uality Assessment Tasks UNIT 3 n OUTCOMES 1, 2 & 3ghsmethods12.weebly.com/.../2/4/9/6/24960942/practice_application_task.pdf · Page 3 of 33 201 Pulished y AT Pemission fo copying

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Introduction

NAME:

UNIT 3 ■ OUTCOMES 1, 2 & 3VCE® Mathematical Methods

SCHOOL-ASSESSED COURSEWORKQATs VCE Maths Methods SCHOOL-ASSESSED COURSEWORK UNIT 3 – OUTCOMES 1, 2 & 3

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Published by QATs. Permission for copying in purchasing school only.

UNIT 3 OUTCOMES 1, 2 & 3

VCE®MATHEMATICAL METHODS

SCHOOL ASSESSED COURSEWORK

Introduction

Outcome 1

Define and explain key concepts as specified in the content from the areas of study, and apply a range of related mathematical routines and procedures.

Outcome 2

Apply mathematical processes in non-routine contexts, including situations requiring problem-solving, modelling or investigative techniques or approaches, and analyse and discuss these applications of mathematics.

Outcome 3

Select and appropriately use numerical, graphical, symbolic and statistical functionalities of technology to develop mathematical ideas, produce results and carry out analysis in situations requiring problem-solving, modelling or investigative techniques or approaches.

Task

Application Task

This task will be marked out of 150. The task is comprised of 3 consecutive components of equal value and duration.

It will contribute 50 out of the total marks allocated (50) for School Assessed Coursework in Unit 3.

This task should take 360 minutes to complete, although this duration may be altered at your teacher’s discretion.

You can access your bound reference and an approved CAS technology.

Answer all questions in your bound logbook. This question booklet and your logbook will be collected at the end of each session.

Write your name on this question booklet.

NAME: ___________________________________________

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Task

QATs ■ VCE® Mathematical Methods ■ SCHOOL-ASSESSED COURSEWORK, UNIT 3, OUTCOMES 1, 2 & 3

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QATs VCE®Maths Methods SCHOOL-ASSESSED COURSEWORK UNIT 3 – OUTCOMES 1, 2 & 3

©2016 Ser3MMETHU3A

Published by QATs. Permission for copying in purchasing school only. 2

Task

Background Theory

This Application Task will use simple geometric transformations of tangent lines to investigate the area below a family of curves and the axis without recourse to the Fundamental Theorem of Calculus. Archimedes of Syracuse, investigated the simplest of these problems (the area of a parabolic segment) over ago.

To get started, you must acquaint yourself with some specific terms related to tangent lines. These are defined in the diagram below.

Figure 1 The is the portion of the tangent line between the point of tangency and the axis intercept . The is the segment between the axis intercept and the abscissa ( ), at the point of tangency .


Task



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COMPONENT 1 Time: 120 minutes

In this component you will construct a family of tangent-segments (the tangent segment family) to the function , and then transform the members of this family into a related family spanning an enlarged area.

1. Find the equation of the tangent line to the curve with equation at 2 marks

2. Find the coordinates of the intercept (point in Figure 1), of the tangent from the previous question. 1 mark

3. Hence, find the length of the sub-tangent ( in Figure 1), for the tangent to the curve with equation at 1 mark

4. Define the tangent-segment function ( in Figure 1), to the curve with equation at 2 marks

5. Plot the tangent-segment function from the previous question on Graph 1 (page 7), showing the coordinates of any relevant endpoint(s), correct to (2 d.p.). 2 marks

6. Show that the tangent segment function to the curve with equation at with is given by . 2 marks

7. Complete a table in your logbook, like the one shown below. Then plot the lines with equation on the extra sheet provided (Graph 1). Your graphs must show the coordinates of any relevant endpoint(s) for

. 6 marks

Coordinates of left endpoint

Coordinates of right endpoint


Task


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A transformation is given by the rule

8. Find the image of the point after transformation by 2 marks


10. The function is the line segment joining points and . Find the rule and domain of 3 marks

11. Compare the length of the segment with the length of the segment . 3 marks

12. Hence, provide a geometric description of the transformation given by Justify your answer. 3 marks

Figure 2 The tangent segment family of and its enlargement under the transformation by


Task



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The in the Figure 2 is four times the area of the tangent segment family by the formula, that is

(1)

13. Where does the enlargement factor of in equation 1 come from? 2 marks

14. Hence, show that the area of the tangent segment family is given by the formula

for a suitable choice of the integer 3 marks

15. Express the area under the curve and the axis for as a fraction of the circumscribing rectangle . No credit will be given for the use of integration. 3 marks

In the remainder of this task, . The tangent segment family of and its enlargement is shown in Figure 3, below.

