College Applied Math Vocabulary

Understanding College Applied Math Vocabulary

This deck shows how applied mathematics turns real situations into models that can be computed, tested, and improved. The terms come from four connected areas. They include disease spread models, optimization, numerical methods for differential equations, and interpolation from data.

These topics may seem separate at first. In practice, they share one habit of thought. Start with a real system, choose useful variables and assumptions, build a mathematical model, then judge whether its results are trustworthy.

A model is never the whole world. It is a controlled simplification that helps people make decisions.

In epidemic models, compartments organize a population into groups whose status matters for disease spread. Rates control how people move between groups. Transmission rate and recovery rate work together to shape whether infections grow or shrink.

The basic reproduction number describes spread when nearly everyone is susceptible. The effective reproduction number changes over time as immunity, behavior, treatment, or public health actions change. Equilibrium matters because it describes a long term state of the model.

When studying these terms, draw arrows between compartments and explain in words what causes each arrow. Then think about how raising or lowering one rate changes the whole system. This is more useful than memorizing a formula alone.

Optimization terms focus on choosing the best action while obeying limits. Linear programming can represent decisions about production, shipping, staffing, or budgets. The primal problem states the decision directly.

Its dual problem gives another viewpoint, often based on the value of limited resources. Shadow price helps interpret that value. Weak duality and strong duality explain the relationship between the two viewpoints and help verify an answer.

In integer programming, some choices must be whole numbers, such as the number of trucks or workers. LP relaxation temporarily allows fractions to make the problem easier to analyze.

Branching, bounding, pruning, and the incumbent form a search process that finds a valid best integer solution without checking every possibility. Practice by identifying decision variables, constraints, and the objective before doing any tableau or pivot work.

Numerical analysis and interpolation address a different problem. Sometimes a model has no convenient exact solution, or data are only known at selected points. A grid breaks a continuous domain into locations.

A stencil describes which nearby values are used in a calculation. Consistency, stability, truncation error, and the CFL condition help determine whether repeated calculations approach the correct physical behavior instead of producing meaningless results. Interpolation estimates values between known data points.

Lagrange basis polynomials and divided differences build polynomial estimates, while knots and natural cubic splines create smooth curves across many intervals. Study this deck by sorting terms into the four areas, making a small concept map for each area, and solving short examples. For every result, state what it means in the real situation and state which assumptions could make it unreliable.