Math 275 - Topics in Calculus I

• Prerequisites: Consent of instructor.
• Frequency: Fall (I)
• Student Body: This course is intended for Honors students and it is open to freshmen.
• Credits: 5 (r-N-I)
• Recent Texts: Calculus, by Spivak.
• Course Coordinator: Gloria Maribeffa
• Background and Goals: This course is the first semester of the Calculus Honors sequence developed by the Mathematics Department at the UW. The goal of the sequence is to provide highly motivated and well-prepared students with an opportunity to go beyond the traditional approach to the subject to develop a deeper understanding of this fundamental area of mathematics and to appreciate its power and beauty. The material covers essentially the same topics as the standard first semester calculus course, but the material is discussed in greater depth, and with much more emphasis on mathematical ideas. The course will be challenging, and the student might find it surprisingly different at the beginning. But it is also meant to be a lot of fun and to provide the students with the kind of clear and precise thinking that is characteristic of mathematics and that will be useful for them in almost any subject they pursue.
• Alternatives: Math 221-222.
• Subsequent Courses: Math 276

Content coverage:

• Introduction
  • Real numbers and the completeness axiom
  • Mathematical induction
  • Some inequalities
• Integral Calculus
  • Functions
  • Area as a set function and step functions
  • Integrals of step functions
  • Integrals of general functions
  • Integrals of monotonic functions
  • Integrals of powers and polynomials
  • Properties of the integral
• Applications of Integration
  • Area between two graphs
  • Integrating trig functions
  • Integrals in polar coordinates
  • Calculation of volume
  • Work and average value
  • Indefinite integrals
• Continuous Functions
  • Limits of functions
  • Continuity
  • Properties of limits
  • Composite functions and continuity
  • Intermediate value theorem
  • Inverse functions and their properties
  • Extrema and uniform continuity
  • Integrability of continuous functions
• Differential Calculus
  • Differentiating a function
  • Algebra of derivatives
  • The derivative as a slope
  • The chain rule
  • Applications of the chain rule
  • Extreme values of functions
  • The mean value theorem
  • First and second derivative tests
  • Curve sketching
• Integration and Differentiation
  • The derivative of an indefinite integral
  • Primitive functions
  • Integration by substitution
  • Integration by parts
• The Logarithm, The Exponential, and Inverse Trigonometric Functions
  • Definition of logarithm as an integral
  • Differentiation and integration formulas
  • The exponential function
  • Differentiation and integration formulas
  • Inverse trigonometric functions