# ABU MATH105 Differential and Integral Calculus: Study Guide

> A study guide for ABU Zaria MATH105 (Differential and Integral Calculus): functions, limits, first principles, differentiation and integration, with practice.

Source: https://practicecalculus.com/courses/abu-zaria-math-105
Updated: 2026-11-03

## At a glance

- **Course code:** MATH105
- **Credit units:** 2
- **Semester:** 100 level, first semester
- **Prerequisites:** O level Mathematics
- **Covers:** Odd, even and periodic functions and their graphs, an intuitive treatment of limits and continuity, differentiation from first principles, techniques of differentiation, higher derivatives, techniques of integration and evaluating definite integrals
- **Next course:** MATH104 Conic Sections and Applications of Calculus (second semester); MATH201 Mathematical Methods I at 200 level
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MATH105 is the first calculus course at Ahmadu Bello University, Zaria: 2 credit units in the first semester of 100 level.
- It covers functions and their symmetries, limits and continuity (intuitively), differentiation and integration.
- Its only prerequisite is O level Mathematics.
- MATH105 is the stated prerequisite for several 200-level courses, including MATH201 Mathematical Methods I and MATH203 Real Analysis I.

**ABU MATH105**, Differential and Integral Calculus, is the first calculus course at Ahmadu Bello University, Zaria. It is a 2-credit-unit course in the first semester of 100 level in the Department of Mathematics, with O level Mathematics as its only prerequisite. This guide maps every MATH105 topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What ABU MATH105 covers

The department's course outline has three parts. The first is functions of a real variable: odd, even and periodic functions, their symmetries and graphs, and limits and continuity treated intuitively rather than with formal proofs. The second is differentiation, from first principles and then by the general techniques, including higher derivatives. The third is integration: integration as the inverse of differentiation, the general techniques of integration, and the definite integral, where the outline says the focus is on evaluation.

That is the core of a **Calculus 1** course. Applications such as rates of change, curve sketching, areas and volumes are not in MATH105; ABU teaches them in [MATH104](/courses/abu-zaria-math-104) in the second semester.

## MATH105 topic map

| MATH105 topic | Practice | Read |
| --- | --- | --- |
| Functions of a real variable | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Polynomial functions](/functions/polynomial-functions) |
| Odd, even and periodic functions, graphs | [Key trig values](/learn/calculus-1/functions-warmup/key-trig-values) | [Sine function](/functions/sine-function), [power functions](/functions/power-functions) |
| Limits (intuitive) | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit), [how to evaluate limits](/blog/how-to-evaluate-limits) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity) |
| Differentiation from first principles | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Techniques of differentiation | [The product rule](/learn/calculus-1/derivative-rules/product-rule) | [Product rule](/rules/product-rule), [quotient rule](/rules/quotient-rule), [chain rule](/rules/chain-rule) |
| Higher derivatives | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Power rule](/blog/power-rule) |
| Integration as the inverse of differentiation | [Undoing derivatives](/learn/calculus-1/integrals/undoing-derivatives) | [Antiderivatives](/blog/antiderivatives) |
| Techniques of integration | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [U-substitution](/blog/u-substitution), [integration by parts](/blog/integration-by-parts) |
| Evaluating definite integrals | [Evaluating definite integrals](/learn/calculus-1/integrals/fundamental-theorem-part-2) | [Fundamental Theorem of Calculus](/theorems/fundamental-theorem-of-calculus-part-2) |

## MATH105 and the ABU calculus sequence

| Course | Level and semester | Units | Calculus content |
| --- | --- | --- | --- |
| MATH105 Differential and Integral Calculus | 100, first | 2 | Limits, differentiation, integration |
| MATH104 Conic Sections and Applications of Calculus | 100, second | 2 | Conics, rates of change, curve sketching, areas, volumes |
| MATH201 Mathematical Methods I | 200, first | 3 | Taylor series, reduction formulae, partial differentiation |

MATH105 runs in the same semester as MATH101 Sets and Number System and MATH103 Trigonometry and Coordinate Geometry, which supply the algebra and trigonometry you will use while differentiating. MATH105 is the listed prerequisite for [MATH201](/courses/abu-zaria-math-201), so the techniques you build now carry straight into 200 level.

## Example problems at the MATH105 level

**1. Odd or even?** Show that $f(x) = x^3 - x$ is odd.

$$f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x)$$

So the graph is symmetric about the origin.

**2. First principles.** Differentiate $f(x) = \dfrac{1}{x}$ from the definition.

$$f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h} = \lim_{h \to 0} \frac{-h}{h\,x(x + h)} = \lim_{h \to 0} \frac{-1}{x(x + h)} = -\frac{1}{x^2}$$

**3. A higher derivative.** If $y = x^4 - 3x^2$, then

$$\frac{dy}{dx} = 4x^3 - 6x, \qquad \frac{d^2y}{dx^2} = 12x^2 - 6$$

**4. Evaluating a definite integral.**

$$\int_1^2 (3x^2 + 1)\,dx = \Big[x^3 + x\Big]_1^2 = (8 + 2) - (1 + 1) = 8$$

## How to do well in ABU MATH105

- **Practice first principles until it is mechanical.** Expanding, cancelling $h$ and taking the limit is the same pattern every time. Use [the limit definition](/learn/calculus-1/the-derivative/limit-definition) lesson for repetition.
- **Learn the standard derivatives and integrals cold.** With only 2 credit units, the course moves quickly. Keep the [calculus formula sheet](/formulas) beside you while practicing.
- **Check symmetry before you sketch.** Testing $f(-x)$ for odd or even functions saves time and catches graphing errors.
- **Watch for common slips.** Forgetting the chain rule factor and dropping the constant of integration are the classic ones. Read [common calculus mistakes](/blog/common-calculus-mistakes).
- **Rehearse under exam conditions** with [practice exams](/practice), and follow our plan for [how to prepare for a calculus exam](/blog/how-to-prepare-for-a-calculus-exam).

> **Tip:** when integrating, differentiate your answer to check it. If you get back the original integrand, the answer is right.

## Frequently asked questions

### What is MATH105 at ABU Zaria?

MATH105 Differential and Integral Calculus is a 2-credit-unit, first-semester, 100-level course in the ABU Department of Mathematics. It covers functions of a real variable and their graphs, limits and continuity, differentiation from first principles and by standard techniques, higher derivatives, integration techniques and definite integrals.

### What comes after MATH105?

In the second semester, MATH104 Conic Sections and Applications of Calculus applies differentiation and integration to rates of change, curve sketching, approximations, areas and volumes. At 200 level, MATH201 Mathematical Methods I lists MATH105 as its prerequisite.

### Is MATH105 hard?

Limits are treated intuitively and definite integrals are for evaluation only, so the course focuses on technique. Students who practice differentiation and integration regularly, rather than only before exams, usually do well.

### What should I revise before MATH105?

Revise O level algebra, indices, logarithms and the trigonometric ratios. Being able to factorize and simplify fractions quickly makes first-principles questions much easier.

### How is MATH105 different from MATH104?

MATH105 teaches how to differentiate and integrate. MATH104 uses those skills for conic sections, rates of change, curve sketching, small increments, Newton's approximation, areas and volumes, and first-order differential equations.