angl/exam.pdf).
 + *  Derivatives -- exercises for exam ([PDF](angl/derivatives_exam.pdf))
 + *  Curve sketching -- exercises for exam, ([PDF, exercises from 15 to 34, skip 30](pdf/mm2.pdf))
 +-   You have to register to the exam in the [University Information
 +    System](http://is.mendelu.cz/?lang=en) (UIS). The deadline for
 +    submitting or canceling registration is about 24 hours before the
 +    exam (you will see the exact deadline in UIS).
 +
 +Literature 
 +----------
 +- Literature authored by Robert Mařík (precalculus, calculus, polynomials)
 +  * Paper-friendly textbook ([PDF](angl/paper.pdf), approx 90 pages)
 +  * Shortened version of textbook ([PDF](angl/short_math.pdf), approx 20 pages)
 +  * Screen-friendly textbook ([PDF](angl/screen.pdf))
 +  * [Presentations, real world applications, solved problems,
 +quizzes](frvs/index.html)
 +  * for slides from lectures see below
 +- Literature authored by Simona Fišnarová (integral calculus, linear algebra)
 +  *   [Home page](http://user.mendelu.cz/fisnarov/?page=3)
 +
 +
 +Lectures  by Robert Mařík (January 2014)
 +---------------------------------------
 +
 +1. **Precalculus, Calculus;
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img width="80" src="slides.png"/>PDF slides from the lecture](angl/mathematics_2014_01_07.pdf)** 
 + * Functions: basic elementary functions
 +      ([short list](frvs/funkce.pdf), [graphs](frvs/ref-cards/grafy_el.pdf)),
 +      elementary functions, increasing/decreasing, one-to-one,
 +      inverse, ([texbook/pages from 15 to 24](angl/screen.pdf), [interactive exercises on inverse functions](frvs/tests/inverzni-funkce.pdf))
 + * Introduction to limits 
 + \([MIT online course](http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-2-limits/ "video lectures")\)    
 +     * Rough geometrical idea ([animations](frvs/limits-animations.pdf))
 +     * Definition of limit ([texbook](angl/screen.pdf))
 +    * Applications of limits ([texbook](angl/screen.pdf))
 +        * Continuity, continuity of elementary functions ([wikipedia](http://en.wikipedia.org/wiki/Continuous_function))
 +        * Theorem of Bolzano  ([textbook/page 49](angl/screen.pdf), [wikipedia](http://en.wikipedia.org/wiki/Intermediate_value_theorem))
 +        * Solving inequalitites using Theorem of Bolzano ([textbook/page 52](angl/screen.pdf), [interactive exercises](frvs/tests/ner.pdf))
 +        * Derivatives: 
 +   * definition ([texbook/page 57](angl/screen.pdf)), 
 +   * real world interpretation ([texbook/page 61](angl/screen.pdf)),
 +                  * geometric interpretation ([texbook/page 58](angl/screen.pdf)), 
 +   * ([derivatives on SOS math](http://www.sosmath.com/calculus/diff/der00/der00.html))
 +
 +2. **Calculus;
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img width="80" src="slides.png"/>PDF slides from the lecture](angl/mathematics_2014_01_14.pdf)** 
 +   * Evaluating derivatives 
 + * [refcard](frvs/ref-cards/formulas.pdf), 
 + * [solved exercises](frvs/derivace.pdf), 
 + * [interactive exercises](frvs/tests/der1.pdf), 
 + * [online calculator with step by step solution](http://um.mendelu.cz/maw-html/index.php?lang=en&form=derivace), 
 +   * Linear approximation
 +   * Related rates prolems
 +2. **Calculus;
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img width="80" src="slides.png"/>PDF slides from the lecture](angl/mathematics_2014_01_21.pdf)** 
 +   * Minima and maxima of function
 +   * Curve sketching: [problems for final exam (from 15 to 34, skip 30)](pdf/mm2.pdf), [solved problems](frvs/prubeh.pdf)
 +   * Derivatives [exercises for exam](angl/derivatives_exam.pdf)
 +2. **Calculus;
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img width="80" src="slides.png"/>PDF slides from the lecture](angl/mathematics_2014_01_28.pdf)** 
