Differentiation study guide final 2007

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					        Study guide for
        Differentiation

        Kia ora, welcome

        This module is about differential calculus. It will help you revise your knowledge of
        introductory calculus, learn some new techniques of differentiation, and use
        derivatives to solve a variety of problems.


        Read right through this study guide to ensure you know everything you need to
        do to prepare for different unit and achievement standards.



        What is in this pack?
        This pack has this study guide and three booklets:
        MX704 Learning about differentiation
        MX705 Developing skills in differentiation
        MX707 Solving problems using differentiation.



        How can I learn best?
        Read through the lessons carefully for understanding and try the steps in the
        examples yourself to make sure you follow the reasoning.

        You can help your own learning by self-marking each set of exercises. These are
        designed to develop your understanding of the key learning concepts and full
        answers are provided. Have the Formulae and tables booklet you received in the first
        posting readily available. If you mislay this, contact your teacher.

        Work through and answer as many questions at a time as you can manage. If you are
        unsure on the topic, or if the question has a long answer, it is useful to check the first
        question or two to make sure you are on the right track.

        There is no point in copying the answers without trying the questions, but if you find
        you have made an error, correct it and continue the work from that point.


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    A good way to improve your learning, if you have gone astray, is to attempt a
    question again to make sure that you understand the main points.

    There are summary points and a review exercise at the end of each booklet. These
    can help you revise before attempting the assignment or the unit standard assessment
    task. Because the unit standard task is open book, before starting it is worthwhile to
    look at the questions and go back through the relevant learning.




    What if I get stuck?

    Sometimes reading through the lesson again, or working through a solution in the
    answer guide is enough, but if you need further help you can

          Contact the maths help-desk, week days, 9 am to 4 pm
              Phone:    0800 835 2788
              Fax:       0800 329 2788
              Email:     maths.helpdesk@tcs.ac.nz
          Contact your maths teacher by phone, email or letter.




    Want to access differentiation online?
    If you have online access, check out The Correspondence School maths website on
    differentiation to look at what is recommended for this topic. Links to past
    examination questions with answers and help with the graphics calculator can be
    found here, too. There is reference to specific sites in this study guide.

    The Maths Department website is at www.tcs.ac.nz
    Click on School Websites
    Click on Maths
    Click on Online courses and support
    Click on Mathematics with calculus
    Click on Differentiation
    The log-in name is parklands and the password is cspark1
    (both lowercase)




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        What standards can I gain?
        You may gain Unit Standard 5265, Differentiate functions and use calculus to solve
        problems, worth six credits at Level 3.

        This module will also help you prepare for the external Achievement Standard 90635
        (C3.1), Find and use derivatives to solve problems, worth six credits at Level 3.

        If you gain both of these, they will both appear on your record of learning. However,
        you will be able to count the credits for only one of these on your NCEA certificate.


        Preparing for Unit Standard 5265
        Ideally, you should complete all the work in this pack. Then you should do the unit
        standard assessment task in MX707 and send it to your teacher.

        However, if you are not considering entry for the external achievement standard on
        differentiation, and you are under time pressure, you can omit any or all of the
        following:
        - MX705 lesson 6, review question 9, assignment questions 5, 6
        - MX707 lesson 8 and assessment question 8
        - the work on graphs of derived functions in this study guide.



        Preparing for Achievement Standard 90635
        Full details of Achievement Standard 90635 from NZQA are given later in this study
        guide. There is an internal examination at the end of August or the beginning of
        September that will help you prepare for this standard.

        Ideally, you should complete all the work in this pack. However, you may be under
        time pressure or just wanting a taste of the topic. You may be happy to achieve the
        standard without a merit or excellence grade.

        To achieve the standard you need to be able to differentiate simple functions
        including use of the chain rule, product rule and quotient rules and be able to solve
        straightforward problems using differentiation. This learning is covered in:
        MX704, MX705 and in MX707 lessons 2 to 7. (For achieved only, you can omit
        learning to do with points of inflection). Then, practise using review questions 1 to 4
        and assignment questions 1, 2, 3, 5, 6 and 7.

        For achieved with merit, you also need to work with parameters and parametric
        equations, be able to differentiate implicitly, be able to differentiate functions from
        first principles, solve more challenging optimisation problems, solve problems
        involving related rates of change, identify features of graphs, sketch graphs of
        polynomials and sketch graphs of derived functions. You should do all the work in


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    this pack, as well as the work on sketching graphs of derived functions in this study
    guide.

