Understand the order completeness properties of real numbers. 🔢
Learn basic properties of limits, infinite limits, and indeterminate forms. ➗
Explore continuous functions, types of discontinuities, and continuity of composite functions. 🔄
Know Rolle's Theorem, Lagrange's Mean Value Theorem, Cauchy's Mean Value Theorem, their geometric interpretation, and applications. 📐
Study hyperbolic and inverse hyperbolic functions of a real variable and their derivatives. ✨
📝 Instructions
🧑🏫 For the Paper-Setter
The question paper will consist of three sections: A, B, and C.
Sections A and B will have four questions each from their respective syllabus sections.
Section C will contain one compulsory question with eleven short answer-type questions covering the entire syllabus uniformly.
Sections A and B: Each question will carry 10 marks.
Section C: This section will carry a total of 20 marks. ✍️
🎓 For the Candidates
Candidates are required to attempt five questions in total.
Select two questions from each of Sections A and B.
Additionally, candidates must answer the compulsory question from Section C. 🖋️
📘 Syllabus
Section A 🔢
Properties of Real Numbers:
Order property of real numbers, bounds, lub and glb, order completeness property, and the Archimedean property of real numbers. 🔄
Limits:
Definition of the limit of a function, basic properties of limits, infinite limits, and indeterminate forms. ➗
Continuity:
Continuous functions, types of discontinuities, continuity of composite functions, continuity of f(x), sign of a function in a neighborhood of a point of continuity, intermediate value theorem, and maximum and minimum value theorem. 🌟
Section B 🌟
Mean Value Theorems:
Rolle's Theorem, Lagrange's Mean Value Theorem, Cauchy's Mean Value Theorem, their geometric interpretation, and applications. Taylor's and Maclaurin's theorems with various forms of remainders and their applications. 📐
Hyperbolic Functions:
Hyperbolic and inverse hyperbolic functions of a real variable and their derivatives, successive differentiations, and Leibnitz's theorem. ✨
📚 Books Recommended
D. Murray & M.R. Spiegel: *Theory and Problems of Advanced Calculus*, Schaum's Outline Series. 📖
P.K. Jain and S.K. Kaushik: *An Introduction to Real Analysis*, S. Chand & Co. 🧾