Figure 3 The tangent segment family of and its enlargement under the transformation by


Task


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16. Find the equation of the tangent line to the function at 2 marks

17. Hence, find the coordinates of the point in Figure 3. 2 marks

18. Define the tangent segment function to the function at 2 marks


19. Find the image and of the point and the point respectively, after transformation by 2 marks

20. Show that

in Figure 3, for a suitable choice of the integer 2 marks

21. Hence, express the area under the curve and the axis for (that is, region ) as a fraction of the circumscribing rectangle in Figure 3. No credit will be given for the use of integration. 3 marks

Total: 50 marks

End of Component 1


Task


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Ser1MMETHU3A

QATs VCE Maths Methods SCHOOL-ASSESSED COURSEWORK UNIT 3 – OUTCOMES 1, 2 & 3

©2016 Ser3MMETHU3A

Published by QATs. Permission for copying in purchasing school only.


In this component you will construct a family of tangent-segments (the tangent segment family) to the curve with equation , and then transform the members of this family into a related family spanning an enlarged area.

Figure 4 The tangent segment family and its enlargement, .

1. Find the equation of the tangent line to the curve with equation at . 2 marks

2. Verify that the coordinates of the intercept (point in Figure 4), of the tangent from the previous question are given by . 2 marks

3. Define the tangent-segment function to the curve with equation at . 2 marks

4. Show that the tangent segment function to the curve with equation at is given by . 4 marks


Task



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5. Hence, define and sketch the tangent segment function to the curve with equation at . Be sure to include the coordinates of any relevant points along with the relevant portion of on your sketch. 4 marks

6. Complete a table in your logbook, like the one below, showing the coordinates of the endpoints of the tangent segment from question 4 for . (Express endpoints in terms of ) 6 marks

Coordinates of left endpoint Coordinates of right endpoint




9. Define the line segment function joining points and . 3 marks

10. Use your CAS technology to compare the length of the segment with the length of the segment . 3 marks

11. Hence, provide a geometric description of the transformation given by Justify your answer. 3 marks

12. Show that in the Figure 4 is related to the area of the tangent segment family by the formula

for a suitable choice of the integer 2 marks

13. Hence, show that the area of the tangent segment family is given by the formula

6 marks


Task


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14. Express the area under the curve as a fraction of the circumscribing rectangle . No credit will be given for the use of integration. 5 marks

15. Hence, find the area bounded by the curve , the axis and the line as a function of and (see Figure 5). No credit will be given for the use of integration. 2 marks

Figure 5 The area

16. Use your CAS technology to verify your answer to the previous question. 2 marks

Total: 50 marks

End of Component 2


Task



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In component 2, transformations of the plane were used to express the area between the curve with equation (where ) and the axis (see shaded area in Figure 6 below) as the fraction

of the area of the rectangle circumscribing this region.

Figure 6

In this final component, a different transformation of the plane will be used to express the area between the curve with equation (where , ) and the axis, as a fraction of the relevant rectangle circumscribing this region.

1. In component 2 the tangent segment at on the curve with equation

(where was expressed as with the rule and the domain

Hence, find the tangent segment function to the curve with equation at . 3 marks

2. Find the line segment with equation which is the image of the tangent-segment at after the transformation

(that is, after a translation of units in the negative direction, parallel to the axis). 4 marks

3. Show that

. Hence, define and describe a dilation transformation that is

equivalent to the translation transformation . 3 marks


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4. Hence, find the equation which is the image of the equation after the transformation

