Introductory Calculus - Lecture 1

Practical Information

• Lectures: 16 lectures total.
  • Schedule: Mondays and Wednesdays at 10 AM.
• Instructor: Dan Ciubotaru.
• Resources:
  • Lecture notes written by Cath Wilkins (available online).
  • Eight problem sheets covered in four tutorials.
  • Recommended book: Mary Boas's Mathematical Methods in Physical Sciences.

Syllabus Overview

• First Half (7-8 Lectures):
  • Focus on Differential Equations (ODEs & PDEs).
  • Techniques: Combination of science and art.
• Second Half:
  • Line and double integrals (compute arc lengths, areas).
  • Calculus of functions in two variables (introduction to multivariable calculus).
  • Topics include surfaces, gradients, Taylor's theorem, critical points, Lagrange multipliers.

Course Relevance

• Useful for other prelim courses (multivariable calculus, dynamics, PDEs).
• Complements Analysis courses.
• Forms the groundwork for Part A Applied Mathematics options.

Differential Equations

• Definition: Involves independent variable (x), dependent variable (y), and derivatives of y with respect to x.
• Order: Determined by the highest derivative.
• Example: Simplest ODE is dy/dx = f(x) (solvable by integration).
• Application Examples:
  • Mechanics: Newton's second law as a differential equation.
  • Electrical circuits: RLC circuit described by a second-order differential equation.
  • Exercise: Write a DE for radioactive decay.

Integration Techniques

• Integration by Parts:
  • Derived from the product rule.
  • Definite and indefinite integration forms.
  • Example problems: Integrating x^2sin(x) and 2x - 1 ln(x^2+1).
• Challenges:
  • Integration by parts may require iterative solving (e.g., e^xsin(x)).
  • Recursive formulas for integrals like cos^n(x).

Solving Differential Equations

• Separable Equations: dy/dx = a(x)b(y).
  • Technique: Separate variables and integrate.
  • Example problem provided.
• Considerations:
  • Be cautious dividing by functions like b(y) - may miss solutions when b(y) = 0.

Conclusion

• Focus on groundwork: Understanding integration techniques is crucial.
• Importance of practice and familiarity with integration for solving DEs.
• Next Lecture: More on differential equations.

Ensure to review and practice integration techniques, as they are fundamental to solving differential equations throughout this course.