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Graduate School
Calculus I
Futaba FUJIE Associate Professor
Department: Institute of Liberal Arts & Sciences
Class Time:  2015 Spring Thursday 
Recommended for:  School of Science 
Course Overview
Key Features
Although basics of single variable calculus are covered in high school mathematics, there are actually a lot more to consider in order to really understand the subject. For example, what does it really mean that a function is continuous at a certain point? While graphs are helpful, there are situations where the function in question is so complicated that it is impossible to draw its graph. Intuition is important, but it is essential to understand how things are mathematically defined and use them as building blocks. Learning mathematics is in some sense learning foreign languages. By going through proofs of theorems together in classroom, I hope that you feel comfortable with "math grammar" so that you can enjoy mathematics more.
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Syllabus
Course Objectives
Analysis is the field of mathematics that describes and analyzes quantitative changes and its central method is differential and integral calculus. While calculus is an essential research tool in natural sciences, it has recently found a number of applications in social sciences as well. The aim of the first half of this oneyear course is to provide a solid understanding of single variable calculus.
Course Content
The topics covered in this course are listed below.
 the limit of a sequence, the limit of a function and its continuity
 key words:
 the limit of a sequence,
 the limit of a function,
 the monotone convergence theorem,
 basic properties of continuous functions
 differential calculus of one variable
 key words:
 the derivative of a function and its geometric properties,
 basic properties of derivatives of functions,
 the inverses and derivatives of elementary functions,
 the mean value theorem,
 derivatives of higher order,
 Taylor's theorem,
 the limit of an indeterminate form
 integral calculus of one variable
 key words:
 definite integral,
 indefinite integral,
 the fundamental theorem of calculus,
 improper integral
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Page last updated October 14, 2016
The class contents were most recently updated on the date indicated. Please be aware that there may be some changes between the most recent year and the current page.