Precalculus: Mathematical Modeling

Overview

This course helps to build the foundational material to use mathematics as a tool to model, understand, and interpret the world around us. This is done through studying functions, their properties, and applications to data analysis. Concepts of precalculus provide the set of tools for the beginning student to begin their scientific career, preparing them for future science and calculus courses. This course is designed for all students, not just those interested in further mathematics courses. Students interested in the natural sciences, computer sciences, psychology, sociology, or similar will genuinely benefit from this introductory course, applying the skills learned to their discipline to analyze and interpret their subject material. Students will be presented with not only new ideas, but also new applications of an old subject. Real-life data, exercise sets, and regular assessments help to motivate and reinforce the content in this course, leading to learning and mastery.

Syllabus

Module 1: Linear Modeling

Data is all around us, Trillions of terabytes of data are generated and recorded daily by just using a smartphone, driving a car, or using a credit card. In this course, students examine how data is created, obtained, examined, and used to shape everyday life. To understand an analyze this data, researchers in diverse fields conjecture models based on their data. With more data, they modify and update the model accordingly. The model is then tested to validate results and conclusions. This module will help you to visualize large data sets, present different models for data, and statistics to measure how good the model is to the data. We will also focus on how to communicate the results of the models and common pitfalls to avoid overreaching or overstating your conclusions.

Module 2: Exponential Modeling

Perhaps the most important function of this course and your future courses in calculus, the exponential function is introduced to model many natural phenomena. For example, this function is used to measure population growth, the spread of a disease, and the elimination of drug from the body. Types of interest and present value calculations will require the understanding and use of the exponential functions. We will allow the exponential to have any positive base, but the natural exponential, that with e = 2.718.. will be our main object of study.

Modeling with Other Functions

In this module, we will use other functions to model specific behavior, such as polynomial, periodic and power functions.

Module 4: Dimensional Analysis

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric charge) and units of measure (such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. In this module, we will study dimensional analysis, or more specifically the factor-label method and apply it to measure the strength of an explosion.