Unit 4

10 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How does an investment pay out a regular income, and when can it last forever?

Model annuities and perpetuities using recurrence relations and analyse the regular payments they support.

Annuities that pay a regular income from a lump sum, the recurrence model, and perpetuities that pay forever, in TCE Mathematics Applications.

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How does a fund grow when we both earn interest and add regular contributions?

Model annuity investments and superannuation where regular contributions are added to a fund that earns compound interest.

Annuity investments and superannuation modelled by a recurrence that adds interest then a regular contribution, with growth of deposits plus interest in TCE Mathematics Applications.

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How does a loan balance fall as repayments are made, and how is interest split out?

Model reducing-balance loans with recurrence relations and amortisation tables.

Reducing-balance loans, recurrence relations, amortisation tables splitting interest from principal, and total interest paid in TCE Mathematics Applications.

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How do we model quantities that grow or decay by a constant factor over time?

Model and analyse linear and geometric growth and decay using recurrence relations and rules.

Linear (arithmetic) versus geometric (exponential) growth and decay, recurrence relations, closed-form rules, and applications in TCE Mathematics Applications.

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How do we connect every site in a network at the lowest total cost?

Find minimum spanning trees of weighted networks using a systematic algorithm.

Trees, spanning trees and minimum spanning trees, with Prim's algorithm applied to the minimum connector problem in TCE Mathematics Applications.

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What is the greatest amount that can flow through a network from source to sink?

Determine the maximum flow through a network using the maximum-flow minimum-cut theorem.

Sources, sinks, edge capacities, cuts, and the maximum-flow minimum-cut theorem for finding the greatest flow through a network in TCE Mathematics Applications.

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How do we assign workers to tasks to minimise total cost or time?

Solve allocation problems using the Hungarian algorithm to find an optimal assignment.

The assignment problem, bipartite matching, and the Hungarian algorithm for optimal allocation of workers to tasks in TCE Mathematics Applications.

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How do we schedule a project and find the shortest time in which it can finish?

Use activity networks, forward and backward scanning, and float to identify the critical path of a project.

Activity networks, earliest and latest start times, float, and the critical path that fixes the minimum project completion time in TCE Mathematics Applications.

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How do we describe and model the relationship between two numerical variables?

Analyse bivariate data using scatterplots, correlation, and least-squares regression lines.

Scatterplots, correlation coefficient, the coefficient of determination, and least-squares regression for prediction in TCE Mathematics Applications.

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How do we describe patterns in data over time and forecast future values?

Analyse time series using smoothing and trend lines, then forecast future values.

Trend, seasonal and irregular components, moving-average smoothing, seasonal indices, and trend-line forecasting in TCE Mathematics Applications.

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