ABU MATH104 Conic Sections and Applications of Calculus Guide

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 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 and the ABU sequence

MATH104 runs alongside MATH102 Algebra and MATH106 Vectors and Dynamics. If differentiation still feels slow, revisit the MATH105 guide before tackling applications, and see the MATH201 guide 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).

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

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

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 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 guide shows the method step by step.
• Keep formulas handy. The calculus formula sheet has the area and volume formulas used in this course.
• Practice under time pressure with practice exams, and avoid the slips listed in 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.