Differential Calculus: From Practice to Theory
Credit: Cover of "Differential Calculus: From Practice to Theory" textbook, adapted from a work by Crockett Johnson, used with permission
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Differential Calculus: From Practice to Theory covers all of the topics in a typical first course in differential calculus. Initially it focuses on using calculus as a problem solving tool (in conjunction with analytic geometry and trigonometry) by exploiting an informal understanding of differentials (infinitesimals). As much as possible large, interesting, and important historical problems (the motion of falling bodies and trajectories, the shape of hanging chains, the Witch of Agnesi) are used to develop key ideas. Only after skill with the computational tools of calculus has been developed is the question of rigor seriously broached. At that point, the foundational ideas (limits, continuity) are developed to replace infinitesimals, first intuitively then rigorously. This approach is more historically accurate than the usual development of calculus and, more importantly, it is pedagogically sound.
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Credit: Image adapted from a figure by Christopher Griffin and is licensed under CC BY-NC-SA 3.0 US
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This is a set of lecture notes for MATH 555, Penn State's graduate Numerical Optimization course. Numerical Optimization is the study of maximizing or minimizing functions through numerical techniques. Generally, it's rare to optimize anything other than through numerical techniques (unless of course you're talking about something really simple). Numerical optimization is used every day and is built on techniques from multi-variable calculus, optimization theory (obviously) numerical linear algebra (for algorithm efficiency) and other branches of mathematics.
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