Covenant University MAT212 Mathematical Methods I: Study Guide

MAT212 Mathematical Methods I is the 200 level calculus methods course in the Covenant University Industrial Mathematics programme. It is a 3-unit course in the Alpha semester, taken alongside MAT211 Real Analysis I and MAT213 Differential Equations I. It extends the single-variable calculus of MAT121 to functions of two variables and adds the series and coordinate tools used across applied mathematics. This guide maps each MAT212 topic to free practice.

What MAT212 covers

The department's 200 level course list describes MAT212 as covering partial differentiation and its applications, including the classification of critical points of functions of two variables, and Lagrange multipliers. It also covers changing from Cartesian to polar, spherical and cylindrical coordinates, Taylor's and Maclaurin's series, differential coefficients of the nth order, and Leibnitz's rule, with an application to the solution of differential equations.

In a standard sequence these topics sit in Calculus 3 (partial derivatives, optimisation, coordinate systems) and the end of Calculus 2 (Taylor series).

MAT212 topic map

MAT212 in the Covenant sequence

MAT211 Real Analysis I, in the same semester, studies convergence of series and Taylor's theorem from a more theoretical angle, so the two courses reinforce each other. MAT213 Differential Equations I is also taken in the Alpha semester.

Example problems at the MAT212 level

These are original practice problems.

1. Partial derivatives. For f(x, y) = x^3y^2 - 4xy,

2. A saddle point. For f(x, y) = x^2 - y^2, the only critical point is (0, 0), and D = f_{xx}f_{yy} - f_{xy}^2 = (2)(-2) - 0 = -4 < 0, so it is a saddle point.

3. Leibnitz's rule. The third derivative of x^2 e^x is

4. Changing coordinates. The point with Cartesian coordinates (1, 1, \sqrt{2}) has cylindrical coordinates r = \sqrt{2}, \theta = \frac{\pi}{4}, z = \sqrt{2}, and spherical radius \rho = \sqrt{1 + 1 + 2} = 2.

How to do well in MAT212

• Treat partial derivatives as ordinary ones. Hold the other variable constant and use the rules from MAT121. If they feel rusty, revisit the chain rule and product rule.
• Follow a fixed routine for critical points. Solve f_x = f_y = 0, compute D, then classify. Writing every step avoids most errors.
• Memorise a few Maclaurin series. e^x, \sin x, \cos x and \ln(1 + x) cover most problems; keep them on your formula sheet.
• Mix topics when you revise. Use practice to switch between series, partial derivatives and optimisation, the way an exam does.