    For achieved with excellence, problems could include among other things, a proof,
    establishing a model before solving a problem and related rates of change involving
    more than two related rates. Refer to the standard description attached.



    How will I be assessed?
    You can assess your own progress by doing the activities and reviews and marking
    them yourself.

    You can achieve the unit standard by doing the unit standard assessment task in
    MX707. Your teacher will assess this. If you don’t achieve it on the first attempt,
    your teacher will provide you with a further assessment opportunity to do so.

    Your teacher will also mark the assignment at the end of each booklet. If you are
    going to sit the external achievement standard at the end of the year, this will help
    you identify areas in which you need to do extra work.



    National Qualifications Authority Entry
    If you are not already enrolled in a secondary school, in July, you will be sent an
    entry form for national qualifications. If you wish to have the internally or
    externally assessed standards you achieve recognised on your Record of Learning
    you will need to enter and pay the entry fee. You will also need to state what
    externally assessed standards you want to sit examinations in. 90635 Calculus is one
    of these.



    Can I use a graphics calculator?
    Yes you can. The manual that came with your calculator will help you become
    familiar with its use. If you have a Casio graphics calculator CFX-9750 PLUS let
    your teacher know as you can be sent help directly relevant to this topic. The same
    help is on the Mathematics with Calculus website – see above.

    You must still show the derivatives needed to solve differentiation problems.



    Graphs of derived functions and on-line learning
    Find these after the information on the external achievement standard that follows.


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Subject Reference                       Calculus 3.1 (90635 Version 2)
Title                                   Differentiate functions and use derivatives to solve problems
Level            3                      Credits        6              Assessment            External
Subfield         Mathematics
Domain           Calculus
Registration date           16 November 2005           Date version published          16 November 2005



This achievement standard involves differentiating functions and using derivatives to solve
problems.

              Achievement Criteria             Explanatory Notes
               Differentiate functions         Types of functions will be selected from:
                 and use derivatives to            power
                 solve problems.                   exponential (base e only)
                                                   logarithmic (base e only)
                                                   trigonometric (including reciprocal functions).
                                                Differentiation of functions may include the use of the
                                                  chain rule and product and quotient rules for expanded
                                                  polynomials:
                                                  - chain rule with polynomials in expanded form such as
                                                      i        ( x 2  5 x) 7
                                                        ii      3
                                                                   2x  3
Achievement




                                                        iii     7e 2 x
                                                        iv      ln( 2 x  7)
                                                        v       sin 5x
                                                   - product and quotient rules for combinations of
                                                     straightforward functions, at least one of which is in
                                                     expanded polynomial form, such as
                                                        i       x 2 sin x
                                                        ii     (2 x 3  4)e x
                                                                 2x
                                                        iii
                                                                x3
                                                  Problems may include:
                                                    optimisation of a given function
                                                    rates of change which may involve kinematics
                                                    finding equations of normals and tangents
                                                    locating maxima and minima of polynomial
                                                       functions.




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                           Achievement Criteria                Explanatory Notes
                            Demonstrate knowledge              Knowledge, concepts and techniques of differentiation
                              of advanced concepts and            will be selected from the following types:
                              techniques of                        differentiation from first principles of polynomial
                              differentiation and solve              functions of degree 3
                              differentiation problems.            sketching the graph of a derived function from a
                                                                     given graph
                                                                   differentiation of combinations of functions
                                                                     including:
                                                                     i     products, such as (3 x 2  7) 3 (4 x  8) or
                                                                           x 2 sin x
                                                                                                 x
                                                                      ii    quotients, such as
                                                                                              1 x2
Achievement with Merit




                                                                      iii implicit differentiation such as x 2  3 y 2  15
                                                                      iv parametric differentiation for first derivative
                                                                         only
                                                                    identifying features of given graphs involving a
                                                                      selection from:
                                                                      i    limits
                                                                      ii differentiability
                                                                      iii discontinuity
                                                                      iv gradients
                                                                      v concavity
                                                                      vi turning points
                                                                      vii points of inflection
                                                                    sketching graphs to demonstrate knowledge of the
                                                                      above features.
                                                                  Problems may involve:
                                                                    interpretation of features of graph
                                                                    modelling of a situation
                                                                    optimisation
                                                                    related rates of change, involving two directly
                                                                      related rates.
                                Solve more complex               Problems may involve:
                                 differentiation problem(s).        establishing a model
        Achievement with




                                                                    a proof
           Excellence




                                                                    testing the nature of turning points and verifying
                                                                      points of inflection
                                                                    related rates of change involving more that two
                                                                      related rates, eg dh/dt = dh/dθ.dθ/dv.dv/dt
                                                                    the use of higher derivatives including parametric
                                                                      and implicit differentiation techniques.




