Calculus through Data & Modeling: Precalculus Review

Go to Course: https://www.coursera.org/learn/calculus-through-data-and-modelling-precalculus-review

Introduction

### Course Review: Calculus through Data & Modeling: Precalculus Review on Coursera Are you eager to sharpen your mathematical skills before diving into the complexities of calculus? "Calculus through Data & Modeling: Precalculus Review," offered on Coursera, presents an excellent opportunity for students to bolster their foundational knowledge in mathematics. #### Course Overview This course is designed to provide an application-oriented and investigative approach to essential mathematical concepts that are vital for success in both single and multivariable calculus. At the heart of the curriculum lies the in-depth study of various functions – including polynomial, rational, exponential, logarithmic, and trigonometric functions. These topics are not merely abstract; the course emphasizes their usefulness in modeling and analyzing real-world data, ensuring that learners can make connections between mathematics and its practical applications. Equipped with either a graphing calculator or computer software, participants will get hands-on experience that enhances their learning and connection to the material. #### Detailed Syllabus Review **1. Exponential and Logarithmic Functions** This module provides a thorough review of fundamental concepts, focusing on how these functions are used to model natural phenomena. You’ll learn how exponential functions can describe relationships where a consistent change in one variable leads to proportional changes in another. Real-world applications in areas such as population growth, finance, and various scientific fields are examined, making the learning process both relevant and engaging. **2. Trigonometric Functions** Incorporating an exploration of cyclic behaviors through trigonometric functions, this module illustrates their significant role in various periodic phenomena, from ocean waves to heartbeats. Students will learn to apply and construct functions that model these oscillatory behaviors, which is crucial for understanding more complex calculus concepts. **3. Vectors in Space** Delving into the world of vectors, the course takes you through classical Euclidean geometry, highlighting their importance in representing physical quantities with both magnitude and direction. This module not only covers the theoretical aspects of vectors but also their applications in physics, making the content universally applicable across scientific disciplines. **4. Equations of Lines and Planes** Building on multidimensional analytic geometry, students will learn to apply vectors to craft algebraic equations describing lines and planes. This unit emphasizes visualization and manipulation of these equations, laying a robust foundation for future studies in multivariable calculus. **5. Precalculus Review Final Exam** To wrap up the course, a comprehensive final exam is provided, allowing students to identify their strengths and weaknesses. This is an excellent opportunity for self-assessment, guiding learners on where they might need additional resources or practice. #### Course Recommendations **Who Should Take This Course?** This course is particularly suited for students who wish to solidify their precalculus knowledge before advancing to calculus. Whether you are transitioning from high school mathematics, returning to education after a break, or simply looking to refresh your skills, this course is designed with you in mind. **Pros:** - Applications-oriented learning connects mathematics to real-world scenarios. - Strong emphasis on functions equips students with the tools needed for calculus. - Interactive components (use of graphing calculators and software) enhance comprehension. - A well-structured syllabus focused on critical mathematical concepts. **Cons:** - The pace may be quick for those without any background in mathematics. - Limited peer interaction compared to in-person courses might affect networking opportunities. In conclusion, "Calculus through Data & Modeling: Precalculus Review" on Coursera stands out as a vital stepping stone for anyone serious about pursuing calculus. By combining theoretical knowledge with practical applications, this course effectively prepares students for the challenges ahead in their mathematical journey. I highly recommend enrolling in this course to build a strong foundation in precalculus that will contribute to your success in calculus and beyond.

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.

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.

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.

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.

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.