Delhi University DSC-5 Calculus (BSc Hons Maths): Study Guide

DSC-5: Calculus is a Semester II core paper in the B.Sc. (Hons.) Mathematics programme at the University of Delhi (DU), under the Undergraduate Curriculum Framework (UGCF) 2022. It carries 4 credits, with 3 lectures and 1 tutorial a week. This guide follows the official syllabus published by the DU Department of Mathematics and maps each topic to free practice.

Where DSC-5 fits

The UGCF 2022 B.Sc. (Hons.) Mathematics programme spreads calculus over several core papers rather than one big course.

Eligibility is Class XII pass with Mathematics, and the stated prerequisite is DSC-2 Elementary Real Analysis.

What DSC-5 covers

The learning outcomes are the notion of limits, continuity and uniform continuity; the geometric properties of continuous functions on closed and bounded intervals; applications of the derivative, relative extrema and mean value theorems; and higher order derivatives, Taylor's theorem, indeterminate forms and tracing of curves.

• Unit I, Limits and continuity (15 hours): limits of functions by the epsilon-delta and sequential approaches, algebra of limits, the squeeze theorem, one sided limits, infinite limits and limits at infinity, continuous functions and their properties on closed and bounded intervals, and uniform continuity.
• Unit II, Differentiability and mean value theorems (15 hours): differentiability, algebra of differentiable functions, the chain rule, relative extrema, the interior extremum theorem, Rolle's theorem, the mean value theorem and its applications, and the intermediate value theorem for derivatives.
• Unit III, Successive differentiation, Taylor's theorem and curve tracing (15 hours): higher order derivatives and the nth derivative, Leibnitz's theorem, Taylor's theorem and series for e^x, \sin x and \cos x, indeterminate forms and L'Hôpital's rule, concavity and inflexion points, singular points, asymptotes, and tracing rational functions and polar equations.

DSC-5 topic map

Example problems at the DSC-5 level

These are original practice problems, not past DU questions.

1. An epsilon-delta limit. To show \lim_{x \to 2}(3x - 1) = 5, note |(3x - 1) - 5| = 3|x - 2|. Given \varepsilon > 0, choose \delta = \varepsilon/3.

2. Leibnitz's theorem. The nth derivative of x^2 e^x is

3. Mean value theorem. For f(x) = x^3 on [0, 2], f'(c) = \frac{8 - 0}{2} = 4, so 3c^2 = 4 and c = \frac{2}{\sqrt{3}}, which lies in (0, 2).

How to do well in DSC-5

• Write definitions word for word. Proof questions on continuity and uniform continuity are marked on precision, so learn the epsilon-delta statements exactly.
• Balance proofs with computation. Use Limits practice and applications practice to keep your computing fast.
• Trace curves step by step: domain, symmetry, intercepts, asymptotes, increasing and decreasing intervals, then concavity.
• Study a little every day with practice and read how to study calculus for a weekly routine.