# ABU MATH104 Conic Sections and Applications of Calculus Guide

> A study guide for ABU Zaria MATH104: conic sections, rates of change, curve sketching, small increments, Newton's method, areas and volumes, with practice.

Source: https://practicecalculus.com/courses/abu-zaria-math-104
Updated: 2026-11-03

## At a glance

- **Course code:** MATH104
- **Credit units:** 2
- **Semester:** 100 level, second semester
- **Prerequisites:** O level Mathematics
- **Covers:** Parabola, ellipse and hyperbola in Cartesian and parametric form with tangents and normals, rates of change, curve sketching with asymptotes and turning points, small increments and errors, Newton's approximation, areas and volumes, first-order differential equations
- **Next course:** MATH202 Elementary Differential Equations I (lists MATH104 as its prerequisite)
- **Practice here:** [Calculus 1](/learn/calculus-1), from [critical points](/learn/calculus-1/applications-of-derivatives/critical-points) to [volumes](/learn/calculus-1/applications-of-integrals/volumes-disks)

## Key takeaways

- MATH104 is a 2-credit-unit, second-semester, 100-level course at Ahmadu Bello University, Zaria.
- Half the course is conic sections: parabolas, ellipses and hyperbolas with their tangents and normals.
- The other half applies calculus: rates of change, curve sketching, small increments, Newton's approximation, areas and volumes.
- It also introduces first-order differential equations and is the listed prerequisite for MATH202.

**ABU MATH104**, Conic Sections and Applications of Calculus, is the second-semester, 100-level course at Ahmadu Bello University, Zaria, where calculus is put to work. It carries 2 credit units and follows [MATH105 Differential and Integral Calculus](/courses/abu-zaria-math-105) from the first semester. This guide maps each MATH104 topic to free practice, so you can learn the applications by solving problems with instant feedback.

## What ABU MATH104 covers

The department's course outline splits MATH104 into two strands. The conics strand studies the parabola, ellipse, hyperbola and rectangular hyperbola, their Cartesian and parametric equations, eliminating the parameter, and finding tangents and normals.

The calculus strand covers rates of change (velocity, acceleration and other rates), curve sketching with asymptotes, maxima and minima, small increments, approximations and errors, Newton's approximation, and simple applications of integration to areas and volumes. It finishes with first-order differential equations.

In standard terms, this is the **applications half of Calculus 1**, plus analytic geometry of conics and a first look at differential equations.

## MATH104 topic map

| MATH104 topic | Practice | Read |
| --- | --- | --- |
| Conics in parametric form | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Quadratic function](/functions/quadratic-function) |
| Tangents and normals to curves | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) | [Implicit differentiation](/blog/implicit-differentiation) |
| Implicit differentiation of conics | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Implicit differentiation rule](/rules/implicit-differentiation) |
| Velocity, acceleration and rates of change | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates](/blog/related-rates) |
| Maxima and minima | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Optimization problems](/blog/optimization-problems) |
| Curve sketching and asymptotes | [Limits at infinity](/learn/calculus-1/limits/limits-at-infinity) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Small increments, approximations and errors | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| Newton's approximation | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Linear approximation](/blog/linear-approximation) |
| Areas | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves) |
| Volumes | [Volumes with disks](/learn/calculus-1/applications-of-integrals/volumes-disks) | [Volumes of revolution](/blog/volumes-of-revolution) |
| First-order differential equations | [Separable differential equations](/learn/calculus-2/improper-integrals-and-applications/separable-equations) | [Antiderivatives](/blog/antiderivatives) |

## MATH104 and the ABU sequence

| Course | Semester | Role |
| --- | --- | --- |
| MATH105 Differential and Integral Calculus | 100 level, first | Learn to differentiate and integrate |
| MATH104 Conic Sections and Applications of Calculus | 100 level, second | Apply calculus to geometry, motion, approximation, areas and volumes |
| MATH201 Mathematical Methods I | 200 level, first | Taylor series, reduction formulae, arc length, partial differentiation |
| MATH202 Elementary Differential Equations I | 200 level, second | Builds on MATH104's first-order equations |

MATH104 runs alongside MATH102 Algebra and MATH106 Vectors and Dynamics. If differentiation still feels slow, revisit the [MATH105 guide](/courses/abu-zaria-math-105) before tackling applications, and see the [MATH201 guide](/courses/abu-zaria-math-201) for what comes next.

## Example problems at the MATH104 level

**1. Eliminating a parameter.** The curve $x = t^2$, $y = 2t$ has $t = \frac{y}{2}$, so $x = \frac{y^2}{4}$, that is, $y^2 = 4x$: a parabola.

**2. Tangent and normal to a parabola.** Find the tangent and normal to $y^2 = 8x$ at $(2, 4)$.

Differentiating implicitly, $2y\,\dfrac{dy}{dx} = 8$, so $\dfrac{dy}{dx} = \dfrac{4}{y} = 1$ at $(2, 4)$.

$$\text{Tangent: } y = x + 2, \qquad \text{Normal: } y = -x + 6$$

**3. Small increments.** The radius of a sphere increases from $5$ cm to $5.02$ cm. Estimate the increase in volume.

$$\delta V \approx \frac{dV}{dr}\,\delta r = 4\pi r^2\,\delta r = 4\pi(25)(0.02) = 2\pi \text{ cm}^3$$

**4. Newton's approximation.** Improve $x_0 = 1.5$ as a root of $x^2 - 2 = 0$.

$$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1.5 - \frac{0.25}{3} \approx 1.4167$$

which is already close to $\sqrt{2} \approx 1.4142$.

## How to do well in ABU MATH104

- **Draw every conic.** Sketch the curve, mark the point, then find the tangent. A picture catches sign errors in the gradient immediately.
- **Use the derivative as a gradient.** Tangents, normals, small increments and Newton's method all come from the same idea: the derivative is the slope of the tangent line. Review [tangent lines](/learn/calculus-1/the-derivative/tangent-lines) if this is not yet automatic.
- **Set up rate problems in words first.** Write what is changing, what you know and what you want, then differentiate. Our [related rates](/blog/related-rates) guide shows the method step by step.
- **Keep formulas handy.** The [calculus formula sheet](/formulas) has the area and volume formulas used in this course.
- **Practice under time pressure** with [practice exams](/practice), and avoid the slips listed in [common calculus mistakes](/blog/common-calculus-mistakes).

> **Tip:** for a volume of revolution, always write $V = \pi \int y^2\,dx$ first and only then substitute the curve. Forgetting to square $y$ is the most common error in this topic.

## Frequently asked questions

### What is MATH104 at ABU Zaria?

MATH104 Conic Sections and Applications of Calculus is a 2-credit-unit course in the second semester of 100 level. It covers the parabola, ellipse and hyperbola, then uses calculus for rates of change, curve sketching, approximations and errors, Newton's method, areas, volumes and first-order differential equations.

### Do I need MATH105 before MATH104?

The department lists O level Mathematics as the formal prerequisite. In practice, MATH104 applies the differentiation and integration taught in MATH105 in the first semester, so those skills should be solid.

### What comes after MATH104?

MATH202 Elementary Differential Equations I at 200 level lists MATH104 as its prerequisite. Calculus itself continues in MATH201 Mathematical Methods I.

### Is MATH104 hard?

Conic sections involve a lot of algebra, and word problems on rates of change need careful setup. Students who draw diagrams and practice a few problems daily usually find it very manageable.

### How should I study for MATH104?

Learn the standard equations of each conic and how to differentiate them implicitly or parametrically. Then practice applied problems: a rate of change, a turning point, an area and a volume in each study session.