# MTH 102 Elementary Mathematics II (Calculus): Study Guide

> MTH 102 Elementary Mathematics II (Calculus) study guide: limits, derivatives, curve sketching, integrals, areas and volumes, with free practice.

Source: https://practicecalculus.com/courses/nuc-mth-102
Updated: 2026-11-03

## At a glance

- **Course code:** MTH 102 (Elementary Mathematics II, Calculus)
- **Credit units:** 2 units, compulsory (30 lecture hours)
- **Level:** 100 level, core in the CCMAS B.Sc. Mathematics programme and many other science and computing programmes
- **Covers:** Functions of a real variable and their graphs, limits and continuity, the derivative, techniques of differentiation, extrema and curve sketching, integration, definite integrals, areas and volumes
- **Comes after:** MTH 101 Elementary Mathematics I (Algebra and Trigonometry)
- **Next course:** MTH 201 Mathematical Methods I (200 level)
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MTH 102 is the first calculus course in the NUC CCMAS: a compulsory 2-unit, 100 level course.
- It covers the core of a standard Calculus 1 course, from limits and the derivative to definite integrals, areas and volumes.
- It builds on MTH 101 (algebra and trigonometry) and leads to MTH 201 Mathematical Methods I at 200 level.
- Every MTH 102 topic maps to a free Calculus 1 lesson on PracticeCalculus with instant feedback.

**MTH 102 Elementary Mathematics II (Calculus)** is the first calculus course in the National Universities Commission's Core Curriculum and Minimum Academic Standards (CCMAS). It is a compulsory 2-unit course at 100 level, taken by mathematics students and by students in many other science, computing and engineering-related programmes that follow the CCMAS. Because so many universities use the same code, students often search for it simply as "MTH 102 calculus". This guide maps every MTH 102 topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What MTH 102 covers

The CCMAS outline for MTH 102 starts with functions of a real variable and their graphs, then moves to limits and the idea of continuity. The derivative is introduced as the limit of a rate of change, followed by techniques of differentiation and their use in finding extreme values and sketching curves. The second half treats integration as the reverse of differentiation, methods of integration, definite integrals, and applications to areas and volumes.

The stated learning outcomes are that students can tell the different rules of differentiation and integration apart, explain functions, limits and continuity, and apply definite integrals to areas and volumes.

That is a close match for a standard **Calculus 1** course. Universities add their own material to the CCMAS core (the CCMAS sets about 70 percent of each programme), so your department may go further, but the topics below are the shared foundation.

## MTH 102 topic map

Each row links to a lesson you can practice and to a page that explains the idea.

| MTH 102 topic | Practice | Read |
| --- | --- | --- |
| Functions of a real variable and graphs | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Polynomial functions](/functions/polynomial-functions), [rational functions](/functions/rational-functions) |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [How to evaluate limits](/blog/how-to-evaluate-limits) |
| The idea of continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity) |
| The derivative as a limit of rate of change | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Techniques of differentiation | [Power rule](/learn/calculus-1/derivative-rules/power-rule) | [Product rule](/rules/product-rule), [quotient rule](/rules/quotient-rule), [chain rule](/rules/chain-rule) |
| Extreme values | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Extreme value theorem](/theorems/extreme-value-theorem) |
| Curve sketching | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Integration as the inverse of differentiation | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives), [reverse power rule](/rules/reverse-power-rule) |
| Methods of integration | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [u-substitution](/blog/u-substitution), [integration by parts](/blog/integration-by-parts) |
| Definite integrals | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/blog/fundamental-theorem-of-calculus) |
| Application to areas | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves) |
| Application to volumes | [Volumes with disks](/learn/calculus-1/applications-of-integrals/volumes-disks) | [Volumes of revolution](/blog/volumes-of-revolution) |

