VCAA Bulletin. VCE, VCAL and VET. Supplement 2

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1 VCE, VCAL and VET VCAA Bulletin Regulations and information about curriculum and assessment for the VCE, VCAL and VET No. 87 April 20 Victorian Certificate of Education Victorian Certificate of Applied Learning Vocational Education and Training Supplement 2 Mathematical Methods (CAS) stud advice This advice has been developed in response to queries from teachers during the first ear of full cohort implementation of Mathematical Methods (CAS) Units 3 and 4 in 200, and the VCAA metropolitan and regional workshops for first time teachers of Mathematical Methods (CAS) Units 3 and 4 conducted in 200. Strictl increasing and strictl decreasing functions Students should be familiar with the notions of strictl increasing and strictl decreasing for functions of a single real variable, and be able to identif intervals over which such a function is strictl increasing or strictl decreasing. An eample relating to the interval over which a simple polnomial function is strictl decreasing was included as Question 4 of the sample questions for Mathematical Methods (CAS) Eamination, published on the VCAA website in earl 200: publications/mmcas-sampfin-w.pdf Strictl increasing A function f is said to be strictl increasing when a < b implies f(a) < f(b) for all a and b in its domain. This definition does not require f to be differentiable, or to have a non-zero derivative, for all elements of the domain. If a function is strictl increasing, then it is a one-to-one function and has an inverse function that is also strictl increasing. This publication is also available online at VCAA 20

2 Eample The function f : R R, f () = 3 is strictl increasing with zero gradient at the origin, as illustrated in Figure : Figure : Part of the graph of f 3 The inverse function f : R R, f( ) = is also strictl increasing, with a vertical tangent of undefined gradient at the origin. Eample 2 The hbrid function g with domain [0, ) and rule: 2 g ( ) = > 2 is strictl increasing, and is not differentiable at = 2, as illustrated in Figure 2: Figure 2: Part of the graph of g 2 VCAA BULLETIN VCE, VCAL and VET SUPPLEMENT 2 APRIL 20

3 A function is said to be strictl increasing over an interval when a < b implies f(a) < f(b) for all a and b in the interval. Thus, the function h : R R, h() = 3 is not strictl increasing, as illustrated in Figure 3: Figure 3: Part of the graph of h However, the function is strictl increasing over the interval 0, 3, as illustrated in Figure 4: Figure 4: Interval over which h is strictl increasing Strictl decreasing A function f is said to be strictl decreasing when a < b implies f(a) > f(b) for all a and b in its domain. A function is said to be strictl decreasing over an interval when a < b implies f(a) > f(b) for all a and b in the interval. Eample 3 The function f : R R, f( ) = e + is strictl decreasing over R, as illustrated in Figure 5: Figure 5: Part of the graph of f VCAA BULLETIN VCE, VCAL and VET SUPPLEMENT 2 APRIL 20 3

4 Eample 4 The function g : R R, g() = cos() is not strictl decreasing as illustrated in Figure 6: Figure 6: Part of the graph of g However, g is strictl decreasing over the interval [0, p] as illustrated in Figure 7: Figure 7: The graph of g on the interval [0, p] If a function with rule = f () is strictl increasing over an interval then the function with rule = -f() will be strictl decreasing on the same interval. For eample = sin() is strictl increasing on the interval π π, 2 2 and = -sin() is strictl decreasing over this interval. Maimum and minimum problems in modelling contets Functions are used to model various situations in theoretical and practical contets, and to carr out related analsis including the identification of local and global maimum and minimum values. In such contets the domain and graph of a function are important aspects of analsis. In some cases the analsis can be simplified using a related function. A differentiable function ma or ma not have a maimum value or minimum value over a given interval, and an maimum value or minimum value, where this eists, can be identified b considering zeros of the derivative or endpoints. Depending on whether eact or approimate answers are required, analtical, graphical or numerical methods can be emploed, as applicable, to determine the required values. 4 VCAA BULLETIN VCE, VCAL and VET SUPPLEMENT 2 APRIL 20

