# ABU MATH201 Mathematical Methods I: Study Guide and Practice

> A study guide for ABU Zaria MATH201 (Mathematical Methods I): Leibniz's theorem, Taylor series, reduction formulae, improper integrals and partial derivatives.

Source: https://practicecalculus.com/courses/abu-zaria-math-201
Updated: 2026-11-03

## At a glance

- **Course code:** MATH201
- **Credit units:** 3
- **Semester:** 200 level, first semester
- **Prerequisites:** MATH105 or equivalent
- **Covers:** Successive differentiation and Leibniz's theorem, Taylor and Maclaurin series, tangents, normals and curvature, reduction formulae, arc length and area, improper integrals, partial derivatives, the chain rule, Jacobians, extrema, Lagrange multipliers and linear approximation
- **Earlier courses:** MATH105 Differential and Integral Calculus and MATH104 Conic Sections and Applications of Calculus
- **Practice here:** [Calculus 2](/learn/calculus-2) and [partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) in [Calculus 3](/learn/calculus-3)

## Key takeaways

- MATH201 Mathematical Methods I is a 3-credit-unit, first-semester, 200-level course at Ahmadu Bello University, Zaria.
- Its prerequisite is MATH105 Differential and Integral Calculus or an equivalent.
- The first part applies single-variable calculus: Leibniz's theorem, Taylor and Maclaurin series, curvature, reduction formulae, arc length and improper integrals.
- The second part is partial differentiation: partial derivatives, the chain rule, Jacobians, extrema and Lagrange multipliers.

**ABU MATH201**, Mathematical Methods I, is the first 200-level calculus course at Ahmadu Bello University, Zaria. It carries 3 credit units in the first semester, and its listed prerequisite is [MATH105 Differential and Integral Calculus](/courses/abu-zaria-math-105) or an equivalent. This guide maps every MATH201 topic to free practice, so you can learn each method by solving problems with instant feedback.

## What ABU MATH201 covers

The department's outline has two blocks. The first, applications of calculus, revises the techniques of differentiation, then covers successive differentiation and Leibniz's theorem, Taylor's and Maclaurin's series, tangents and normals to plane curves, and curvature. On the integration side it covers definite integrals, methods of integration, reduction formulae, arc length of a plane curve, the area enclosed by a plane curve and improper integrals.

The second block is partial differentiation: real-valued functions of two and three variables, partial derivatives, the chain rule, the Jacobian, extrema, Lagrange's multipliers, increments, differentials and linear approximations.

In standard terms, the first block is mostly **Calculus 2** (series, integration techniques, arc length and improper integrals) and the second is the differential half of **Calculus 3**.

## MATH201 topic map

| MATH201 topic | Practice | Read |
| --- | --- | --- |
| Revision of differentiation | [Mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) | [Chain rule](/rules/chain-rule), [logarithmic differentiation](/rules/logarithmic-differentiation) |
| Successive differentiation, Leibniz's theorem | [Concavity and the second derivative](/learn/calculus-1/applications-of-derivatives/second-derivative) | [Product rule](/rules/product-rule) |
| Taylor and Maclaurin series | [Common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Tangents, normals and curvature | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) | [Implicit differentiation](/blog/implicit-differentiation) |
| Methods of integration | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts), [partial fractions](/blog/partial-fractions) |
| Reduction formulae | [Powers of sine and cosine](/learn/calculus-2/integration-techniques/trig-integrals) | [Integration by parts rule](/rules/integration-by-parts-rule) |
| Arc length of a plane curve | [Arc length and surface area](/learn/calculus-2/improper-integrals-and-applications/arc-length-and-surface-area) | [Definite integral properties](/rules/definite-integral-properties) |
| Area enclosed by a plane curve | [Area in polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-area) | [Area between curves](/blog/area-between-curves) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives) |
| Chain rule and Jacobian | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Change of variables and the Jacobian](/learn/calculus-3/multiple-integrals/change-of-variables) |
| Extrema and Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Increments, differentials, linear approximation | [Tangent planes and linear approximation](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |

## MATH201 in the ABU sequence

| Course | Level | Units | Prerequisite |
| --- | --- | --- | --- |
| MATH105 Differential and Integral Calculus | 100, first semester | 2 | O level Mathematics |
| MATH104 Conic Sections and Applications of Calculus | 100, second semester | 2 | O level Mathematics |
| MATH201 Mathematical Methods I | 200, first semester | 3 | MATH105 |
| MATH203 Real Analysis I | 200, first semester | 3 | MATH105 |

MATH201 and MATH203 run in the same semester and share the same prerequisite: MATH201 computes with calculus while Real Analysis I proves why it works. If your 100-level skills need work, revisit the [MATH105](/courses/abu-zaria-math-105) and [MATH104](/courses/abu-zaria-math-104) guides.

