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  • Course Introduction

    Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT. Solving an algebra problem, like \( y = 2x + 5 \), merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate \( R \), such as \( Y = X_0+Rt \), where \( t \) is elapsed time and \( X_0 \) is the initial deposit. With compound interest, things get complicated for algebra, as the rate \( R \) is itself a function of time with \( Y = X_0 + R(t)t \). Now we have a rate of change which itself is changing. Calculus came to the rescue, as Isaac Newton introduced the world to mathematics specifically designed to handle those things that change.

    Calculus is among the most important and useful developments of human thought. Even though it is over 300 years old, it is still considered the beginning and cornerstone of modern mathematics. It is a wonderful, beautiful, and useful set of ideas and techniques. You will see the fundamental ideas of this course over and over again in future courses in mathematics as well as in all of the sciences (e.g., physical, biological, social, economic, and engineering). However, calculus is an intellectual step up from your previous mathematics courses. Many of the ideas you will gain in this course are more carefully defined and have both a functional and a graphical meaning. Some of the algorithms are quite complicated, and in many cases, you will need to make a decision as to which appropriate algorithm to use. Calculus offers a huge variety of applications and many of them will be saved for courses you might take in the future.

    This course is divided into five learning sections, or units, plus a reference section, or appendix. The course begins with a unit that provides a review of algebra specifically designed to help and prepare you for the study of calculus. The second unit discusses functions, graphs, limits, and continuity. Understanding limits could not be more important, as that topic really begins the study of calculus. The third unit introduces and explains derivatives. With derivatives, we are now ready to handle all of those things that change mentioned above. The fourth unit makes visual sense of derivatives by discussing derivatives and graphs. The fifth unit introduces and explains antiderivatives and definite integrals. Finally, the reference section provides a large collection of reference facts, geometry, and trigonometry that will assist you in solving calculus problems long after the course is over.

  • Unit 1: Preview and Review

    While a first course in calculus can strike you as something new to learn, it is not comparable to learning a foreign language where everything seems different. Calculus still depends on most of the things you learned in algebra, and the true genius of Isaac Newton was to realize that he could get answers for this something new by relying on simple and known things like graphs, geometry, and algebra. There is a need to review those concepts in this unit, where a graph can reinforce the adage that a picture is worth one thousand words. This unit starts right off with one of the most important steps in mastering problem solving: Have a clear and precise statement of what the problem really is about.

    Completing this unit should take you approximately 7 hours.

  • 1.1: Preview of Calculus

  • 1.1.1: Practice Problems

  • 1.1.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • The Slope of a Tangent Line; pages 1-2.
      • The Area of a Shape; pages 3-4.
      • Limits; page 4.
      • Differentiation and Integration; page 4.

  • 1.2: Lines in the Plane

  • 1.2.1: Practice Problems

  • 1.2.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • The Real Number Line; page 1.
      • The Cartesian Plane; page 2.
      • Increments and Distance between Points in the Plane; pages 2-3.
      • Slope between Points in the Plane; pages 4-6.
      • Equations of Lines; page 6.
      • Two-Point and Slope-Intercept Equations; pages 6-7.
      • Angles between Lines; page 8.
      • Parallel and Perpendicular Lines; pages 8-9.
      • Angles and Intersecting Lines; page 10.

  • 1.3: Functions and Their Graphs

  • 1.3.1: Practice Problems

  • 1.3.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Definition of a Function; page 1.
      • Function Machines; page 2.
      • Functions Defined by Equations; pages 2-3.
      • Functions Defined by Graphs and Tables of Values; pages 3-4.
      • Creating Graphs of Functions; pages 4-5.
      • Reading Graphs; pages 6-8.

  • 1.4: Combinations of Functions

  • 1.4.1: Practice Problems

  • 1.4.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Multiline Definition of Functions; page 1.
      • Wind Chill Index Sample; pages 1-3.
      • Composition of Functions - Functions of Functions; pages 3-4.
      • Shifting and Stretching Graphs; pages 5-6. 
      • Iteration of Functions; pages 6-7.
      • Absolute Value and Greatest Integer; pages 7-9.
      • Broken Graphs and Graphs with Holes; pages 10-11.

