SURGE AND LOGISTIC MODELS

PART 1
A surge model has an equation of the form where and are positive constants. The independent variable is usually time, .
This model is used extensively in the study of medicinal doses where there is an initial rapid increase to a maximum and then a slower decay to zero.
1 On the same set of axes, graph the following.
i)
ii)
iii)

2 Record any observations from Question 1 above, particularly the effect of Aon the graph of .

3 On the same set of axes, graph for values of and 5.

4 From observation of the graphs in step 3 make a conjecture about how the value of b affects the x-coordinates of the stationary point and point of inflection for these graphs.

5 Prove your conjecture made above.

6   Comment on the significance of this model in relation to the study of medicinal doses by relating the specific features of the graph. State the limitations to such a model
(A minimum of 4 key points should be made)
PART 2
A logistic model has the form    where and C are positive constants. The dependent variable   again is usually time, .
This model is useful in limited growth problems, that is, when growth cannot go beyond a particular value for some reason.
1   On the same set of axes, graph   for values of C= 30, 10 and 5.

2 Comment upon the effect of C on the specific features of the graph.

3 On the same set of axes, graph for values of A = 1, 3, 5 and 10.
Comment on the effect that A has on the specific features of the graph.

4 On the same set of axes, graph for values of = 1, 2, 3 and 10.
Comment on the effect that b has on the specific features of the graph.

5 Discuss your findings on the logistic model.Comment on the significance of this model in relation to limited growth problems by relating the specific features of the graph. State the limitations or assumptions for such a model in a real life context.
(A minimum of 3 key points must be made).

PART 3Investigate one of the following scenarios or develop a scenario of your own that will allow for suitable analysis using either a surge or a logistic model.

For this section write your response as a report which includes an introduction, a main body and a conclusion.
1.   Recently 150 students attended the school formal. One student started spreading a rumour that her best friends had broken up and this news spread quickly throughout the venue.
Choose an appropriate function in general terms that might model this situation, given that Nrepresents the number of studentswho knew about the rumour aftert hours.
•   State the values of any of the constants that could be determined from the information given.
•   Draw a sketch of a graph of the functions showing as much detail as known.
•   Indicate the point where the rumour is spreading at the greatest rate.
Justify all your decisions and discuss any limitations for your model
2.   When students enter the school at the end of the lunch break there is an initial rush of students moving into the building that gradually reduces.

Select a suitable function that would model the movement of students into the main building were N represents the number of students entering per minute after t minutes.

•   State the values of any constants for this model with evidence to justify your choices.
•   Draw a sketch of the graph of the function showing as much detail as known.
•   Indicate the point where the students are entering the building at the greatest rate.
Justify all your decisions and discuss any limitations of your model.

Your report will be Assessed using the following Criteria

Mathematical Knowledge and Skills and Their Application
The specific features are as follows:
MKSA1   Knowledge of content and understanding of mathematical concepts and relationships.
MKSA2   Use of mathematical algorithms and techniques (implemented electronically where appropriate) to find solutions to routine and complex questions.
MKSA3   Application of knowledge and skills to answer questions in applied and theoretical contexts.
Mathematical Modelling and Problem-solving
The specific features are as follows:
MMP1   Application of mathematical models.
MMP2   Development of solutions to mathematical problems set in applied and theoretical contexts.
MMP3   Interpretation of the mathematical results in the context of the problem.
MMP4   Understanding of the reasonableness and possible limitations of the interpreted results, and recognition of assumptions made.
MMP5   Development and testing of conjectures, with some attempt at proof.
Communication of Mathematical Information
The specific features are as follows:
CMI1   Communication of mathematical ideas and reasoning to develop logical arguments.
CMI2   Use of appropriate mathematical notation, representations, and terminology.

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