Nepal Mathematics Centre-2(C-1)-Curve sketching

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8/31/2011 1 The World of NMC

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8/31/2011 2 NAMASTE gd:]t HjHjnkf GOOD MORNING Nxf:;f]M km\ofkm'NhLM ;nfdflnsd

I. Curve Sketching:

8/31/2011 3 I. Curve Sketching Prof. Dr. R.M Shreshtha Principal H.N. Upadhyaya, Lecturer R. Gyanwali Session I(B) : Resource Materials Study Session I(C) : Interactive Discussion Session I(D) : Intensive Workshop Session I(A) : Curriculum Review

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8/31/2011 4 I. Introduction: This course deals with the fundamentals of advanced mathematical concepts. It also tries to consolidate the concepts and skills learnt in Mathematics course in school level. It is desirable at the end of each unit sufficient related problems be solved. Mathematics XI Higher Secondary School Curriculum (2067/2010) Full Marks 100 Teaching hours 150 Academic Session I(A)

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8/31/2011 5 Unit 3: Curve Sketching 10 hrs Odd and even functions, Periodicity of a function, symmetry (about x – axis, y – axis and origin) of Elementary functions, Monotonocity of a function, Sketching graphs of polynomial functions , Trigonometric, Exponential, Logarithmic functions (Simple cases only) II. Specific Objectives : On completion of this course students will be able to: 4. sketch the curves

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8/31/2011 6 Digression Curve-Sketching Lesson 1

Curve Sketching Unit:

8/31/2011 7 WELCOME Curve Sketching Unit to

Course Content Overview:

8/31/2011 8 Course Content Overview Curve sketching course content begins with odd and even functions plus basic characteristics like periodicity, symmetry and monotonocity of functions followed by sketching of algebraic and non-algebraic functions. (note: reciprocal and rational functions do not come under polynomial functions)

Specific Objective: On completion of this course students will be able to: “sketch the curves”:

8/31/2011 9 Specific Objective: On completion of this course students will be able to: “sketch the curves” Elaboration needed This single objective does not reflect the entire content of the unit: For being able to sketch a curve properly, we need additional information about the basic functions.

Here is something that we need to know about basic function and techniques of sketching the curves.:

8/31/2011 10 Nothing can be done if we first do not know “ What we mean by Curve Sketching”. Curve sketching must NEVER be understood simply as plotting some points and then joining them by a smooth curve. Here is something that we need to know about basic function and techniques of sketching the curves.

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8/31/2011 11 Learning Objectives (In A nutshell ) To define basic functions To draw a list of basic functions To enumerate the basic features of basic functions To explain what we mean by “Curve Sketching” After completing this unit, students will be able:

D I S A I M I S:

8/31/2011 12 O M A I N N T E R C E P T S Y M M E T R Y S Y M P T O T E S N T E R V A L S A X M I N N F L E C T I O N K E T C H D I S A I M I S To interpret the words in the acronym: with special reference to simple algebraic and non-algebraic functions

D I S A I M I S:

8/31/2011 13 O M A I N N T E R C E P T S Y M M E T R Y S Y M P T O T E S N T E R V A L S A X M I N N F L E C T I O N K E T C H D I S A I M I S To find of simple algebraic and non-algebraic functions

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8/31/2011 14 To apply the basic steps of curve sketching in sequential order To plot certain crucial points To connect the points by a curve that exhibits significant features of the graph of the function under investigation

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8/31/2011 15 Resource Materials Study Some Similar Courses elsewhere with special reference to objectives, content and extent b) Teaching materials available c) Instructional strategies d) Assessment strategies Academic Session I(B)

Glimpses at Extracts from Resource Materials:

8/31/2011 16 Glimpses at Extracts from Resource Materials

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Aim of Lesson:

8/31/2011 21 Aim of Lesson Next slide To introduce what an asymptote is, the difference in a horizontal and vertical asymptote and how to find these.

Example (continued):

8/31/2011 22 Example (continued) f(x)= x 3 - 12x + 1 X Y Cubic function Maximum at (-2,17) -2 20 Minimum at (2,-15) 2 -20 Crosses y-axis at (0,1)

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8/31/2011 24   x y O (0, 1) .7 2.7  .7  2.7 CURVE SKETCHING ? + + + + + + + + + + + + + +  2 2  3 Steps 1. Domain: x :   to +  0 2. Range: y :  3 to +   3. y-intercept: x = 0 y =1 x-intercept: y = 0 x  .7 ;  2.7 4. Critical points at : x =  2, 0, 2

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8/31/2011 26 Inter-active Discussion Academic Session I(C) First Stage Problems Identification (a) 1) Teaching and Learning What problem (s) needs more focus ? Clarity of objectives Yes/ No/ Don’t know b) Specification of details of content Yes/ No/ Don’t know c) Extension of content including problem solving Yes/No/Don’t’know

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8/31/2011 27 Inter-active Discussion Academic Session I(C) Second Stage Problems Identification (b) 2) Management of the allotted 10 hours What problem (s) are most crucial ? Availability of full 10 hours Yes/No/ Don’t know b) Sufficiency of full 10 hours Yes/ No/ Don’t know Necessity of additional problems classes Yes/No/Don’t know

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8/31/2011 28 Inter-active Discussion Academic Session I(C) Third Stage(a) Problems Identification (c) 3) Elaboration of the Evaluation Scheme Will it be better to subdivide 2 marks questions into shorter questions carrying 1 or .5 marks each? Yes/No/Don’t know Will it be advisable to further subdivide a 4 mark question into ( 2 + 2), (3 + 1), (1 + 1 + 1 +1), etc. ? Yes/No/Don’t’know Can we state that a certain fixed number of correct steps carry full marks? Yes/No/Don’t know