  • 1.5: Mathematical Language

  • 1.5.1: Practice Problems

  • 1.5.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Equivalent Statements; page 1.
      • The Logic of "And" and "Or"; page 1.
      • Negation of a Statement; page 2.
      • "If-Then" Statements; pages 2-3.
      • Contrapositive of "If-Then" Statements; page 4.
      • Converse of "If-Then" Statements; pages 4-5.

  • Tópico 18

  • Unit 2: Functions, Graphs, Limits, and Continuity

    The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are a kind of Siamese twins in calculus problems, as we always hope they show up together.

    A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice - the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house. When the car hit the brakes, the student's intuitive mind concluded that at the rate at which the velocity was decreasing (assuming continuity), there was no way the car could stop in time and it would hit the house (the limiting value). Oops. He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house. The gravel represented a discontinuity in his calculations, so his limiting value was not correct.

    Completing this unit should take you approximately 19 hours.

  • 2.1: Tangent Lines, Velocities, and Growth

  • 2.1.1: Practice Problems

  • 2.1.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • The Slope of a Tangent Line; pages 1-3.
      • Average Velocity and Instantaneous Velocity; pages 3-5.
      • Average Population Growth Rate and Instantaneous Population Growth Rate; pages 5-7.

  • 2.2: The Limit of a Function

  • 2.2.1: Practice Problems

  • 2.2.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Informal Notion of a Limit; pages 1-3.
      • Algebra Method for Evaluating Limits; pages 4-6.
      • Table Method for Evaluating Limits; pages 4-6.
      • Graph Method for Evaluating Limits; pages 4-6.
      • One-Sided Limits; pages 6-7.

  • 2.3: Properties of Limits

    • Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 2.3: Properties of Limits" URL
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      Read this section on pages 1-8 to learn about the properties of limits. Work through practice problems 1-6. For the solutions to these problems, see page 14.

    • RootMath: "Solving Limits (Rationalization)" Página
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      Watch this video on finding limits algebraically. Be warned that removing \( x-4 \) from the numerator and denominator in Step 4 of this video is only legal inside this limit. The function \( \frac{x - 4}{x - 4} \) is not defined at \( x = 4 \); however, when \( x \) is not 4, it simplifies to 1. Because the limit as \( x \) approaches 4 depends only on values of \( x \) different from 4, inside that limit \( \frac{x - 4}{x - 4} \) and 1 are interchangeable. Outside that limit, they are not! However, this kind of cancellation is a key technique for finding limits of algebraically complicated functions.

    • Khan Academy: "Calculating Slope of Tangent Line Using Derivative Definition" Página
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      Watch this video on limits as the slopes of tangent lines.

      An earlier Khan Academy video (not used in this course) defined the limit that gives the slope of the tangent line to a curve as \( y = f(x) \) at a point \( x = a \) and called it the derivative of \( f(x) \) at \( a \). The text will introduce this term in Unit 3.

  • 2.3.1: Practice Problems

  • 2.3.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Main Limit Theorem; page 1.
      • Limits by Substitution; page 2.
      • Limits of Combined or Composed Functions; pages 2-4.
      • Tangent Lines as Limits; page 4 and page 5.
      • Comparing the Limits of Functions; page 5 and page 6.
      • Showing that a Limit Does Not Exist; pages 6-8.

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  • 2.4: Continuous Functions

  • 2.4.1: Practice Problems

  • 2.4.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Definition and Meaning of Continuous; page 1.
      • Graphic Meaning of Continuity; pages 1-4.
      • The Importance of Continuity; page 5.
      • Combinations of Continuous Functions; pages 5-6.
      • Which Functions Are Continuous?; pages 6-8.
      • Intermediate Value Property; page 8 and page 9.
      • Bisection Algorithm for Approximating Roots; pages 9-11.

  • 2.5: Definition of a Limit

  • 2.5.1: Practice Problems

  • 2.5.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Intuitive Approach to Defining a Limit; pages 1-7.
      • The Formal Definition of a Limit; pages 7-10.
      • Two Limit Theorems; pages 10-11.

