# Covenant University MAT212 Mathematical Methods I: Study Guide

> Study guide for MAT212 Mathematical Methods I at Covenant University: partial derivatives, critical points, Lagrange multipliers and Taylor series.

Source: https://practicecalculus.com/courses/covenant-university-mat-212
Updated: 2026-11-03

## At a glance

- **Course code:** MAT212 (Mathematical Methods I)
- **Credit units:** 3 units
- **Semester:** Alpha semester, 200 level (Industrial Mathematics course list)
- **Covers:** Partial differentiation and its applications, classifying critical points of functions of two variables, Lagrange multipliers, polar, spherical and cylindrical coordinates, Taylor and Maclaurin series, nth derivatives and Leibnitz's rule
- **Same semester:** MAT211 Real Analysis I and MAT213 Differential Equations I
- **Next in the sequence:** MAT222 Mathematical Methods II (Omega semester, 200 level)
- **Practice here:** [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) in Calculus 3

## Key takeaways

- MAT212 Mathematical Methods I is a 3-unit course in the Alpha semester of 200 level in the Covenant University Industrial Mathematics list.
- It moves calculus to functions of two variables: partial derivatives, critical points and Lagrange multipliers.
- It also covers coordinate changes, Taylor and Maclaurin series, and Leibnitz's rule for nth derivatives.
- It follows MAT121 Calculus and MAT122 Intro. to Vector Analysis, and is followed by MAT222 Mathematical Methods II.

**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 $n$th 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 topic | Practice | Read |
| --- | --- | --- |
| Functions of two variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial differentiation | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Partial derivatives](/blog/partial-derivatives) |
| Higher partial derivatives | [Second partial derivatives](/learn/calculus-3/partial-derivatives/second-partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| Applications of partial differentiation | [Tangent planes and linear approximation](/learn/calculus-3/partial-derivatives/tangent-planes) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Classifying critical points | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Optimization problems](/blog/optimization-problems) |
| Lagrange multipliers | [Lagrange multipliers](/learn/calculus-3/partial-derivatives/lagrange-multipliers) | [Lagrange multipliers](/blog/lagrange-multipliers) |
| Polar coordinates | [Polar coordinates](/learn/calculus-2/power-series-parametric-polar/polar-coordinates) | [Double integrals](/blog/double-integrals) |
| Spherical and cylindrical coordinates | [Cylindrical and spherical coordinates](/learn/calculus-3/multiple-integrals/cylindrical-spherical) | [Double integrals](/blog/double-integrals) |
| Taylor and Maclaurin series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| nth derivatives and Leibnitz's rule | [The product rule](/learn/calculus-1/derivative-rules/product-rule) | [Product rule](/rules/product-rule) |

## MAT212 in the Covenant sequence

| Course | Semester | Units | Focus |
| --- | --- | --- | --- |
| [MAT121](/courses/covenant-university-mat-121) Calculus | Omega, 100 level | 3 | Single-variable differentiation and integration |
| [MAT122](/courses/covenant-university-mat-122) Intro. to Vector Analysis | Omega, 100 level | 2 | Vectors, grad, div and curl |
| MAT211 Real Analysis I | Alpha, 200 level | 3 | Sequences, series and convergence tests, Taylor's theorem |
| MAT212 Mathematical Methods I | Alpha, 200 level | 3 | Partial differentiation, Lagrange multipliers, coordinates, Taylor series |
| MAT222 Mathematical Methods II | Omega, 200 level | 3 | Fourier series and transforms, partial differential equations |

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 $f(x, y) = x^3y^2 - 4xy$,

$$f_x = 3x^2y^2 - 4y, \qquad f_y = 2x^3y - 4x, \qquad f_{xy} = 6x^2y - 4 = f_{yx}$$

**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

$$\sum_{k=0}^{3}\binom{3}{k}(x^2)^{(k)}(e^x)^{(3-k)} = e^x\left(x^2 + 3\cdot 2x + 3\cdot 2\right) = e^x(x^2 + 6x + 6)$$

**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](/rules/chain-rule) and [product rule](/rules/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](/formulas).
- **Mix topics when you revise.** Use [practice](/practice) to switch between series, partial derivatives and optimisation, the way an exam does.

## Frequently asked questions

### What is MAT212 at Covenant University?

MAT212 Mathematical Methods I is a 3-unit, 200 level course in the Alpha semester. It covers partial differentiation and its applications, classifying critical points of functions of two variables, Lagrange multipliers, changing between Cartesian, polar, spherical and cylindrical coordinates, Taylor and Maclaurin series, nth order derivatives and Leibnitz's rule.

### What is the difference between MAT212 and MAT222?

MAT212 Mathematical Methods I is multivariable differential calculus and series. MAT222 Mathematical Methods II, in the Omega semester, turns to Sturm-Liouville problems, orthogonal functions, Fourier series and transforms, and partial differential equations solved by separation of variables.

### Is MAT212 hard?

Partial derivatives feel familiar if your MAT121 differentiation is strong. Classifying critical points and setting up Lagrange multiplier systems take the most practice, because they combine several steps that each have to be right.

### How do I prepare for MAT212?

Revise the MAT121 differentiation rules, especially the chain rule, and the Maclaurin series of common functions. Then practise partial derivatives and the second-derivative test on many small examples before tackling Lagrange multipliers.