1. Indeterminate Forms and L’Hôpital’s Rule:
This chapter covers topics related to indeterminate forms and L’Hôpital’s Rule, along with the study of improper integrals. It explores the convergence and divergence of integrals, along with the properties of Beta and Gamma functions. Additionally, the chapter delves into various applications of definite integrals, including finding volumes using cross-sections, computing lengths of plane curves, and determining areas of surfaces of revolution.
2. Sequences and Series:
In this chapter, the focus is on sequences and series. It begins by discussing the convergence and divergence of sequences, followed by the Sandwich Theorem and the Continuous Function Theorem for Sequences. Bounded Monotonic Sequences are explored, leading to the analysis of infinite series, such as geometric series, telescoping series, and the nth-term test for divergent series. Further topics include combining series, the Harmonic Series, the Integral test, the p-series, the Comparison test, the Limit Comparison test, the Ratio test, Raabe’s Test, the Root test, and the Alternating series test. The concepts of Absolute and Conditional convergence are covered, along with the Power series and the determination of their Radius of convergence. The chapter concludes with the study of the Taylor and Maclaurin series
3. Fourier Series:
This chapter revolves around the Fourier Series of 2𝑛 periodic functions and Dirichlet’s conditions for representation by a Fourier series. It introduces the concept of the orthogonality of the trigonometric system and explores the Fourier Series of functions with a period of 2𝑛. The chapter also covers the Fourier Series of even and odd functions, along with Half range expansions.
4. Functions of Several Variables:
The focus of this chapter is on the functions of several variables. It begins by discussing limits and continuity, including tests for the non-existence of a limit. The chapter further explores partial differentiation, the Mixed Derivative Theorem, differentiability, the Chain Rule, Implicit Differentiation, Gradient, Directional Derivative, tangent plane, and normal line. The concept of total differentiation is covered, leading to the study of local extreme values and the Method of Lagrange Multipliers.
5. Multiple Integrals:
This chapter delves into the concept of multiple integrals. It covers double integrals over rectangles and general regions, exploring double integrals as volumes. The chapter also includes topics such as changing the order of integration, double integration in polar coordinates, finding areas by double integration, and the application of triple integrals in rectangular, cylindrical, and spherical coordinates. The Jacobian and multiple integrals by substitution are also discussed.
6. Matrices and Linear Equations:
The final chapter is devoted to matrices and linear equations. It covers elementary row operations in matrices, as well as row echelon and reduced row echelon forms. The concept of rank by echelon forms is explored, leading to the study of matrix inverses using the Gauss-Jordan method. The chapter also includes techniques for solving systems of linear equations using Gauss elimination and Gauss-Jordan methods. Additionally, the concepts of eigenvalues and eigenvectors are introduced, along with the Cayley-Hamilton theorem and the process of matrix diagonalization.