5 marks

5. Show that the image of the tangent-segment at after the transformation

is given by . 2 marks

6. Complete a table like the one below in your logbook, showing the coordinates of any relevant endpoint(s) for

. Express your answers in terms of . 6 marks

End points

Figure 7 The definition of Region 1, Region 2 and Region 3 for

7. Find the area of the bounding rectangle in Figure 7. 2 marks

8. Show that . No marks will be given for using integration. 1 mark

9. Find the coordinates of point in Figure 7. 2 marks 10. Find the area of the bounding rectangle in Figure 7. 2 marks

11. Use an argument based upon transformations to show that

for a suitable value of the integer


Task



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No marks will be given for using integration. 4 marks

12. Hence, show that . No marks will be given for using integration. 4 marks

13. Express the area of as a fraction of the area of the bounding rectangle in Figure 7. 2 marks

Figure 8 The definition of Region 3 for

14. It can be shown that the area of Region 3 is given by . Find the rule

1 mark

15. Use your CAS technology to write

with in the form

where are first order polynomial expression of 3 marks

16. Hence, express

as a fraction of the bounding box 3 marks

17. Hence, express

as a function of and 1 mark

18. Use your CAS technology to calculate 183

0

.y dy Hence, verify the correctness to your answer

to the previous question. 2 marks

Total: 50 marks

End of Component 3



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TeacherAdvice


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Teacher Advice

Key knowledge and key skills

The following key knowledge is the focus of this task:

graphs and identification of key features of graphs of the following functions: o power functions,

transformation from to , where and , and f is one of the functions specified above, and the inverse transformation

the relation between the graph of an original function and the graph of a corresponding transformed function (including families of transformed functions for a single transformation parameter)

composition of functions, where composition is defined by given (the notation may be used, but is not required)

solution of literal equations and general solution of equations involving a single parameter review of average and instantaneous rates of change, tangents to the graph of a given

function and the derivative function the key features and properties of a function or relation and its graph and of families of

functions and relations and their graphs, including o the effect of transformations on the graphs of a function or relation o the matrix representation of points and transformations of the plane

the concepts of domain, maximal domain, range and asymptotic behaviour of functions functional relations that describe properties, symmetry and equivalence inferences from analysis and their use to draw valid conclusions related to a given context. domain and range requirements for specification of graphs of functions and relations, when

using technology the role of parameters in specifying general forms of functions and equations similarities and differences between formal mathematical expressions and their

representation by technology the selection of an appropriate functionality of technology in a variety of mathematical

contexts. The following key skills are the focus of this task:

describe the effect of transformations on the graphs of a function or relation apply matrices to transformations of functions and their graphs find the rule of a composite function and give its domain and range communicate conclusions using both mathematical expression and everyday language, in

particular, the interpretation of mathematics with respect to the context. distinguish between exact and approximate presentations of mathematical results produced

by technology, and interpret these results to a specified degree of accuracy use technology to carry out numerical, graphical and symbolic computation as applicable produce results using a technology which identify examples or counter-examples for

propositions produce tables of values, families of graphs and collections of other results using

technology, which support general analysis in problem-solving, investigative and modelling contexts



TeacherAdvice


©2016 Ser3MMETHU3A


use appropriate domain and range specifications to illustrate key features of graphs of functions and relations.

Extract from VCAA VCE®Mathematics (2015-2018) Study Design. © VCAA. Used with permission.

The copyright in this material is owned by the Victorian Curriculum and Assessment Authority (VCAA). Used with permission. The VCAA does not endorse this publication and makes no warranties regarding the correctness or accuracy of its content. To the extent permitted by law, the VCAA excludes all liability for any loss or damage suffered or incurred as a result of accessing, using or relying on the content.

NOTE: This task is sold on condition that it is NOT placed on any school network or social media site (such as Facebook, Wikispaces etc.) at any time. NOT FOR USE BY PRIVATE TUTORS.



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TeacherAdvice


©2016 Ser3MMETHU3A


This task contributes 50 of the 50 marks for School Assessed Coursework in Unit 3.

The coursework scores for this task are:

Outcome 1 Outcome 2 Outcome 3 Total

15 marks 20 marks 15 marks 50 marks

Reproduced by permission from the VCE®Mathematics Assessment Handbook (2015-2018) © VCAA.

Selected content from the VCE®Study Designs and VCE®Advice for Teachers is owned by the Victorian Curriculum and Assessment Authority (VCAA) and reproduced with permission.