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General explanatory notes
1   This achievement standard is derived from Mathematics in the New Zealand Curriculum,
    Learning Media, Ministry of Education, 1992:
     achievement objectives p. 86
     suggested learning experiences pp. 25, 27, 29, 87
     sample assessment activities pp. 88–89
     mathematical processes pp. 24, 26, 28.

2   The use of appropriate technology is expected but candidates must be able to demonstrate
    the skill of differentiation.


                                    Quality Assurance
1   Providers and Industry Training Organisations must be accredited by the Qualifications
    Authority before they can register credits from assessment against achievement standards.

2   Accredited providers and Industry Training Organisations assessing against achievement
    standards must engage with the moderation system that applies to those achievement
    standards.

    Accreditation and Moderation Action Plan (AMAP) reference                   0226




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    Online learning activities
    If you have access to the Internet completing these short activities as you go along
    will enhance your learning. Answers to activities 1, 2 and 3 are on the next page.
    Answers to the work on graphs of derived functions are at the end of the study guide.



    Activity 1
    When to do this: Before MX704 on page 12 on finding the gradient of a tangent to
    y  x2 .

    What to do:
       Select Differentiation – Limits and curves, select the function y  x 2 ,
          (x ^ 2), and arrange the screen so you can see the box beneath the graph.
       Move the point Q as far as you can from P and see what happens to the
          gradient of PQ (the secant) as it gets closer and closer to P from either
          direction.

    Questions to answer:
       Why is the gradient of the secant undefined when it becomes a tangent at P?
       Is the gradient of the tangent at P, given as 12, what you would have expected
          from what you observed as Q approaches P from above and below?




    Activity 2
    When to do this: Before MX704 lesson 3 on finding a formula for gradients of
    tangents.

    What to do:
       Select Differentiation – Differentiation from first principles. When you get to
          this interactive activity, use the “helping hand” to explain what you need to
          do to complete the activity.

    Questions to answer:
       What is the gradient of the tangent to y  x 2 where x = 1?
       What is the gradient of the tangent to y  x 2 where x = 2?
       Write down a formula for finding the gradient of a tangent to a curve from
          first principles.



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            Activity 3

            When to do this: After MX704 Exercise 5A on a quicker way of finding f (x) .
                  If you are already proficient at techniques of differentiation from your work
                  in year 12, you can omit this.

            What to do:
               Select Differentiation – Leibnitz notation and polynomial differentiation.
                  There are two click and drag activities to help with differentiating expressions
                  with negative powers and other expressions.

            Questions to answer:
                             1
               If f ( x)  , what is f (x) ?
                             x
               If f ( x)  ( x  3)(x  2) , what is f (x) ?




Answers to Activities 1, 2 and 3

Activity 1
       The gradient of the secant given on the graph when it becomes a tangent at P is undefined
        because the denominator of the fraction that the gradient of the secant is being calculated
        from is zero at P.
              y  y1
         m 2          is undefined when the denominator is zero.
              x 2  x1
       The closer Q gets to P, the closer the gradient of the secant is to 12. So even though the
        gradient of the secant is undefined at P, it is expected that the gradient of the tangent at P
        is 12.

Activity 2
       The gradient of the tangent to y  x 2 when x = 1 is 2
       The gradient of the tangent to y  x 2 when x = 2 is 4
       A formula for finding the gradient of a tangent to a curve from first principles is on page
        18 of MX704. Return to MX704 page 16 to practice using the formula.

Activity 3
        f ( x)  1x 2
        f ( x)  2 x  1 … (expand brackets before differentiating)




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Sketching graphs of derived functions
Complete this work at any time after completing MX707 if you are entering for the external
standard 90635 Calculus. This work is now a requirement of the standard for Merit.