5 Eample 5 Let d be the function that models the distance from the origin to an point on the graph of = 4 where > 0, as illustrated in Figure 8: Figure 8: Part of the graph of = 4 showing a line segment from the origin to a point on the curve Consideration of the geometr of the situation indicates that d will have a minimum value but no maimum value. This function can be defined eplicitl d: R R, d ( ) = + 2. The minimum value for the distance will occur when the 32 minimum value of the square of the distance also occurs. That is when 2 = 0 for > 0, which gives d 3 min = 8 when = 2. In measurement contets where distance or length is involved, the modelling function is often of the form f( ) = u ( ) uʹ( ) and its derivative is of the form f ʹ( ) = with zeros occurring when u () = 0 and u() > 0. Question 2 from Mathematical 2 u ( ) Methods (CAS) Eamination in 2007 and Question 3 from Section 2 of Mathematical Methods (CAS) Eamination 2 in 200 are related tpes of problems. In the latter case the modelling function for volume has a single maimum value which is a local maimum, but no minimum value as shown in Figure 9: T π π Figure 9: The graph of T :, cos () 2cos ( ) 4 2 R, T() = 3 VCAA BULLETIN VCE, VCAL and VET SUPPLEMENT 2 APRIL 20 5

6 Eample 6 Let A be the function that models the total enclosed area when a 00cm piece of wire is cut into two pieces, where one piece is used to form the perimeter of a square, and the other piece is used to form the circumference of a circle. Consideration of the geometr of the situation indicates that A will have both a minimum value and a maimum value. This function can be defined 2 2 ( 00 ) eplicitl and is readil recognisable as a quadratic function A: [ 0, 00] R, A ( ) = + where cm is the length of 6 4π the piece of wire used to form the perimeter of the square. Figure 0 shows that the local minimum corresponds to the minimum value of A, and the left endpoint of the domain at = 0 corresponds to the maimum value of A: A ( 00 ) Figure 0: The graph of A: [ 0, 00] R, A ( ) = + 6 4π Question from Section 2 of Mathematical Methods (CAS) Eamination 2 in 2007 and Question 3 from Section 2 of Mathematical Methods(CAS) Eamination 2 in 2008 are related tpes of problems. Eample 7 The function f : R R, f( ) = in Eample 3 has no maimum value and no minimum value. The graph of f is asmptotic e + to = from below as - and asmptotic to = 0 from above as. 6 VCAA BULLETIN VCE, VCAL and VET SUPPLEMENT 2 APRIL 20

7 Student responses to tasks involving questions such as these (or variations and generalisations) where problemsolving, modelling or investigative techniques or approaches are required, provide them with the opportunit to demonstrate achievement of aspects of all three outcomes. In particular, where CAS technolog is used, aspects of Outcome 3 can be addressed. As stated on page 35 of the stud design, for Outcome 3 students should be to be able to: select and appropriatel use a computer algebra sstem and other technolog to develop mathematical ideas, produce results and carr out analsis in situations requiring problemsolving, modelling or investigative techniques or approaches. To achieve this outcome, students will draw on knowledge and skills outlined in all the areas of stud. This includes ke knowledge and ke skills as specified on pages of the stud design: Ke knowledge This knowledge includes: eact and approimate specification of mathematical information such as numerical data, graphical forms and general or specific forms of solutions of equations produced b use of a computer algebra sstem; domain and range requirements for a computer algebra sstem s specification of graphs of functions and relations; the role of parameters in specifing general forms of functions and equations; the relation between numerical, graphical and smbolic forms of information about functions and equations and the corresponding features of those functions or equations; the similarities and differences between formal mathematical epressions and their computer algebra sstem representation; the appropriate selection of a technolog application, in particular, computer algebra sstems, in a variet of mathematical contets. Ke skills These skills include the abilit to: distinguish between eact and approimate presentations of mathematical results produced b a computer algebra sstem, and interpret these results to a specified degree of accurac; produce results using a computer algebra sstem which identif eamples or counter-eamples for propositions; produce tables of values, families of graphs or collections of other results using a computer algebra sstem, which support general analsis in problem-solving, investigative or modelling contets; use appropriate domain and range specifications to illustrate ke features of graphs of functions and relations; identif the relation between numerical, graphical and smbolic forms of information about functions and equations and the corresponding features of those functions or equations; specif the similarities and differences between formal mathematical epressions and their computer algebra sstem representation, in particular, equivalent forms of smbolic epressions; make appropriate selections for a computer algebra sstem and other technolog applications in a variet of mathematical contets, and provide a rationale for these selections; relate the results from a particular application to the nature of a particular mathematical task (investigative, problem solving or modelling) and verif these results; specif the process used to develop a solution to a problem using a computer algebra sstem, and communicate the ke stages of mathematical reasoning (formulation, solution, interpretation) used in this process. VCAA BULLETIN VCE, VCAL and VET SUPPLEMENT 2 APRIL 20 7

VCE Mathematical Methods (CAS)

VCE Mathematical Methods (CAS) Written eamination 1 from 2010 (inclusive) From 2010 the Mathematical Methods (CAS) eamination 1 will be based on content from the areas of study for Mathematical Methods