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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. MXPDF www.correspondence.school.nz The Correspondence School 1 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) 2 www.correspondence.school.nz The Correspondence School MXPDF 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 MXPDF www.correspondence.school.nz The Correspondence School 3 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. 4 www.correspondence.school.nz The Correspondence School MXPDF 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 x3 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. MXPDF www.correspondence.school.nz The Correspondence School 5 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. 6 www.correspondence.school.nz The Correspondence School MXPDF 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 MXPDF www.correspondence.school.nz The Correspondence School 7 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. 8 www.correspondence.school.nz The Correspondence School MXPDF 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) MXPDF www.correspondence.school.nz The Correspondence School 9 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. 10 www.correspondence.school.nz The Correspondence School MXPDF 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. MXPDF www.correspondence.school.nz The Correspondence School 11 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 12 www.correspondence.school.nz The Correspondence School MXPDF 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) MXPDF www.correspondence.school.nz The Correspondence School 13 (e) (f) (g) (h) (i) (j) (k) 14 www.correspondence.school.nz The Correspondence School MXPDF 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) MXPDF www.correspondence.school.nz The Correspondence School 15 (e) (f) (g) (h) (i) dy The graph of e x is the dx same as the graph of y e x . (j) (k) 16 www.correspondence.school.nz The Correspondence School MXPDF MXPDF www.correspondence.school.nz The Correspondence School 17

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