Step By Step Calculus » 1.0 - Preface

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MuchLearning’s Step-by-Step Calculus series represents a motivated compromise between the optimal pedagogical calculus ordering and the background needs for early calculus-based courses like physics, statics and probability. Our calculus classroom experience suggests that the most naturally related topics must come together to make the transition between topics smooth and help students build their understanding of calculus in a systematic way. We also find that, in most cases, teaching the general problem first makes it easier for students to understand the specialized version of the same problem. We have chosen to write this new series of calculus books to make a thoughtful change in the ordering of the topics and to portray new approaches to some of the existing concepts.
While we feel that this source can be used for any serious calculus course due to its adaptability, we have tried to keep engineering, science and business students particularly in mind. In contrast to other popular calculus sources, we introduce, motivate and work with such things as Riemann sums, tangent lines, parametric curves and functions of multiple variables from the early sections on. This approach immediately forces students to start comprehending the scope of the calculus that they will learn, to anticipate some of what is coming next and to take the material seriously from the onset. We recognize that calculus can be challenging for most students so we have striven to keep related items (such as area, volume and surface area calculations) together to facilitate quicker, deeper understanding.
Rational functions are vital for Engineering and Applied Science due to their role in transfer function techniques. Hence, we have included rational simplification methods like Euclidean division and carefully developed other areas like partial fraction expansions. Since partial derivatives are essential in early courses like physics as well as probability and we consider functions of several variables from the start, we introduce the partial and ordinary derivatives together immediately after covering limits. Moreover, since the definite integral is defined as a (triangular) series limit, we feel that it is natural to consider series before studying the properties of the integral in depth. This ordering has the desired byproduct of allowing use of (series-based) Taylor polynomials, transfer functions and moment generating functions far earlier than typically allowed in other areas of science, engineering and business. Furthermore, it tends to consolidate the treatments of the indefinite and definite integrals and thereby facilitate learning. These and other ordering choices have been made to promote understanding and efficiency of instruction.
The material covered in the Step-by-Step Calculus series is typically covered in three somewhat aggressive or four more leisurely courses. In the three-course case, the first course usually ends after the chapter on Series, contains only a few of the optional sections and requires a few of the early sections to be skimmed or assigned as high school review reading. The second course would include up to the Volumes and Two Dimensional Integration chapter as well as a few of the optional sections. The third course contains the remainder of the material.
We believe that learning then mastering calculus requires extensive problem solving. To help students build strong problem-solving skills, while maintaining details and rigorous proofs, this Step-by-Step series breaks down seemingly difficult methods to straightforward step-by-step procedures. We also provide a multitude of various applications. We believe that this linkage between the theory of calculus and its uses will help students better understand and love calculus. As we think the best way to master calculus is by solving more and more problems, we have taken an example-driven approach throughout the books. In the same spirit, this series is accompanied by a series of workbooks with hundreds of problems of various difficulty levels that will give students the opportunity to master each topic. We have kept the workbook series separate from this series to provide convenience and flexibility for students. MuchLearning also provides supplementary online resources, including an online version of this series, solutions to workbook problems, audible lectures, problems of all difficulty levels, and many more features, to facilitate students learning and mastering calculus.