## Example problems at the MATH201 level

**1. Leibniz's theorem.** Find the third derivative of $y = x^2 e^x$.

Only the first three terms survive, because the third derivative of $x^2$ is zero:

$$\frac{d^3y}{dx^3} = x^2 e^x + 3(2x)e^x + 3(2)e^x = e^x\left(x^2 + 6x + 6\right)$$

**2. A Maclaurin series.** For $f(x) = \ln(1 + x)$, the derivatives at $0$ give

$$\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots, \qquad -1 < x \le 1$$

**3. A reduction formula.** With $I_n = \int_0^{\pi/2} \sin^n x\,dx$, integration by parts gives $I_n = \dfrac{n - 1}{n} I_{n - 2}$. Since $I_0 = \dfrac{\pi}{2}$,

$$I_4 = \frac{3}{4} \cdot \frac{1}{2} \cdot \frac{\pi}{2} = \frac{3\pi}{16}$$

**4. An improper integral.**

$$\int_0^{\infty} e^{-2x}\,dx = \lim_{t \to \infty}\left[-\frac{1}{2}e^{-2x}\right]_0^t = \lim_{t \to \infty}\left(\frac{1}{2} - \frac{1}{2}e^{-2t}\right) = \frac{1}{2}$$

**5. A Jacobian.** For polar coordinates $x = r\cos\theta$, $y = r\sin\theta$,

$$\frac{\partial(x, y)}{\partial(r, \theta)} = \begin{vmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{vmatrix} = r\cos^2\theta + r\sin^2\theta = r$$

## How to do well in ABU MATH201

- **Make integration by parts automatic.** Reduction formulae, many improper integrals and Leibniz-style patterns all lean on it. Drill [integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) and [repeated parts](/learn/calculus-2/integration-techniques/repeated-parts-and-liate).
- **Memorize a few Maclaurin series.** Knowing $e^x$, $\sin x$, $\cos x$ and $\ln(1 + x)$ by heart lets you build many others quickly. The [calculus formula sheet](/formulas) lists them.
- **Hold variables fixed deliberately.** In partial differentiation, write which variables are constant before differentiating. It prevents the most common mistake in the second block.
- **Review the classic errors** in [common calculus mistakes](/blog/common-calculus-mistakes), especially sign errors in Taylor series and limits of improper integrals.
- **Practice with timed sets** from [practice exams](/practice) before each test.

> **Tip:** when an improper integral looks hard, compare it with $\int_1^{\infty} \frac{1}{x^p}\,dx$ first. Many convergence questions are settled by that one comparison.

## Frequently asked questions

### What is MATH201 at ABU Zaria?

MATH201 Mathematical Methods I is a 3-credit-unit course in the first semester of 200 level. It covers applications of calculus such as Leibniz's theorem, Taylor and Maclaurin series, curvature, reduction formulae, arc length and improper integrals, followed by partial differentiation of functions of two and three variables.

### What is the prerequisite for MATH201?

The department lists MATH105 Differential and Integral Calculus, or an equivalent course, as the prerequisite.

### Is MATH201 the same as Calculus 3?

Partly. The partial differentiation half, with the chain rule, extrema and Lagrange multipliers, is Calculus 3 material. The first half, with Taylor series, reduction formulae, arc length and improper integrals, matches Calculus 2.

### Is MATH201 hard?

It covers a lot in one semester and expects fast, accurate differentiation and integration from 100 level. The ideas are not difficult individually, so regular practice is the key.

### How should I prepare for MATH201?

Make sure integration by parts and substitution feel easy, since reduction formulae and improper integrals build on them. Then preview partial derivatives, which are ordinary derivatives with other variables held constant.