Surge function in modelling medicinal doses

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. • 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 • • • • • • • • 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. • 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. 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 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. • • • • • 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: • • • • 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.