Introductory Calculus: Oxford Mathematics 1st Year Student Lecture

The lecturer, Dan Ciubotaru, begins the first lecture of Introductory Calculus by sharing practical information about the course structure, including the number of lectures, availability of lecture notes online, and tutorial schedules. He introduces himself and sets expectations for the course difficulty, suggesting that it may start off easy but will progress in complexity. The syllabus is briefly introduced, with a focus on differential equations from physical sciences and integration techniques. The lecturer also mentions the textbook 'Mathematical Methods in Physical Sciences' by Mary Boas as a recommended resource.

The lecturer continues by detailing the syllabus, emphasizing the first half of the course will focus on differential equations, including ordinary and partial differential equations (ODEs and PDEs). He provides examples of how these equations arise in various fields, such as mechanics and electrical circuits. The second half of the course will cover line and double integrals, leading to the introduction of multivariable calculus, including topics like gradient, normal vectors, Taylor's theorem, critical points, and Lagrange multipliers. The lecturer also discusses the interconnection between this course and others in the mathematics program, highlighting the relevance of the material to future studies.

The lecturer delves into the applications of differential equations (DEs), defining ordinary differential equations (ODEs) and providing examples from mechanics and electrical circuits. He explains how Newton's second law leads to a second-order differential equation and how an RLC electrical circuit gives rise to a second-order inhomogeneous equation with constant coefficients. The lecturer also poses an exercise for the students to formulate a differential equation for the decay of radioactive material.

The lecturer reviews integration techniques, focusing on integration by parts, which is derived from the product rule. He works through examples to demonstrate the process, including integrating functions like x^2sin(x) and 2x - 1 times the natural log of (x^2 + 1). The lecturer emphasizes the importance of these techniques for solving differential equations and encourages students to practice more if they are not fully comfortable with the examples provided.

The lecturer introduces the concept of recursive formulas for solving certain types of integrals that do not simplify in a single step. He provides an example involving the integral of cosine to the power of n times x, leading to a recursive formula for I(n) in terms of I(n-1) and I(n-2). The lecturer then transitions to discussing separable differential equations, showing how to solve them through a step-by-step example involving the equation x*(y^2 - 1) + y*(x^2 - 1) = 0, and highlights the importance of considering all possible solutions, including those where the denominator might be zero.

In the concluding paragraph, the lecturer summarizes the content covered in the first lecture and sets the stage for future lectures, where more differential equations will be explored. He reminds students of the importance of understanding integration techniques for solving these equations and encourages them to review the material and prepare for the next session.