# Assessment Type 2 Mathematical Investigation

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
• 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

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
• 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

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

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Assessment Type 2 Mathematical Investigation

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