Assessment Type 2 Mathematical Investigation

Stage
2 Mathematical Methods

Assessment
Type 2: Mathematical Investigation

Topic
1: Further Differentiation and Applications

Surge
and Logistic Models

Part
1: The Surge Function

A
surge function is in the form where A
and b
are positive constants.

• On
  the same axes, graph and for the case where and
• Determine
  the coordinates of the stationary point and point of inflection and
  label these on the graph.

• Repeat
  the investigation for three different values of while maintaining
  .
• Include
  your graphs in the report and summarise the findings in a suitable
  table.
• State
  the effect of changing the value of on the graph of .

• Using
  a similar process investigate the effect of changing the value of
  on the graph of .
• Make
  a conjecture on how the value of b effects the x-coordinates of the
  stationary point and the point of inflection of the graph of .
• Prove
  your conjecture.

• Comment
  on the suitability of the surge function in modelling medicinal
  doses by relating the features of the graph to the effect that a
  medicinal dose has on the body.

Discuss
any limitations of the model.

At
least four key points should be made.

Part
2: The Logistic Function

A
logistic function is in the form where and are constants and the
independent variable t is usually time; .

• Investigate
  the effect that the values of and have on the graph of the
  logistics function.
• Discuss
  your findings on the logistic model.

• Relate
  the specific features of the logistic graph to a limited growth
  model.

At
least three key points should be made.

Part
3: Modelling using Surge and Logistic Functions

Using
either a surge or a logistic function (or both) develop a model to
investigate one of the following scenarios.

• Movements
  of students into the school building at the end of lunch.
• A
  crowd leaving a sports venue.
• The
  limited growth of a population.
• pH
  levels during an acid-base titration.
• Repeat
  doses of a medicine.
• The
  spread of information in a group of people.
• Traffic
  density during peak hour.
• The
  acceleration of a car.
• A
  suitable alternative of your choosing.

Select
a suitable function that would model your chosen scenario with the
dependent and independent variables clearly defined.

• State
  the values of any constants for this model with evidence to support
  your choices.
• Draw
  a sketch of the graph of the function showing as much detail as
  known.
• Discuss
  the significance of the key features of the graph including the
  reasonableness of the model and of your conclusions.
• Justify
  all your decisions and discuss any limitations of your model.

Investigation
Report

The format of the investigation report
may be written or multimodal.

The report should include the following:

• an
  introduction – an outline of the problem and the context

• the
  results and analysis, including

• relevant
  data and/or information

• mathematical
  calculations and results, using appropriate representations

• the
  analysis and interpretation of results, including consideration of
  the reasonableness and limitations of the results

• a
  conclusion – summary of your findings

A
bibliography and appendices, as appropriate, may be used.

The
investigation report, excluding bibliography and appendices if used,
must be a maximum
of
15
A4 pages
if written, or the equivalent in multimodal form. The maximum page
limit is for single-sided A4 pages with minimum font size 10. Page
reduction, such as 2 A4 pages reduced to fit on 1 A4 page, is not
acceptable. Conclusions, interpretations and/or arguments that are
required for the assessment must be presented in the report, and not
in an appendix. Appendices are used only to support the report, and
do not form part of the assessment decision.

Performance
Standards for Stage 2 Mathematical Methods

–	Concepts and Techniques	Reasoning and Communication
A	Comprehensive knowledge and understanding of concepts and relationships. Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts. Successful development and application of mathematical models to find concise and accurate solutions. Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems.	Comprehensive interpretation of mathematical results in the context of the problem. Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations. Proficient and accurate use of appropriate mathematical notation, representations, and terminology. Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments. Effective development and testing of valid conjectures, with proof.
B	Some depth of knowledge and understanding of concepts and relationships. Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts. Some development and successful application of mathematical models to find mostly accurate solutions. Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems.	Mostly appropriate interpretation of mathematical results in the context of the problem. Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations. Mostly accurate use of appropriate mathematical notation, representations, and terminology. Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Mostly effective development and testing of valid conjectures, with substantial attempt at proof.
C	Generally competent knowledge and understanding of concepts and relationships. Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in a variety of contexts. Successful application of mathematical models to find generally accurate solutions. Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems.	Generally appropriate interpretation of mathematical results in the context of the problem. Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations. Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy. Generally effective communication of mathematical ideas and reasoning to develop some logical arguments. Development and testing of generally valid conjectures, with some attempt at proof.
D	Basic knowledge and some understanding of concepts and relationships. Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in some contexts. Some application of mathematical models to find some accurate or partially accurate solutions. Some appropriate use of electronic technology to find some accurate solutions to routine problems.	Some interpretation of mathematical results. Drawing some conclusions from mathematical results, with some awareness of their reasonableness or limitations. Some appropriate use of mathematical notation, representations, and terminology, with some accuracy. Some communication of mathematical ideas, with attempted reasoning and/or arguments. Attempted development or testing of a reasonable conjecture.
E	Limited knowledge or understanding of concepts and relationships. Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems. Attempted application of mathematical models, with limited accuracy. Attempted use of electronic technology, with limited accuracy in solving routine problems.	Limited interpretation of mathematical results. Limited understanding of the meaning of mathematical results, their reasonableness or limitations. Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy. Attempted communication of mathematical ideas, with limited reasoning. Limited attempt to develop or test a conjecture.

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