 +   * Taylor polynomial ([presentation](frvs/taylor.pdf), [application](frvs/einstein.pdf), [wikipedia](http://en.wikipedia.org/wiki/Taylor_polynomial#Taylor.27s_theorem_in_one_real_variable))
 +   * Nonlinear equations
 +     * Root finding using bisection of interval ([wikipedia](http://en.wikipedia.org/wiki/Bisection_method))
 +     * Root finding using Newton-Raphson method  ([wikipedia](http://en.wikipedia.org/wiki/Newton%27s_method))
 +   * Algebraic equations, solvability and multiplicity of the solutions [textbook, Chapter 3, only topics covered by the presentation from the lecture](angl/screen.pdf)
 +   * Least squares method ([presentation](frvs/mnc.pdf))
 +   * Summary
 +
 +   
 +
 +Outline of the lectures from January 2013 by Robert Mařík
 +---------------------------------------
 +
 +1. **Precalculus, Calculus;
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img width="80" src="slides.png"/>PDF slides from the lecture](angl/frrms_2013_01_04.pdf)** 
 + * Functions: basic elementary functions
 +      ([short list](frvs/funkce.pdf), [graphs](frvs/ref-cards/grafy_el.pdf)),
 +      elementary functions, increasing/decreasing, one-to-one,
 +      inverse, ([texbook/pages from 15 to 24](angl/screen.pdf), [interactive exercises on inverse functions](frvs/tests/inverzni-funkce.pdf))
 + * Introduction to limits 
 + \([MIT online course](http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-2-limits/ "video lectures")\)    
 +     * Rough geometrical idea ([animations](frvs/limits-animations.pdf))
 +     * Definition of limit ([texbook](angl/screen.pdf))
 +    * Applications of limits ([texbook](angl/screen.pdf))
 +        * Continuity, continuity of elementary functions ([wikipedia](http://en.wikipedia.org/wiki/Continuous_function))
 +        * Theorem of Bolzano  ([textbook/page 49](angl/screen.pdf), [wikipedia](http://en.wikipedia.org/wiki/Intermediate_value_theorem))
 +        * Solving inequalitites using Theorem of Bolzano ([textbook/page 52](angl/screen.pdf), [interactive exercises](frvs/tests/ner.pdf))
 +        * Derivatives: 
 +   * definition ([texbook/page 57](angl/screen.pdf)), 
 +   * real world interpretation ([texbook/page 61](angl/screen.pdf)),
 +          * geometric interpretation ([texbook/page 58](angl/screen.pdf)), 
 +   * ([derivatives on SOS math](http://www.sosmath.com/calculus/diff/der00/der00.html))
 +2. **Calculus II;
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img  width="80" src="slides.png"/>PDF slides from the lecture](angl/frrms_2013_01_08.pdf)**
 +   * Evaluating derivatives 
 + * [refcard](frvs/ref-cards/formulas.pdf), 
 + * [solved exercises](frvs/derivace.pdf), 
 + * [interactive exercises](frvs/tests/der1.pdf), 
 + * [online calculator with step by step solution](http://um.mendelu.cz/maw-html/index.php?lang=en&form=derivace), 
 + * [more online calculators (in Czech)](akademie/diferencialni-pocet.html))
 +   * Applications of derivatives
 +     * Differentiability implies continuity   ([wikipedia](http://en.wikipedia.org/wiki/Continuous_function#Relation_to_differentiability_and_integrability), [textbook/page 62](angl/screen.pdf))
 + * Tangent ([wikipedia](http://en.wikipedia.org/wiki/Tangent_line#More_rigorous_description)), linear aproximation ([textbook/page 70](angl/screen.pdf), [external link](http://www.supermath.info/Calc85to101.pdf))
 +3. **Calculus III (Applications of derivatives);
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img width="80"  src="slides.png"/>PDF slides from the lecture](angl/frrms_2013_01_10.pdf)**
 +  * Related rates of change ([wikipedia](http://en.wikipedia.org/wiki/Related_rates)), see also external links to exercises on related rates at the bottom of the page
 +  * Local extrema: 
 +       * definition, 
 +    * existence of derivative at local extremum implies stationary point, 
 +    * 1-st derivative test 
 +    * links: [textbook/Chapter Caculus - Extremal problems](angl/screen.pdf), [wikipedia](http://en.wikipedia.org/wiki/Local_maximum), [solved exercises](frvs/minmax.pdf), [interactive exercises](frvs/tests/loc11.pdf), 
