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

> Study guide for DSC-5 Calculus in the University of Delhi BSc (Hons) Mathematics, UGCF 2022: limits, continuity, mean value theorems and curve tracing.

Source: https://practicecalculus.com/courses/university-of-delhi-dsc-5
Updated: 2026-11-08

## At a glance

- **Course code:** DSC-5: Calculus (Discipline Specific Core 5)
- **Programme:** B.Sc. (Hons.) Mathematics, Undergraduate Curriculum Framework (UGCF) 2022
- **Credits:** 4 credits (3 lecture + 1 tutorial)
- **Semester:** Semester II
- **Eligibility:** Class XII pass with Mathematics
- **Prerequisites:** DSC-2: Elementary Real Analysis
- **Covers:** Limits and continuity, uniform continuity, differentiability and mean value theorems, higher derivatives, Taylor's theorem, indeterminate forms and curve tracing
- **Next course:** [DSC-8: Riemann Integration](/courses/university-of-delhi-dsc-8) in Semester III
- **Practice here:** [Limits](/learn/calculus-1/limits/what-is-a-limit) and [applications of derivatives](/learn/calculus-1/applications-of-derivatives/critical-points) in Calculus 1

## Key takeaways

- DSC-5 Calculus is a 4 credit Semester II core paper in the University of Delhi B.Sc. (Hons.) Mathematics programme under UGCF 2022.
- Its prerequisite is DSC-2 Elementary Real Analysis, so limits are treated with epsilon-delta and sequential arguments.
- The three units are limits and continuity, differentiability and mean value theorems, and higher derivatives, Taylor's theorem and curve tracing.
- Integration is not in DSC-5; it is studied in DSC-8 Riemann Integration in Semester III.

**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.

| Semester | Core paper | What it covers |
| --- | --- | --- |
| I | DSC-2 Elementary Real Analysis | Real numbers, sequences and series (prerequisite for DSC-5) |
| II | **DSC-5 Calculus** | Limits, continuity, derivatives, Taylor's theorem, curve tracing |
| II | DSC-6 Ordinary Differential Equations | First and higher order ODEs |
| III | [DSC-8 Riemann Integration](/courses/university-of-delhi-dsc-8) | The Riemann integral, Fundamental Theorems, improper integrals |
| IV | [DSC-11 Multivariate Calculus](/courses/university-of-delhi-dsc-11) | Partial derivatives, multiple integrals, Green, Stokes and Gauss |

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 $n$th 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

| DSC-5 topic | Practice | Read |
| --- | --- | --- |
| Limits of functions | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [limit laws](/rules/limit-laws) |
| Squeeze theorem | [Squeeze theorem](/learn/calculus-1/limits/squeeze-theorem) | [Squeeze theorem](/theorems/squeeze-theorem) |
| One sided and infinite limits | [One sided and infinite limits](/learn/calculus-1/limits/one-sided-and-infinite-limits) | [Evaluating limits practice](/practice-problems/evaluating-limits) |
| Limits at infinity | [Limits at infinity](/learn/calculus-1/limits/limits-at-infinity) | [Limits at infinity](/blog/limits-at-infinity) |
| Continuity on closed intervals | [Intermediate value theorem](/learn/calculus-1/limits/intermediate-value-theorem) | [Extreme value theorem](/theorems/extreme-value-theorem) |
| Differentiability | [Differentiability](/learn/calculus-1/the-derivative/differentiability) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Chain rule | [Chain rule](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule) |
| Relative extrema | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Fermat's theorem](/theorems/fermats-theorem) |
| Rolle's and mean value theorems | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Rolle's theorem](/theorems/rolles-theorem), [mean value theorem](/theorems/mean-value-theorem) |
| Taylor's theorem and series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor's theorem](/theorems/taylors-theorem) |
| Indeterminate forms | [L'Hôpital's rule](/learn/calculus-1/applications-of-derivatives/lhopitals-rule) | [L'Hôpital's rule](/blog/lhopitals-rule) |
| Concavity and inflexion points | [Second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching](/blog/curve-sketching-with-derivatives) |
| Asymptotes and tracing rational functions | [Increasing and decreasing](/learn/calculus-1/applications-of-derivatives/increasing-and-decreasing) | [Rational functions](/functions/rational-functions) |
| Tracing polar equations | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Curve sketching practice](/practice-problems/curve-sketching) |

## 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 $n$th derivative of $x^2 e^x$ is

$$\frac{d^n}{dx^n}(x^2 e^x) = e^x\left(x^2 + 2nx + n(n - 1)\right)$$

**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](/learn/calculus-1/limits/limits-practice) and [applications practice](/learn/calculus-1/applications-of-derivatives/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](/practice) and read [how to study calculus](/blog/how-to-study-calculus) for a weekly routine.

## Frequently asked questions

### What is DSC-5 Calculus at Delhi University?

DSC-5 Calculus is the fifth Discipline Specific Core course of the B.Sc. (Hons.) Mathematics programme under the University of Delhi Undergraduate Curriculum Framework 2022. It is taught in Semester II with 3 lectures and 1 tutorial a week, for 4 credits.

### What are the units of DU DSC-5 Calculus?

Unit I is limits and continuity (15 hours), Unit II is differentiability and mean value theorems (15 hours), and Unit III is successive differentiation, Taylor's theorem and tracing of plane curves (15 hours).

### Which books are prescribed for DU DSC-5?

The essential readings are Anton, Bivens and Davis, Calculus (10th edition); Bartle and Sherbert, Introduction to Real Analysis (4th edition); Gorakh Prasad, Differential Calculus (19th edition); and Ross, Elementary Analysis. Apostol's Calculus Vol. 1 and Ghorpade and Limaye are suggested readings.

### Is DU DSC-5 more theoretical than school calculus?

Yes. It uses the epsilon-delta definition of a limit, uniform continuity, and proofs of Rolle's and the mean value theorem. You still need to compute, especially nth derivatives, Taylor expansions and curve sketches, so practise both proofs and problems.