Raise your hand if you’ve been stressed about your math scores. Don’t worry – we’ve all been there. Based on 15 years of experience teaching at the University of Texas at Arlington, author David Andrew Smith uses *Calculus 1* to dispel the myth that advanced math can’t be friendly and proves that clear, sensible examples can actually save you study time.

You shouldn’t need to spend days in the library burrowed in your Calculus textbook. Therefore, *Calculus 1* will help you save time and improve in your Calculus class with straightforward examples beyond your lectures. Without question, this book provides everything you need to study Calculus. Additionally, students can use *Calculus 1* as a class supplement or as a more manageable alternative to your textbook.

This book is perfect for undergraduate and junior college students, as well as high school students studying for AP exams. An upgrade from the first edition, Edition 2 includes more exercises and helpful examples laid out in an easy-to-follow fashion.

This book is perfect for undergraduate and junior college students, as well as high school students studying for AP exams. An upgrade from the first edition, Edition 2 includes more exercises and helpful examples laid out in an easy-to-follow fashion.

## Calculus 1 Content

Smith guides you through four core principles and chapters:

- Limits and Continuity
- Differentiation
- Applications of Differentiation
- Integration

Each chapter includes the fundamentals, figures, diagrams, tables, and plenty of examples to help work through your equations. To tie it all together, the last chapter on Integration will deliver you an “a-ha moment.” It walks you through theory, proof and examples of the two fundamental theorem of Calculus.

*Calculus 1* helps you make sense of the theory. You’ll see that it’s not as complex as you may think – in fact, it’s quite possible to master. Without Calculus, we wouldn’t have things that transformed our world like search engines, weather models, safe and strong bridges, economic forecasts, and more.

Ebook (pdf file) 245 pages.

## Calculus 1 begins with limits

This book begins with an intuitive introduction to limits using tables and graphs. Next, we introduce techniques for finding limits which involve algebra, rationalization, geometric limits, and also the squeeze theorem. Then, before studying the precise definition of the limit, we study continuous functions. Specifically, we study one-sided and two-sided continuity, removable continuity, and continuity on an interval. We also discuss and explore the Intermediate Value Theorem. At the end of the first chapter, Limits and Continuity, we discuss rates of change; that is, the average rate of change and instantaneous rate of change. We also discuss the tangent line problem.

## Differentiation is the heart of calculus

In the next chapter, Differentiation, we introduce the derivative of a function using a limit definition. Following this we develop the basic rules of differentiation, including the product and quotient rules. Derivatives of trigonometric functions and the chain rule are then studied. Importantly, implicit differentiation and the derivatives of inverse functions are then discussed. The chapter ends by investigating related rate problems and linearization and differentials.

## There are many areas of science and engineering that are applications of differentiation

The third chapter of Calculus 1, Applications of Differentiation, begins by covering extreme values such as relative extrema and absolute extrema. The infamous Mean Value Theorem is then covered in detail. Next, we discuss the First Derivative Test, the Second Derivative Test, and the Concavity Test. These tests culminate in the study of curve sketching, applied optimization problems, and parametric equations. Indeterminate forms and l’Hospital’s Rule are studied at the end of this chapter.

## Integration is the soul of calculus

The final chapter, Integration, begins with the concept of anti-derivatives followed by introducing integration by substitution. Readers then study sigma notation, Riemann sums, and using partitions to find area. Next, the definite integral is motivated and defined and the connection between the area under a curve and the tangent line problem is then discussed. The culmination of the reader’s concentrated work is brought out in the two fundamental theorems of calculus. This chapter then ends with the study of numerical integration, including the midpoint rule, the trapezoidal rule, and Simpson’s rule.

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