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  • Unit 3: Derivatives

    In this unit, we start to see calculus become more visible when abstract ideas such as a derivative and a limit appear as parts of slopes, lines, and curves. Then, there are circles, ellipses, and parabolas that are even more geometric, so what was previously an abstract concept can now be something we can see. Nothing makes calculus more tangible than to recognize that the first derivative of an automobile's position is its velocity and the second derivative of that position is its acceleration. We are at the very point that started Isaac Newton on his quest to master this mathematics, what we now call calculus, when he recognized that the second derivative was precisely what he needed to formulate his Second Law of Motion \( F = MA \), where \( F \) is the force on any object, \( M \) is its mass, and \( A \) is the second derivative of its position. Thus, he could connect all the variables of a moving object mathematically, including its acceleration, velocity, and position, and he could explain what really makes motion happen.

    Completing this unit should take you approximately 42 hours.

  • 3.1: Introduction to Derivatives

  • 3.1.1: Practice Problems

  • 3.1.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Slopes of Tangent Lines; pages 1-2.
      • Tangents to \( y = x^2 \); pages 2-5.

  • 3.2: The Definition of a Derivative

  • 3.2.1: Practice Problems

  • 3.2.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Formal Definition of a Derivative; pages 1-2.
      • Calculations Using the Definition; pages 2-6.
      • Tangent Line Formula; page 4.
      • \( \sin \) and \( \cos \) Examples; pages 4-5.
      • Interpretations of the Derivative; pages 6-8.
      • A Useful Formula: \( D(xn) \); pages 8-10.
      • Important Definitions, Formulas, and Results for the Derivative, Tangent Line Equation, and Interpretations of \( f'(x) \); page 10.

  • 3.3: Derivatives, Properties and Formulas

  • 3.3.1: Practice Problems

  • 3.3.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Which Functions Have Derivatives?; pages 1-3.
      • Derivatives of Elementary Combination of Functions; pages 3-6.
      • Using the Differentiation Rules; pages 7-8.
      • Evaluative a Derivative at a Point; page 9.
      • Important Results for Differentiability and Continuity; page 9.


  • 3.4: Derivative Patterns

  • 3.4.1: Practice Problems

  • 3.4.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • A Power Rule for Functions: \( D(f n(x)) \): To review this topic, focus on pages 1 and 2.
      • Derivatives of Trigonometric and Exponential Functions: To review this topic, focus on pages 3-6.
      • Higher Derivatives - Derivatives of Derivatives: To review this topic, focus on pages 6-7.
      • Bent and Twisted Functions: To review this topic, focus on pages 7-8.
      • Important Results for Power Rule of Functions and Derivatives of Trigonometric and Exponential Functions; page 9.

  • 3.5: The Chain Rule

  • 3.5.1: Practice Problems

  • 3.5.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Chain Rule for Differentiating a Composition of Functions; page 1.
      • The Chain Rule Using Leibnitz Notation Form; page 2.
      • The Chain Rule Composition Form; pages 2-5.
      • The Chain Rule and Tables of Derivatives; pages 5-6.
      • The Power Rule for Functions; page 7.

  • Tópico 53

  • 3.6: Some Applications of the Chain Rule

  • 3.6.1: Practice Problems

  • 3.6.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Derivatives of Logarithms; pages 1-2.
      • Derivative of \( ax \); pages 2-3.
      • Applied Problems; pages 3-5.
      • Parametric Equations; pages 5-6.
      • Speed; page 8.

  • 3.7: Related Rates

  • 3.7.1: Practice Problems

  • 3.7.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • The Derivative as a Rate of Change; pages 1-7.

  • 3.8: Newton's Method for Finding Roots

  • 3.8.1: Practice Problems

  • 3.8.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Newton's Method Using the Tangent Line; pages 1-3.
      • The Algorithm for Newton's Method; pages 3-5.
      • Iteration; page 5.
      • What Can Go Wrong with Newton's Method?; pages 5-6.
      • Chaotic Behavior and Newton's Method; pages 6-8.

  • 3.9: Linear Approximation and Differentials

  • 3.9.1: Practice Problems

  • 3.9.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Linear Approximation and Its Process; pages 1-4.
      • Applications of Linear Approximation to Measurement Error; pages 4-6.
      • Relative Error and Percentage Error; pages 6-7.
      • The Differential of a Function; pages 7-8.
      • The Linear Approximation Error; pages 8-10.

