Bishop's review was harshly critical see Criticism of nonstandard analysis. The book was first reviewed by Errett Bishop, noted for his work in constructive mathematics. Thus the microscope is used as a device in explaining the derivative. The "infinitesimal microscope" is used to view an infinitesimal neighborhood of a standard real. The standard part function "rounds off" a finite hyperreal to the nearest real number. The derivative of ƒ is then the ( standard part of the) slope of that line (see figure). Similarly, an infinite-magnification microscope will transform an infinitesimal arc of a graph of ƒ, into a straight line, up to an infinitesimal error (only visible by applying a higher-magnification "microscope"). When one examines a curve, say the graph of ƒ, under a magnifying glass, its curvature decreases proportionally to the magnification power of the lens. Similarly, an infinite-resolution telescope is used to represent infinite numbers. In his textbook, Keisler used the pedagogical technique of an infinite-magnification microscope, so as to represent graphically, distinct hyperreal numbers infinitely close to each other. The usual definitions in terms of ε–δ techniques are provided at the end of Chapter 5 to enable a transition to a standard sequence. Keisler defines all basic notions of the calculus such as continuity (mathematics), derivative, and integral using infinitesimals. Keisler also published a companion book, Foundations of Infinitesimal Calculus, for instructors, which covers the foundational material in more depth. Keisler's textbook is based on Robinson's construction of the hyperreal numbers.
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