 +    * see also external links to exercises on min/max problems at the bottom of the page
 +  * Curve sketching: [problems from 15 to 34, skip 30](pdf/mm2.pdf), [solved problems](frvs/prubeh.pdf)
 +  * Derivatives -- exercises for exam ([PDF](angl/derivatives_exam.pdf))
 +4. **Other selected topics;
 +&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; [<img  width="80"  src="slides.png"/>PDF slides from the lecture](angl/frrms_2013_01_22.pdf)**
 +   * Algebraic equations, solvability and multiplicity of the solutions [textbook, Chapter 3, only topics covered by the presentation from the lecture](angl/screen.pdf)
 +   * Curve sketching: [exercises](pdf/mm2.pdf), [solved problems](frvs/prubeh.pdf)
 +   * Nonlinear equations
 +     * Root finding using bisection of interval ([wikipedia](http://en.wikipedia.org/wiki/Bisection_method))
 + * Root finding using Newton-Raphson method  ([wikipedia](http://en.wikipedia.org/wiki/Newton%27s_method))
 +   * Taylor polynomial ([presentation, Chapters 1 and 3 only](frvs/taylor.pdf), [wikipedia](http://en.wikipedia.org/wiki/Taylor_polynomial#Taylor.27s_theorem_in_one_real_variable))
 +   * Least squares method ([presentation, Chapters 1 and 3 only](frvs/mnc.pdf))
 +
 +
 +Other resources collected for course in January 2013
 +---------------
 +-   Online video tutorials
 +    *  [Patrick Jones](http://patrickjmt.com/#calculus)
 + *  MIT
 +    * [Gilbert Strang](http://www.academicearth.org/lectures/big-picture-of-calculus)
 +    * [David Jerison](http://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010)
 + *  [Calculus I in 20 Minutes](http://www.youtube.com/watch?index=0&playnext=1&feature=PlayList&v=EX_is9LzFSY&list=PLB44D811A2E0DE413)
 +-   Applications:
 +    [http://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx](http://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx)
 +-   Min/max problems (external links)
 +    -   [link 1](http://www.math.wfu.edu/tutorials/Math111/optimization.pdf)
 +        (skip 22)
 + -   [link 2](http://www.saylor.org/site/wp-content/uploads/2011/11/4-5AppliedMaxMinProblems1.pdf)
 + -   [link 3](http://www.scribd.com/doc/18488241/XII-Application-of-the-Derivatives-Assignment)
 +        from 37 to 60
 + -   [link 4](http://www.whitman.edu/mathematics/calculus/calculus_06_Applications_of_the_Derivative.pdf)
 +        page 11-13, problems 2-27 (solved problems are from page 5) -
 +        page numbers reffer to PDF file, not numbers printed in the top
 +        of the pages.
 +- Related rates problems (external links)
 +    - [link 1](http://teachers.misd.k12.wa.us/hs/calculus/documents/4.6_related_rates.pdf)
 + - [link 2](http://fredmath.net/Calculus/cal-I/lecture2/sec2.8/sec-2.8.pdf)
 + - [link 3](http://campuses.fortbendisd.com/campuses/documents/Teacher/2010/teacher_20101112_1338_2.pdf)
 +   - Water runs into a conical tank at the rate of 9 ft^3/min. The tank stands vertex down and has a height of 10
 +feet and a base radius of 5 feet. How fast is the water level rising when the water is 6 feet deep?
 +   - Two truck convoys leave a depot at the same time. Convoy A travels east at 40 mph and convoy B travels
 +north at 30 mph. How fast is the distance between the convoys changing a) in 6 minutes b) in 30 minutes
 +   - Two commercial jets at 40,000 ft. are both flying at 520 mph towards an airport. Plane A is flying south and
 +is 50 miles from the airport while Plane B is flying west and is 120 miles from the airport. How fast is the
 +distance between the two planes changing at this time?
 +    - [link 4](http://www.saylor.org/site/wp-content/uploads/2011/11/3-7RelatedRates1.pdf)
 +
 +</markdown>
  
mathematics.txt · Poslední úprava: 2020/03/05 07:36 (upraveno mimo DokuWiki)

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