 Activity 4
 If you have access to the Internet, go to Differentiation – The meaning of the derivative and
 work through this ‘click and drag’ activity that explains how the graph of the derived function
 of y  x 2 is obtained.

 Then go to question 1 below.




 Activity 5

 Look at the graphs of y  x 2 and its derived function drawn
 below each other. Using these graphs and your knowledge of
 differentiating functions, which of the following statements do
 you think are correct?

        The derived function of a parabola is a line.
        Stationary points on f (x) (maximum or minimum)
         become x intercepts on f (x) .
        When f (x) is increasing, f (x) is above the x-axis.
        When f (x) is decreasing, f (x) is below the x-axis.




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  Activity 5
  Go to Differentiation – Graphs and derivatives and watch the short slide show that shows what
  happens to points at maximum and minimum values of a function when the gradient function is
  drawn.

  Then go to Question 2 below.




  Activity 7 Above is part of a cubic curve with its derived function (the parabola) drawn as a
                                                                 dy
  dotted line on the same axes. (For this, the vertical axis is the axis). Using these graphs and
                                                                 dx
  your knowledge of differentiating functions, which of the following statements do you think are
  correct?

           The derived function of a cubic is a parabola.
           Stationary points on f (x) (maximum or minimum) become x intercepts on f (x) .
           Points of inflection on f (x) become turning points of f (x) .
           When f (x) is increasing, f (x) is above the x-axis.
           When f (x) is decreasing, f (x) is below the x-axis.



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In questions 1 and 2 all of the bullet points are correct. The information about graphs of derived
functions is summarised below. There are several bullet points added for the sake of completion.
You will be able to verify these for yourself as you complete the following activities.



                          Features of graphs of derived functions

        The derived function of a cubic is a parabola.
        Stationary points on f (x) (maximum or minimum) become x intercepts on f (x) .
        Points of inflection on f (x) become turning points of f (x) .
        Stationary points of inflection on f (x) become turning points on the x-axis of f (x) .
        When f (x) is increasing, f (x) is above the x-axis.
        When f (x) is decreasing, f (x) is below the x-axis.
        Points of inflection on f (x) become turning points of f (x) .
        If there is a discontinuity or spike or abrupt change in slope on f (x) , f (x) is undefined.
        Vertical asymptotes on f (x) stay vertical asymptotes on f (x) .
        Horizontal asymptotes on f (x) stay horizontal asymptotes on . f (x)
        At a spike, an abrupt change in slope or a discontinuity, f (x) is undefined.



A summary of graph shapes will help, too.

Graph of f (x)        Graph of f (x)
quartic               cubic
cubic                 parabola
parabola              line
Line gradient m       Horizontal line through m


More detail about graph shapes, extrapolating from the bullets above –

              Graph of f (x)                                Graph of f (x)
Quartic going up first                            Cubic going down first
Quartic going down first                          Cubic going up first
Cubic going up first                              Parabola going down first
Cubic going down first                            Parabola going up first
Parabola going up first (inverted)                Line going down (negative gradient)
Parabola going down first (cup shaped)            Line going up (positive gradient)
Line with gradient m                              Horizontal line through m
Line with gradient 0                              Line along the x axis




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 Activity 8
 Go to Differentiation – Practice sketching derived functions. There is a lot of ‘click and drag’
 practice here where you will choose pairs of graphs and their derived functions. Practise until
 you feel familiar with most of the features summarised. Remember that you may be asked to
 draw the graph of a derived function from a given graph and not just select an appropriate
 function from a list, so do try to do this as well and use the click and drag function to check your
 answers.




Question 1
For each of the following, sketch the graphs of the derived functions. It is best to sketch the
derived functions below the graph of their functions as in question 1 above. The answers have
them on the same axes, however, as in question 2 above.


(a)                                                  (b)




(c)                                                  (d)




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(e)                                  (f)




(g)




(h)                                                 (i)




(j)


                                             (k)




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Answers to Question 1

                                                                                          dy
The derived functions are dotted on each graph and for these the vertical axis label is      .
                                                                                          dx


(a)                                                  (b)




(c)                                                  (d)




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(e)                                    (f)




(g)




(h)                                                      (i)




                                                   dy
                                     The graph of      e x is the
                                                   dx
                                     same as the graph of y  e x .

(j)

                                             (k)




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