Introductory Calculus - Lecture 1

Practical Information

• Lectures: 16 lectures, twice a week (Mondays and Wednesdays at 10am)
• Lecture Notes: Online, written by Cath Wilkins
• Lecturer: Dan Ciubotaru
• Problem Sheets: 8 problem sheets, first two are online
  • Covered in four tutorials (four hours in total)
• Reading List: Online
  • Recommended Book: Mary Boas's Mathematical Methods in Physical Sciences

Syllabus Overview

• First Half: Differential Equations (7-8 lectures)
  • Ordinary Differential Equations (ODEs)
  • Partial Differential Equations (PDEs)
  • Techniques for solving differential equations
• Second Half: Calculus Topics
  • Line and double integrals (compute arc lengths, areas)
  • Calculus of functions in two variables
  • Topics: various surfaces, gradients, normal vectors, Taylor's theorem in two variables, critical points, Lagrange multipliers

Relation to Other Courses

• Useful for Multivariable Calculus, Dynamics, PDEs, Fourier Series
• Related to Analysis (particularly Analysis 2, next term)
• Foundational for applied mathematics options in Part A (e.g., Differential Equations, Fluid and Waves)

Differential Equations

Ordinary Differential Equations (ODEs)

• Equation involving an independent variable (x) and a dependent variable (y)
• Example: Simplest form is dy/dx = f(x), solvable by direct integration
• Mechanics Example: Newton's second law (F = ma)
  • Second-order differential equation: a = d²r/dt²
• Electrical Engineering Example: RLC circuit
  • Second-order differential equation: L d²Q/dt² + R dQ/dt + (1/C)Q = V
• Exercise: Write a differential equation for the rate of radioactive decay

Separable Differential Equations

• Form: dy/dx = a(x)b(y)
• Separate variables and integrate: 1/b(y) dy = a(x) dx
• Example: Solve x(y²-1) + y(x²-1) dy/dx = 0 for x in (0,1)
  • Resulting general solution: (1-y²)(1-x²)=C

Integration Techniques Review

Integration by Parts

• Derived from the product rule: ∫ f(x)g'(x) dx = f(x)g(x) - ∫ f'(x)g(x) dx
• Example: ∫ x² sin(x) dx
  • Apply integration by parts multiple times
  • Result: -x² cos(x) + 2x sin(x) - 2 cos(x) + C
• Be aware of cases where integration by parts doesn't simplify: Example, ∫ e^x sin(x) dx
  • Recursive handling

Recursive Formula for Certain Integrals

• Example: I(n) = ∫ cos^n(x) dx
  • Derive formula I(n) in terms of I(n-2) using integration by parts
  • General formula: I(n) = (1/n)cos^(n-1)(x)sin(x) + ((n-1)/n)I(n-2)

Substitution

• Example: 2x - 1 ln(x²+1) dx
  • Set up for integration by parts
  • Break down into simpler integrals

Miscellaneous Techniques

• Exercise: Rate at which a radioactive substance decays is proportional to the remaining number of atoms
• Being comfortable with various integration techniques is crucial for solving differential equations

Conclusion

• Recap of the simplest types of differential equations solvable by direct integration or separation of variables
• Preview of what's coming next: More differential equations in the following lecture