The VCAA does not endorse this publication and makes no warranties regarding the correctness or accuracy of its content. To the extent permitted by law, the VCAA excludes all liability for any loss or damage suffered or incurred as a result of accessing, using or relying on the content.

Assessment planning

The task should be conducted in Term 1 after the background material on functions and derivatives has been taught thoroughly, but before the study of integration.

This is a rigorous task in both its length and the mathematical understanding it requires. Teachers should use the questions in the task and their particular requirements as the basis of teaching and learning in the lead up time to its commencement.

The task would ideally be given in 3 x 120 minute blocks or modified at teacher discretion and given session times in the school. Students should complete the graphs in Component 1 on a separate sheet (Graph 1) but answer all remaining questions in their bound logbook.

This question booklet, the student logbook, and Graph 1 should be collected at the end of each session.



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Solution Pathway

Below are suggested responses. Teachers should consider the merits of alternative responses.

Component 1

1 , so equation of the tangent line is

(M1) Reasonable attempt to sub into point slope formula (A1) All Correct

O1 O3

2

(A1) All Correct

O3

3

(A1) All Correct O1

4 (A1) Rule (A1) Domain

O1 O1

5 Line segment joining (C1) End points stated on graph (A1) Shape

O3 O3

6 Rearranging

(M1) Attempt to sub ( (M1) Attempt to solve for

O2 O2



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7

(A1) (C1) Plot (A1) (C1) Plot (A1) (C1) Plot

O3 O3 O3 O3 O3 O3



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8

(M1) Reasonable attempt to apply mapping (with or without matrix algebra) (A1)

O3 O1

9

(M1) Evidence of attempt to transform (with or without matrix algebra) (A1)

O3 O1

10 The line segment joining and is (M1) Or equivalent O2



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SolutionPathway


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is the image of under the said transformation

method (A1)Rule (A1) Domain required

O2 O2

11

is twice as long as

(M1) Reasonable attempt (M1) Reasonable attempt (A1)

O1 O1 O2

12

The line segment is collinear to but is twice as long, with the location of point invariant under the transformation. is uniform dilation from of scale factor

(R1) Or any equivalent reason (A1) Dilation with scale factor (A1) Coordinates of centre of the dilation

O2 O2 O2

13

is a uniform dilation from of scale factor , so the is magnified by a factor of

(R1) Or any equivalent reason (A1)

O2 O1

14

(A1) (A1) (A1)

O2 O2 O2

15

(M1) Seen or used (M1) Seen or used (A1)

O2 O2 O2

16

, so equation of the tangent line is

(M1) Seen or used (A1)

O1 O1



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17

Point is the of the tangent line from the previous question.


O1 O3

18

(A1) (A1)

O1 O1

19

(A1) (A1)

O3 O3

20

is a uniform dilation of from , so area is magnified by

M1 Seen or used A1

O2 O1

21

(M1) Seen or used (M1) Seen or used (A1) No marks should be awarded without suitable method

O2 O2 O2



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Component 2

1 , so equation of the tangent line is (or )

(M1) Reasonable attempt to sub into point slope formula (A1) All Correct

O1 O3

2

(M1) Explicit statement of correct linear equation (M1) Attempt to sub

O1 O2

3 (A1) Rule

(A1) Domain O1O1

4 Collecting like terms

(M1) Attempt to substitute ( in point-slope formula (A1) (M1) Explicit statement of linear equation (A1)

O1 O2 O2 O2

5 A1 (Rule) A1 (Domain) A1 (Shape) A1 (End points)

O3 O3 O1 O1



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6

(A1) Left End (A1) Right End (A1) Left End (A1) Right End (A1) Left End (A1) Right End

O3 O3 O3 O3 O3 O3

7

(M1) Reasonable attempt to apply mapping (with or without matrix algebra) (A1)

O1 O3

8

(M1) Evidence of attempt to transform (with or without matrix algebra) (A1)

O1 O3

Coordinates of left

endpoint

Coordinates of right endpoint



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9 The line segment joining and is

is the image of under the said transformation

(M1) Or equivalent method (A1)Rule (A1) Domain required

O2 O2 O2

10

2 2

2 2 22 2 2

2

1 0

1or since 0

n

nn

nST a a an

n a a n a a nnn

22

2 2 2

0 1n n

n

S T a a n a

n a a

is times as long as

(A1) (A1) (A1)