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8/31/2011 29 Inter-active Discussion Academic Session I(C) Third Stage (Alternative) Problems Identification (c) 3) Elaboration of the Evaluation Scheme Is it OK to break a) Each 2 marks question into four parts each carrying .5 marks for the determination of any four of Domain, Range, x- intercept, y-intercept, symmetry, asymptote Yes/No/Don’t’know Each 4 marks question into two parts each carrying 2 marks for determination of any two of critical points, interval of increase/decrease, local maximum or minimum, Point of inflection, Concavity Yes/No/Dont’ know

Questionaires related to Introduction:

8/31/2011 30 Questionaires related to Introduction Do you think that this unit deals with the fundamentals of advanced mathematical concepts as mentioned in the Introduction of the course ? Yes/No/ Don’t know Does it consolidate the concept and skills learnt in mathematics course in school level? Yes/No/Don’t know Do you find this unit sufficient in problems-solving ? Yes/No/Don’t know Optional: Your suggestions: ( Only one sentence for each question) 1. 2. 3. Academic Session I(C)

Questionaires related to Objectives:

8/31/2011 31 ● To make distinction between domain and range (i.e., the set of permissible values of x and y ), Yes/No/Don’t know ● To determine - x- and y-intercepts i.e. the points where the graph crosses the axes) Yes/No/Don’t know ●To examine - symmetry , if any, (i.e. odd or even or periodic nature of the function), Yes/No/Don’tknow Questionaires related to Objectives What is your opinion about the single objective ? Good/Satisfactory/ unsatisfactory Will it be appropriate to include the following specific objectives ? Academic Session I(C)

Questionaires related to Objectives:

8/31/2011 32 ●To find asymptote , (i.e., the line to which a curve comes closer and closer but never meeting it) if any Yes/No/Don’t know ●To identify intervals of increase or decrease Or monotonocity (i.e., intervals in which the slope is negative or positive) Yes/No/Don’t’ know Questionaires related to Objectives Will it be appropriate to include the following specific objectives ? Academic Session I(C)

Questionaires related to Objectives:

8/31/2011 33 To determine maximum and minimum (i.e. points where the slope is zero or tangent line is horizontal) or (i.e. local maximum or minimum or concavity upwards or downwards) Yes/No/Don’t know ●To find -point of inflection (i.e., the point where the concavity switches from CU to CD or CD to CU) Yes/No/Don’t know ●To draw - sketch A reasonable looking curve (or a qualitative picture of the graph) capturing certain basic features by putting together the foregoing information. Academic Session I(C) Questionaires related to Objectives

Questionaires related to Course Content:

8/31/2011 34 Questionaires related to Course Content Do you feel that the course content needs more detailed description? Yes/No/ Don’t know What do you assess the difference (s) in content and extent in the published books ? Not problematic/ Problematic/ Don’t’ know Will it be beneficial if the course content and extent is elaborated by incorporating the following detailed course description? a) Odd and even function with reference to following basic functions: linear function, quadratic function, cubic function, biquadratic function, absolute value function, reciprocal functions, elementary exponential, logarithmic and trigonometric Yes/No/Don’t know Academic Session I(C)

Questionaires related to Course Content:

8/31/2011 35 Questionaires related to Course Content Will it be beneficial if the course content and extent is elaborated by incorporating the following detailed course description? b) Periodicity and determination of period, amplitude and phase Yes/No/Don’t know c) Symmetry , finding symmetric of basic functions only Yes/No/Don’t know d) Idea of zeros, poles and holes to determine critical points and asymptotes Yes/No/Don’t know e ) Monotonocity ; determination of interval of increase or decrease using difference quotient or slope related to two consecutive points Yes/No/Don’t know Academic Session I(C)

Questionaires related to Course Content:

8/31/2011 36 Questionaires related to Course Content f ) Determination of local maximum and minimum of basic functions Yes/No/Don’t know g) Point of inflection or second order critical points and their determination Yes/No/Don’t know h) Concavity change Yes/No/Don’t know Will it be beneficial if the course content and extent is elaborated by incorporating the following detailed course description? Academic Session I(C)

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8/31/2011 37 Your suggestions: ( Just a couple of sentences only) 1. 2. 3. Academic Session I(C)

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Challenges and Opportunities:

8/31/2011 39 Challenges and Opportunities General Challenges Changes in content, approach and attitude that will cater the need of rapidly changing science and technology dominated world of work ? Narrowing/ Bridging the ever increasing multi-dimensional gap ( internal as well as international) in teaching , learning and evaluation. Developing a national curriculum frame Something more: 1. .. … …. 2…. … … Academic Session I(D)

Challenges and Opportunities:

8/31/2011 40 Challenges and Opportunities Specific Challenges Clarity in Course overview or introduction Detailed Specification of objectives indicating the content and extent of teaching, learning and evaluation Preparation of Teaching manual Standardization of Teaching and Testing materials plus marking system Something more: … … … … … … … … …. Academic Session I(D)

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8/31/2011 41 Opportunities Learning, applying and earning Academic revitalization Manual plus technological skills enhancement Effective teaching capability and time management Preparing teaching and learning materials plus teaching aids that utilize modern technology, especial computer and e-information) Efficiency in testing and evaluating through improved materials and techniques Something more: 1. .. … …. 2…. … …

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8/31/2011 42 GOOD LUCK EVERYBODY NEPAL MATHEMATICS CENTRE SEE YOU NEXT TIME THANKS

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