Assignment Task
Mathematical Investigation
Topic 1: Further Differentiation and Applications
Surge and Logistic Models
Part 1: The Surge Function
A surge function is in the form fx=Axe-bx where A and b are positive constants.
On the same axes, graph y=f(x) and y=fâ(x) for the case where A=____ and b=____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=____.
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.
Part 2: The Logistic Function
A logistic function is in the form Pt=L1+Ae-bt where L, A and b are constants and the independent variable t is usually time; t?0.
Investigate the effect that the values of L, A and b 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 â 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 â summary of your findings
A bibliography and appendices, as appropriate, may be used.
The investigation report, excluding bibliography and appendices if used
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.
-Concepts and TechniquesReasoning and CommunicationAComprehensive 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.
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