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Stage 2 Mathematical Methods

Assessment Type 2: Mathematical Investigation

Surge and Logistic Models

The Surge Function

A surge function is in the form 𝑓(𝑥) = 𝐴𝑥𝑒 −𝑏𝑥 where A and b are positive constants.

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On the same axes, graph 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑓’(𝑥) for the case where 𝑨 = 𝟏𝟎 and 𝒃 = 𝟒

𝑖. 𝑒. 𝑓(𝑥) = 10𝑥𝑒 −4𝑥

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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 𝑨 while maintaining 𝒃 = 𝟒.

Include your graphs in the report and summarise the findings in a suitable table.

State the effect of changing the value of 𝑨 on the graph of 𝑦 = 𝐴𝑥𝑒 −𝑏𝑥 .

Using a similar process investigate the effect of changing the value of 𝒃 on the graph of 𝑦 = 𝐴𝑥𝑒 −𝑏𝑥 .

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 𝑦 = 𝐴𝑥𝑒 −𝑏𝑥 .

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.

The Logistic Function

𝐿

A logistic function is in the form 𝑃(𝑡) = 1+𝐴𝑒 −𝑏𝑡 where 𝑳, 𝑨 and 𝒃 are constants and the independent

variable t is usually time; 𝑡 ≥ 0.

This model is useful in limited growth problems, that is, when the growth cannot go beyond a particular value

for some reason.

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Investigate the effect that the values of 𝐿, 𝐴 and 𝑏 have on the graph of the logistics function.

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Discuss your findings on the logistic model.

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

At least three key points should be made.

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.

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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 in a 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.

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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.

Make further suggestions to refine your model that may or may not use logistic or surge models to

produce a better model (if necessary).

The report should include the following:

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A possible structure that includes an introduction main body and conclusion

Relevant data and/or information

Mathematical calculations and results, using appropriate representations.

Analysis and interpretation of results, including consideration of the reasonableness and limitations

of the results.

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 singlesided 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.

Your investigation will be assessed using the following assessment design criteria

Concepts and Techniques

CT1

CT2

CT3

CT4

Knowledge and understanding of concepts and relationships.

Selection and application of mathematical techniques and algorithms to find solutions to problems in a

variety of contexts.

Application of mathematical models.

Use of electronic technology to find solutions to mathematical problems.

Reasoning and Communication

RC1 Interpretation of mathematical results.

RC2 Drawing conclusions from mathematical results, with an understanding of their reasonableness and

limitations.

RC3 Use of appropriate mathematical notation, representations, and terminology.

RC4 Communication of mathematical ideas and reasoning to develop logical arguments.

RC5 Development, testing, and proof of valid conjectures.

Performance Standards for Stage 2 Mathematical Methods

A

Concepts and Techniques

Reasoning and Communication

Comprehensive knowledge and understanding of concepts

and relationships.

Comprehensive interpretation of mathematical results in the context of

the problem.

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.

Drawing logical conclusions from mathematical results, with a

comprehensive understanding of their reasonableness and limitations.

Successful development and application of mathematical

models to find concise and accurate solutions.

B

Effective development and testing of valid conjectures, with proof.

Some depth of knowledge and understanding of concepts

and relationships.

Mostly appropriate interpretation of mathematical results in the context

of the problem.

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.

Drawing mostly logical conclusions from mathematical results, with

some depth of understanding of their reasonableness and limitations.

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.

Generally competent knowledge and understanding of

concepts and relationships.

Generally appropriate interpretation of mathematical results in the

context of the problem.

Generally effective selection and application of

mathematical techniques and algorithms to find mostly

accurate solutions to routine problems in a variety of

contexts.

Drawing some logical conclusions from mathematical results, with

some understanding of their reasonableness and limitations.

Generally appropriate and effective use of electronic

technology to find mostly accurate solutions to routine

problems.

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.

E

Mostly accurate use of appropriate mathematical notation,

representations, and terminology.

Mostly appropriate and effective use of electronic

technology to find mostly accurate solutions to routine and

some complex problems.

Successful application of mathematical models to find

generally accurate solutions.

D

Highly effective communication of mathematical ideas and reasoning to

develop logical and concise arguments.

Appropriate and effective use of electronic technology to

find accurate solutions to routine and complex problems.

Some development and successful application of

mathematical models to find mostly accurate solutions.

C

Proficient and accurate use of appropriate mathematical notation,

representations, and terminology.

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.

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.

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.

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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Mathematical Models

practical applications

Logistic Functions

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