## Where MTH 102 sits in the CCMAS sequence

| Course | Level | Focus |
| --- | --- | --- |
| [MTH 101](/courses/nuc-mth-101) Elementary Mathematics I | 100 level | Sets, numbers, sequences and series, quadratics, binomial theorem, complex numbers, trigonometry |
| MTH 102 Elementary Mathematics II | 100 level | Single-variable calculus |
| [MTH 103](/courses/nuc-mth-103) Elementary Mathematics III | 100 level | Vectors, coordinate geometry and dynamics |
| [MTH 201](/courses/nuc-mth-201) Mathematical Methods I | 200 level | Taylor series, partial derivatives, Lagrange multipliers, line and multiple integrals |

MTH 101 supplies the algebra and trigonometry you use in every MTH 102 calculation. MTH 201 then reviews MTH 102 and extends calculus to functions of two and three variables.

## Example problems at the MTH 102 level

These are original practice problems of the kind the course covers.

**1. A limit by factorising.** Evaluate $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$.

$$\lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = \lim_{x \to 3} (x + 3) = 6$$

**2. Turning points.** Find and classify the stationary points of $y = x^3 - 3x$.

$$\frac{dy}{dx} = 3x^2 - 3 = 0 \implies x = \pm 1, \qquad \frac{d^2y}{dx^2} = 6x$$

At $x = 1$, $y'' = 6 > 0$, so $(1, -2)$ is a minimum. At $x = -1$, $y'' = -6 < 0$, so $(-1, 2)$ is a maximum.

**3. A volume of revolution.** Rotate $y = \sqrt{x}$, $0 \le x \le 4$, about the $x$-axis:

$$V = \pi \int_0^4 (\sqrt{x})^2\,dx = \pi \int_0^4 x\,dx = \pi \left[\frac{x^2}{2}\right]_0^4 = 8\pi$$

## How to do well in MTH 102

- **Master the differentiation rules early.** The product, quotient and [chain rule](/rules/chain-rule) appear in almost every later topic, including curve sketching and integration by substitution.
- **Practice a little every day.** Short daily sessions on [practice](/practice) beat one long session before the exam, because calculus skills fade without repetition.
- **Keep a formula sheet beside you.** The [calculus formula sheet](/formulas) puts the standard derivatives and integrals on one page.
- **Watch for algebra slips.** Our list of [common calculus mistakes](/blog/common-calculus-mistakes) covers sign errors, dropped constants and the other errors that cost marks.
- **Plan your revision.** The guide on [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam) shows how to turn your notes into timed practice.

> **Tip:** if limits feel abstract, start with [direct substitution](/learn/calculus-1/limits/direct-substitution) and only reach for factorising when you get $\frac{0}{0}$.

## Frequently asked questions

### What is MTH 102?

MTH 102, Elementary Mathematics II (Calculus), is the 100 level calculus course in the National Universities Commission core curriculum (CCMAS). It covers functions, limits, continuity, differentiation, curve sketching, integration and the use of definite integrals for areas and volumes.

### What topics are in MTH 102 calculus?

The CCMAS outline lists functions of a real variable and their graphs, limits and the idea of continuity, the derivative as the limit of a rate of change, techniques of differentiation, extreme values and curve sketching, integration as the reverse of differentiation, methods of integration, definite integrals, and applications to areas and volumes.

### What comes after MTH 102?

In the CCMAS B.Sc. Mathematics programme, the next calculus course is MTH 201 Mathematical Methods I at 200 level, which adds the mean value theorem, Taylor series, partial derivatives, Lagrange multipliers, line integrals and multiple integrals.

### Is MTH 102 hard?

MTH 102 is many students' first contact with calculus, so the new ideas can feel heavy at first. Most marks are lost on algebra slips and on not knowing which differentiation rule to use, both of which improve quickly with short, regular practice.

### How can I prepare for MTH 102?

Revise the MTH 101 material you will lean on most: indices, quadratic equations and trigonometric values. Then work through limits and derivatives with many short problems. The free Calculus 1 path on PracticeCalculus follows the same order as MTH 102.