Donát Bánki Faculty of Mechanical and Safety EngineeringMATHEMATICS: CALCULUS I.

Lecturers: Ágnes Bércesné Novák PhD, Pál Pentelényi PhD

SyllabusRELIABLE FUNCTION PLOTTER

RELIABLE Calculus pages: http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx

http://archives.math.utk.edu/visual.calculus/

HELP SHEETS FOR THE EXAMS:

http://digitus.itk.ppke.hu/~b_novak/BANKI/kepletgyujtemeny.pdf

http://digitus.itk.ppke.hu/~b_novak/BANKI/der-tabl.pdf

Week 1-2 September 10-11, 17-18 Complex numbers

A good summary:

http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS/upfiles/topics/ess_at_12_cn_stu.pdf

For practicing:

- IF you have Stroud: Engineering Mathematics, go through the 2 Complex numbers chapters, and try to solve all revision problems. In some edition Chapter 1 and 2 deals with comples numbers.

- IF you do NOT have Stroud’s book, you will like these interactive pages.

Theory and practice problems with solution

PLEASE FIND THE YOUR TURN QUIZZES AT THE AND OF EACH CHAPTER AND SOLVE!

Introduction (notion, algebraic form and addition, subtraction, multiplication, division ):

http://www.mathsisfun.com/numbers/complex-numbers.html

Complex plane: http://www.mathsisfun.com/algebra/complex-plane.html

Polar form : http://www.mathsisfun.com/algebra/complex-number-multiply.html

Quiz finder: http://www.mathopolis.com/questions/quiz-search.php

Subtracting n-th root:

http://www.freelearningchannel.com/l/Content/Materials/Mathematics/Trigonometry/textbooks/CK12_Trigonometry/html/6/7.html

Week 3 September 24.: Exponential form. Solving quadratic equations: http://www.mathsisfun.com/algebra/fundamental-theorem-algebra.html

Wednesday: Polynomials - Long division (Seminar)

Solved examples (in Hungarian):

http://digitus.itk.ppke.hu/~b_novak/BANKI/polinomosztas.pdfThursday: Limits of Sequences of Real Numbers

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_1.ppt

Week 3 September 25

Wednesday:

Functions, Elementary functions and their inverses.

Properties: http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_2.ppt

monotonity, concave, convex, inverse trigonometrical functions, etc.

Week 5 October 2-3

Functional limit, continuity:

http://digitus.itk.ppke.hu/~b_novak/BANKI/ Lecture_3.ppt Week 6 October 9-10

Properties of cont. functions

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_3_4.ppt

Week 7 October 15 (3 LECTURES)-16 (3 LECTURES)

Differentiation:

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_4_diff_1.ppt

http://www.math.umn.edu/~garrett/qy/Secant.html

http://www.math.umn.edu/~garrett/qy/TraceTangent.html

Rules for differentitation, derivative of elementary functions and their inverses, physical interpretation (velocity, acceleration)

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_4_diff_1_2_3.ppt

practice problems:

https://math.la.asu.edu/~carlson/dt1.pdf

solution: https://math.la.asu.edu/~carlson/dts1.pdf

*****************************************************--

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/productruledirectory/ProductRule.html

Only problems 1-7

***************************************************************

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/trigderivdirectory/TrigDerivatives.html

here you may find new functions: sec(x):=1/cos(x), csc(x):=1/sin(x) – but it is not in our curricula, you may skip them.

Quiz will be on finding derivatives using sum, product, quotient rule. You supposed to KNOW these 3 rules FROM HEART!

MATH test will be on 6th november.

For this you may find useful the following pages:

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/limcondirectory/LimitConstant.html

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/liminfdirectory/LimitInfinity.html

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/continuitydirectory/Continuity.html

https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/defderdirectory/DefDer.html

NEXT LECTURE: 29 of October

Week 8 October 15

Tuesday:

Implicit differentiation, logarithmic differentiation:

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_4_diff_1_2_3.ppt

Finding minima, maxima:

Rolle’s and Mean Value (Lagrange’s) theorem, applications:

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_6_Rolle_g.ppt

http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-maxmin-2009-1.pdf

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_6_extrema_practical.pptx

Practice problems for the midterm exam:

http://digitus.itk.ppke.hu/~b_novak/BANKI/parctice_midterm.docx

Week 9 October 22-held on October 15

Week 10 October 29

Bernoulli-L’Hospital rule, see the end of the following file:

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_6_Rolle_g.ppt

Hyperbolic functions:

http://digitus.itk.ppke.hu/~b_novak/BANKI/Lecture_7_hyperbolic.ppt

Week 4 October 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

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

MIT FREE VIDEOS: Only till 50th minutes : Lecture 114.9 MB

02: Limits, 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)

Notion of a derivative (function).

HOMEWORK: arc cot (x), and the limit problems.

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.

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

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

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!