This text helps students foster computational skills and intuitive understanding with a careful balance of theory, applications and optional materials. This edition offers revised coverage in several areas and a new section looking at applications to differential geometry, physics and forms of life as well as a large number of new exercises and expansion of the book's historical notes, which help students understand how calculus evolved by profiling important mathematicians. With its contemporary balance between theory, application and historical development the fifth edition gives an insight into how mathematics progresses and is in turn influenced by the natural world.