  • 3.10: Implicit and Logarithmic Differentiation

  • 3.10.1: Practice Problems

  • 3.10.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Implicit Differentiation; pages 1-3.
      • Logarithmic Differentiation; pages 3-5.

  • Tópico 69

  • Unit 4: Derivatives and Graphs

    A visual person should find this unit extremely helpful in understanding the concepts of calculus, as a major emphasis in this unit is to display those concepts graphically. That allows us to see what, so far, we could only imagine. Graphs help us to visualize ideas that are hard enough to conceptualize - like limits going to infinity but still having a finite meaning, or asymptotes - lines that approach each other but never quite get there.

    Graphs can also be used in a kind of reverse by displaying something for which we should take another mathematical look. It is hard enough to imagine a limit going to infinity, and therefore never quite getting there, but the graph can tell us that it has a finite value, when it finally does get there, so we had better take a serious look at it mathematically.

    Completing this unit should take you approximately 29 hours.

  • 4.1: Finding Maximums and Minimums

  • 4.1.1: Practice Problems

  • 4.1.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Methods for Finding Maximums and Minimums; page 1.
      • Terminology: Global Maximum, Local Maximum, Maximum Point, Global Minimum, Local Minimum, Global Extreme, and Local Extreme; page 2.
      • Finding Maximums and Minimums of a Function; pages 3-5.
      • Is \( f(a) \) a Maximum, Minimum, or Neither?; page 5.
      • Endpoint Extremes; pages 5-7.
      • Critical Numbers; page 7.
      • Which Functions Have Extremes?; pages 7-8.
      • Extreme Value Theorem; pages 8-9.

  • 4.2: The Mean Value Theorem and Its Consequences

  • 4.2.1: Practice Problems

  • 4.2.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Rolle's Theorem; pages 1-2.
      • The Mean Value Theorem; pages 2-4.
      • Consequences of the Mean Value Theorem; pages 4-6.

  • 4.3: The First Derivative and the Shape of a Function f(x)

  • 4.3.1: Practice Problems

  • 4.3.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Definitions of the Function; page 1.
      • First Shape Theorem; pages 2-4.
      • Second Shape Theorem; pages 4-7.
      • Using the Derivative to Test for Extremes; pages 7-8.

  • 4.4: The Second Derivative and the Shape of a Function f(x)

  • 4.4.1: Practice Problems

  • 4.4.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Concavity; pages 1-2.
      • The Second Derivative Condition for Concavity; pages 2-3.
      • Feeling the Second Derivative: Acceleration Applications; pages 3-4.
      • The Second Derivative and Extreme Values; pages 4-5.
      • Inflection Points; pages 5-6.

  • Tópico 83

  • 4.5: Applied Maximum and Minimum Problems

  • 4.6: Infinite Limits and Asymptotes

  • 4.6.1: Practice Problems

  • 4.6.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Limits as \( x \) Approaches Infinity; pages 1-4.
      • Using Calculators to Find Limits as \( x \) Goes to Infinity; page 5.
      • The Limit Is Infinite; pages 5-6.
      • Horizontal Asymptotes; pages 6-7.
      • Vertical Asymptotes; pages 7-8.
      • Other Asymptotes as \( x \) Approaches Infinity; pages 8-9.
      • Definition of \( \lim_{x\rightarrow \infty}{x}=k \); pages 9-10.

  • 4.7: L'Hopital's Rule

  • 4.7.1: Practice Problems

  • 4.7.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • A Linear Example; page 1.
      • 0/0 Form of L'Hopital's Rule; page 2.
      • Strong Version of L'Hopital's Rule; pages 2-3.
      • Which Function Grows Faster?; page 4.
      • Other Indeterminate Forms; pages 4-6.

  • Tópico 91

  • Unit 5: The Integral

    While previous units dealt with differential calculus, this unit starts the study of integral calculus. As you may recall, differential calculus began with the development of the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the notion of an area. In fact, we will be able to extend the notion of the area and apply these more general areas to a variety of problems. This will allow us to unify differential and integral calculus through the Fundamental Theorem of Calculus. Historically, this theorem marked the beginning of modern mathematics and is extremely important in all applications.

    Completing this unit should take you approximately 32 hours.

