at Herstmonceux Castle, U.K.

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MATH 121/6.0 - Differential and Integral Calculus

Instructor: Dr. Chris Taylor

Calculus is a branch of mathematics that can describe precisely how one numerical output quantity changes in response to changes in one or more numerical input quantities. Two of its main aspects are differentiation, which concerns the instantaneous rate of change of one output variable with respect to one or more input variables; and integration, which deals with the cumulative change in a given output variable. For example, if a ball is thrown upwards, differentiation can tell us its vertical velocity (rate of change of height with time) at any point in time; and integration can tell us its net change in height over a given period. Many applications of calculus are found in science, engineering, commerce, and medicine; indeed, much of modern technology would be impossible without it. The development of calculus spans many centuries and cultures, but in Europe, calculus in something like its modern form first appeared in the seventeenth century, when Newton and Leibniz independently synthesized differential and integral calculus into a single powerful discipline. Many important developments and applications followed, although calculus was not made fully rigorous until the nineteenth century, with the emergence of the limit concept.

This is a general two-term calculus course, starting with a revision of high- school-level pre-calculus, and the basics of single-variable differentiation and integration, and then moving on to more advanced topics, such as multivariable calculus, differential equations, and various techniques of optimization. Students can take this course with or without high-school calculus experience. The course is not intended as a pure maths course, and so there is more of an emphasis on techniques and applications than on formal proofs.

The expected learning outcomes are as follows:

  • Conceptual understanding and technical mastery of the following main areas of calculus and pre-calculus:
    1. Basics of algebra and arithmetic.
    2. Functions and graphs.
    3. Geometry and trigonometry.
    4. Limits.
    5. Differentiation.
    6. Integration.
    7. Differential equations.
    8. Partial derivatives and vector calculus.
  • An ability to apply knowledge of the topics above to solve extended problems, both abstract and applied.
  • An ability to communicate and present such mathematical problem-solving skills in printed documents that combine explanatory English text with mathematical equations and graphs in a coherent and comprehensible way.