This course is an applications-oriented, investigative approach to the study of the mathematical topics needed for further coursework in single and multivariable calculus. The unifying theme is the study of functions, including polynomial, rational, exponential, logarithmic, and trigonometric functions. An emphasis is placed on using these functions to model and analyze data. Graphing calculators and/or the computer will be used as an integral part of the course.
Overview
Syllabus
- Exponential and Logarithmic Functions
- In this module, we will review some of the key concepts from Precalculus. Exponential and logarithmic functions arise often when modeling natural phenomena, and are important to Calculus. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable. This occurs widely in the natural and social sciences, as in a self-reproducing population, a fund accruing compound interest, or a growing body of manufacturing expertise. Thus, the exponential function also appears in a variety of contexts within physics, chemistry, engineering, mathematical biology, and economics.
- Trigonometric Functions
- Equally important are the trigonometric functions, some of the most well-known examples of periodic or cyclic functions. Common phenomena have an oscillatory, or periodic, behavior. This is observed through ocean waves, sound waves, or even the regular beating of your heart. All these phenomena can be modeled using equations based on the familiar sine and cosine functions. In this module, we will see how to apply and construct functions that permit us to model cyclic behavior.
- Vectors in Space
- In classical Euclidean geometry, vectors are an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment (A, B)) and same direction (e.g., the direction from A to B). Vectors are used both in abstract sense as well as for applications, particularly in physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars, which have no direction. For example, velocity, forces and acceleration are represented by vectors. In this module, we will study vectors specifically in the xy-plane and in "3D" space.
- Equations of Lines and Planes
- Continuing our study of multi-dimensional analytic geometry, vectors are now applied to create algebraic equations to describe common objects like lines and planes in space. This module will test your ability to visualize algebraic equations and to create movement and thus control of these objects in space by performing algebraic manipulations. This will create a solid foundation for our study of multivariable calculus on these higher dimensional objects.
- Precalculus Review Final Exam
- The assessment below will help to identify strengths as weaknesses in your foundational material in order to be successful in single and multivariable differentiable calculus. Use the assessment below as a guide as to where to follow up and seek out more resources and examples.
Taught by
Joseph W. Cutrone, PhD