  • 5.1: Introduction to Integration

  • 5.1.1: Practice Problems

  • 5.1.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Area; pages 1-4.
      • Applications of Area like Distance and Total Accumulation; pages 5-7.

  • 5.2: Sigma Notation and Riemann Sums

  • 5.2.1: Practice Problems

  • 5.2.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Sigma Notation; pages 1-2.
      • Sums of Areas of Rectangles; pages 3-4.
      • Area under a Curve - Riemann Sums; pages 5-8.
      • Two Special Riemann Sums - Lower and Upper Sums; pages 9-10.

  • 5.3: The Definite Integral

  • 5.3.1: Practice Problems

  • 5.3.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • The Definition of the Definite Integral; pages 1-3.
      • Definite Integrals of Negative Functions; pages 3-5.
      • Units for the Definite Integral; pages 5-6.

  • 5.4: Properties of the Definite Integral

  • 5.4.1: Practice Problems

  • 5.4.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Properties of the Definite Integral; pages 1-2.
      • Properties of Definite Integrals of Combinations of Functions; pages 3-5.
      • Functions Defined by Integrals; pages 5-6.
      • Which Functions Are Integrable?; pages 6-7.
      • A Nonintegrable Function; page 8.

  • 5.5: Areas, Integrals, and Antiderivatives

  • 5.5.1: Practice Problems

  • 5.5.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Area Functions as an Antiderivative; pages 1-2.
      • Using Antiderivatives to Evaluate Definite Integrals; pages 2-4.
      • Integrals, Antiderivatives, and Applications; pages 4-6.

  • 5.6: The Fundamental Theorem of Calculus

  • 5.6.1: Practice Problems

  • 5.6.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Antiderivatives; pages 1-3.
      • Evaluating Definite Integrals; pages 4-5.
      • Steps for Calculus Application Problems; pages 6-8.
      • Leibnitz's Rule for Differentiating Integrals; page 9.

  • Tópico 111

  • 5.7: Finding Antiderivatives

    • Washington State Board for Community and Technical Colleges: Dale Hoffman's "Contemporary Calculus, Section 5.7: Finding Antiderivatives" URL
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      Read this section on pages 1-9 to see how one can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or u-substitution (for the most commonly-used variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1-4. For solutions to these problems, see pages 12-13.

    • RootMath: "Integration: U-Substitution" Página
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      Watch these videos on change of variable, also called substitution or u-substitution.

  • 5.7.1: Practice Problems

  • 5.7.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Indefinite Integrals and Antiderivatives; page 1.
      • Properties of Antiderivatives (Indefinite Integrals); pages 2-3.
      • Antiderivatives of More Complicated Functions; pages 3-4.
      • Getting the Constant Right; pages 4-5.
      • Making Patterns More Obvious - Changing Variables; pages 5-8.
      • Changing the Variables and Definite Integrals; pages 8-9.
      • Special Transformations - Antiderivatives of \( \sin^2(x) \) and \( \cos^2(x) \); page 9.

  • 5.8: First Application of Definite Integral

  • 5.8.1: Practice Problems

  • 5.8.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Area between Graphs of Two Functions; pages 1-4.
      • Average (Mean) Value of a Function; pages 4-6.
      • A Definite Integral Application - Work; pages 6-8. 

  • 5.9: Using Tables to Find Antiderivatives

  • 5.9.1: Practice Problems

  • 5.9.2: Review

    • Before moving on, you should be comfortable with each of these topics:

      • Table of Integrals; pages 1-3.
      • Using Recursive Formulas; page 3.

  • Tópico 121

  • Appendix

    By reviewing this course, you will have an invaluable list of references to assist you in solving future calculus problems after this course has ended. It is a standard experience, when solving calculus problems on your own, to react to the new problem with the following: "We did not solve that kind of problem in the course." Ah, but we did, in that the new problem is often a combination, or composition, of two problem types that were covered.

    The course could not cover all possible trigonometric functions you will encounter. If you encounter a need for the derivative of \( \tan(x) \), it is sufficient to recall that \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) and that sine and cosine were covered. You can eventually become so good at this that future calculus problems can almost seem to be little more than plugging into formulas.

    Engineering students who have to take several courses that involve the use of calculus are noted for having a Table of Integrals on their hip wherever they go, such as this one posted on Wikipedia.