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Schaum's Outlines: Calculus

Frank Ayres, Jr., and Elliott Mendelson
McGraw Hill
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
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It’s not marketed this way, but in many ways this is an old-fashioned calculus book, with skimpy but straightforward explanations, few figures, and a strong emphasis on techniques and drill. It reviews analytic geometry, and covers everything that would normally be in first-year calculus, including differentiation and integration, multi-variable and vector calculus, differential equations, and infinite series. The applications are also very traditional, consisting of geometric problems and some items from mechanics.

The book is marketed as a supplement and review for pre-calculus and calculus courses. I think it works well for calculus (but not for pre-calculus, because there is essentially no coverage of high-school algebra or trigonometry). The book is divided into brief (8 to 10-page) chapters, each containing an explanation of the topic, a series of worked examples, and a series of exercises with answers. The exercises are for drill, and are not especially challenging but are typical of what appears on calculus exams. There are essentially no word problems or applications in the exercises.

There is some use of technology. There is a series of thirty 4–5 minutes videos, available online at no cost at the publisher’s web site These videos provide walk-throughs of selected exercises from the book, using an electronic blackboard and an audio narrative. They are not referenced in the text, except on the cover, and require some digging to find on the web. These are well done and provide another medium to reach students. A number of exercises call for a calculator or graphing calculator (they are marked “GC”). Most of these involve graphing a function, and some make good use of the calculator, for example to plot parametric functions. There is also a lot of numeric work, for example using Newton’s method for finding roots and for approximating infinite series. There are no tutorials for the calculator: you are assumed to know how to use the calculator already.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

Linear Coordinate Systems. Absolute Value. Inequalities • Rectangular Coordinate Systems • Lines • Circles • Equations and their Graphs • Functions • Limits • Continuity • The Derivative • Rules for Differentiating Functions • Implicit Differentiation • Tangent and Normal Lines • Law of the Mean. Increasing and Decreasing Functions • Maximum and Minimum Values • Curve Sketching. Concavity. Symmetry • Review of Trigonometry • Differentiation of Trigonometric Functions • Inverse Trigonometric Functions • Rectilinear and Circular Motion • Related Rates • Differentials. Newton’s Method • Antiderivatives • The Definite Integral. Area under a Curve • The Fundamental Theorem of Calculus • The Natural Logarithm • Exponential and Logarithmic Functions • L’Hopital’s Rule • Exponential Growth and Decay • Applications of Integration I: Area and Arc Length • Applications of Integration II: Volume • Techniques of Integration I: Integration by Parts • Techniques of Integration II: Trigonometric Integrands and Trigonometric Substitutions • Techniques of Integration III: Integration by Partial Fractions • Miscellaneous Substitutions • Improper Integrals • Applications of Integration II: Area of a Surface of Revolution • Parametric Representation of Curves • Curvature • Plane Vectors • Curvilinear Motion • Polar Coordinates • Infinite Sequences • Infinite Series • Series with Positive Terms. The Integral Test. Comparison Tests • Alternating Series. Absolute and Conditional Convergence. The Ratio Test • Power Series • Taylor and Maclaurin Series. Taylor’s Formula with Remainder • Partial Derivatives • Total Differential. Differentiability. Chain Rules • Space Vectors • Surface and Curves in Space • Directional Derivatives. Maximum and Minimum Values • Vector Differentiation and Integration • Double and Iterated Integrals • Centroids and Moments of Inertia of Plane Areas • Double Integration Applied to Volume under a Surface and the Area of a Curved Surface • Triple Integrals • Masses of Variable Density • Differential Equations of First and Second Order