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1: Critical Concepts for Calculus

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    116530

    • This page is a draft and is under active development. 

    • Gilbert Strang & Edwin “Jed” Herman
    • OpenStax
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    Calculus is the mathematics that describes changes in functions. In this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functions, and we show the properties of their graphs. We provide examples of equations with terms involving these functions and illustrate the algebraic techniques necessary to solve them. In short, this chapter provides the foundation for the material to come. It is essential to be familiar and comfortable with these ideas before proceeding to the formal introduction of calculus in the next chapter.

    Caution

    Due to time constraints, many professors skip most of the material in this chapter and leave it to you to review on your own. This is not done because your professor is lazy. The reality is that you are expected to have mastery of all topics in elementary algebra (Algebra 1), intermediate algebra (Algebra 2), college algebra, and trigonometry prior to enrolling in a calculus course.

    If you have not had a full course in trigonometry, it is highly recommended that you enroll in one at your college and save calculus for next semester. I cannot emphasize this enough.

    • 1.1: Algebraic Manipulations Critical for Calculus
      We begin our study of calculus by reviewing common algebraic manipulations used throughout this course. We also introduce the Mathematical Mantra.
    • 1.2: Functions
      In this section, we provide a formal definition of a function and examine several ways in which functions are represented—namely, through tables, formulas, and graphs. We study formal notation and terms related to functions. We also define composition of functions and symmetry properties. Most of this material will be a review for you, but it serves as a handy reference to remind you of some of the algebraic techniques useful for working with functions.
    • 1.3: Basic Classes of Functions and Their Graphs
      We begin by reviewing the basic properties of linear and quadratic functions, and then generalize to include higher-degree polynomials. By combining root functions with polynomials, we can define general algebraic functions and distinguish them from the transcendental functions we examine later in this chapter. We finish the section with piecewise-defined functions and take a look at how to sketch the graph of a function that has been shifted, stretched, or reflected from its initial form.
    • 1.4: Inverse Functions
      An inverse function reverses the operation done by a particular function. Whatever a function does, the inverse function undoes it. In this section, we define an inverse function formally and state the necessary conditions for an inverse function to exist. We examine how to find an inverse function and study the relationship between the graph of a function and the graph of its inverse.
    • 1.5: Exponential and Logarithmic Functions
      In this section, we review exponential and logarithmic functions. In addition, we spend time to review the critical Laws of Logarithms and introduce the number \( e \).
    • 1.6: Trigonometry
      Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions. In addition, we review inverse trigonometric functions and their ranges.
    • 1.7: Hyperbolic Functions
      The material in this section is likely not review. Instead, it introduces an important family of functions called the hyperbolic functions. These functions are used throughout calculus and differential equations.
    • 1.8: Chapter 1 Review Exercises

    Thumbnail: The graph of \(f(x)=e^x\) has a tangent line with slope 1 at \(x=0\). (CC BY; OpenStax)

    This page titled 1: Critical Concepts for Calculus is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.