O3 O3 O1

11

The line segment is parallel to but is times as long, with the location of point invariant under the transformation. is a uniform dilation from of scale factor

(R1) Or any equivalent reason (A1) Dilation with scale factor (accept enlargement, but must have scale factor also) (A1) Coordinates of centre of the dilation

O2 O2 O2



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12

is a uniform dilation from of scale factor , so the is magnified by a factor of

(R1) Or any equivalent reason (A1)

O2 O1

13

with scale factor

2

22

1area of area Δ1

1 1 area Δ1

OST OSSn

n ASTn

but

(M1) Seen or used (A1) (M1) Seen or used (A1) (M1) Evidence of substitution for

O2 O2 O2 O2 O2 O1

14

(M1) Seen or used (M1) Seen or used (M1) Seen or used (M1) Seen or used (A1)

O1 O2 O2 O2 O2

15

(M1) Needed (A1) Seen or used

O1 O1



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16

0

1

area of ,

11

,

an

n

OAT x n N

a n Nn

dx

(M1) needed (A1) integral needed

O3 O3

Component 3

1

Substitute in with

gives

(M1) Reasonable attempt to sub (A1) All correct (A1) All correct

O2 O2 O2

2

(M1) (A1) (M1) Seen or used (A1) (No need for primes on either answer)

O2 O2 O2 O2



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3

Dilation by a factor of

from the axis

(M1) Reasonable attempt to combine over a common denominator (A1) Scale factor (A1) Direction Do not accept dilate along an axis

O1 O1 O2

4

(M1) (M1) (A1) (M1) Seen or used (A1) (No need for primes on either answer)

O1 O1 O2O2 O2

5

(M1) Sub

in (M1) (No need for primes on either answer)

O2 O3

6 End points

(A1) Rule (A1) End points (A1) Rule (A1) End points (A1) Rule (A1) End points

O3 O3 O3 O3 O3 O3



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7 (M1)

(A1) O1 O1

8 From component 2,

(M1) Seen or used

O1

9 From Figure 7, it follows that

n n n aan

n x x


O1 O1



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10

(M1) (A1)

O1 O1

11 ( )ny nx is the image of ny x after a dilation by a factor of

1/ n from the y axis.

Consequently, the area between the axis and the curve ny x is multiplied by a factor of 1/ n to give the area

between the axis and the curve ( )ny nx

1area Region 3 area Region 3 area Regio 2( n )n

Solving this last equation for Region 2

area Region 2 1 area Region 3n

(A1) Identify correct dilation transformation (A1) Identify correct effect on area from the axis (M1) Appropriate algebraic statement in terms of areas required (A1)

O2 O2 O1 O3



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12 From Figure 7,

From previous question

Solving for this last equation for Region 3

(M1) Seen or used (M1) Seen or used (A1) (M1) Reasonable attempt to substitute their formulae the for component areas

O2 O2 O3 O1

13 From previous question

(M1) Seen or used (A1) Accept either answer

O2 O2

14

(A1)

O3



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15

11

01 1

11

1

n n na nn

n n

a ay dy

n n

an

1 1

0

11

na nn ay dy

n n

(A1) (A1) A1

O3 O3 O3

16 From question 15

11

0

1 11

nanny dy a

n n

Rearranging

(M1) (M1) (A1)

O2 O1 O2



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17 From question 15,

1 1 1

0 1

nann nny dy a

n

(accept 1 1

1/ 11

nbn

also)

A1

O1

18 Integrating

183

0

y dy using CAS technology gives

183

0

12y dy

From question 15,

1 1 1

018 1 13 3

0

1 / 1

1/ 3 1

1

1 8 12

bn ny

y y

bdyn

d

A1 A1

O3 O3

Documents

uality Assessment Tasks UNIT 3 n OUTCOMES 1, 2 & 3ghsmethods12.weebly.com/.../2/4/9/6/24960942/practice_application_task.pdf · Page 3 of 33 201 Pulished y AT Pemission fo copying