Calculus and Linear Algebra, Volume 1: Vectors in the plane and one-variable calculus by Wilfred Kaplan (PDF)

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⭐While in high-school, we utilized Mary Dolciani: Modern Introductory Analysis. Kaplan’s text refers to that book and that is a good omen. Happily, Dolciani is ample preparation for this introductory calculus tome (first edition, 1970) The preface informs us that the “integration of calculus with linear algebra” is a prime objective. This first volume (subtitled: vectors in the plane and one-variable calculus) is well-written and lucid. Let us take a tour:(1) You will review elementary algebra, geometry and trigonometry (chapter one). Standard material, you conclude with “proof by induction.” (Induction, always a favorite topic).(2) next, vector algebra. Regards linear algebra, an introduction to linear independence (page 59). You conclude with parametrization (parametrization is always a favorite topic).(3) limits and continuity, all is standard. You will conclude with Least Upper Bound Axiom (page 130) and epsilon-delta proofs. A lucid exposition.(4) derivatives: discussion of chain rule, which includes a nice “three-turning-gears problem” (problem #5, page 168). Parametric equations and vectors, revisited.(5) integrals, beginning page 237. Standard material at elementary vantage. There is a nice exposition of partial fraction decomposition. Another pass: Least Upper Bound and Greatest Lower Bound (page 287).A highlight: inequalities for integrals (pages 317-320, Section 4.23).(6) fifth chapter, transcendental functions: sine, cosine, exponential, logarithmic and hyperbolic. These are all introduced at elementary vantage (recall: “radian measure is preferred in this book,” page 32).(7) applications of derivative. Standard material: maxima and minima, elementary plane curves, Taylor’s formula with remainder following which applications of integrals: area, volume, centroids. Also: improper integrals, introductory differential equations, numerical evaluation of integrals.(8) Next up: infinite series (page 566). Both integration and infinite series are introduced much later than I usually recommend. An exposition spanning fifty-pages.(9) In the first volume, linear algebra is introduced only sparingly. Otherwise, what we do have in this first volume is an elementary and lucid account of basic calculus. Offering but few innovations, there is a concentration upon clarity of exposition and foundations. Answers to many of the exercises are included. The exercises are straightforward. No surprises. Now, there are many excellent books from which to learn calculus. While I do not claim that this first volume is particularly innovative, it is useful as refresher of introductory calculus and a source of straightforward problems. It is well written, thus, pleasant to read. Geometry is frequently used to motivate a proof (preface).Thus, recommended for an elementary exposition of basic material.

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