# UNN MTH 242 Mathematical Methods I: Study Guide and Practice

> A study guide for UNN MTH 242 (Mathematical Methods I): Taylor series, partial derivatives, Lagrange multipliers, line and multiple integrals, with practice.

Source: https://practicecalculus.com/courses/unn-mth-242
Updated: 2026-11-03

## At a glance

- **Course code:** MTH 242
- **Credit units:** 3 units
- **Level:** 200 level, second semester (standard four-year programme)
- **Covers:** Review of differentiation and integration, the mean value theorem, Taylor series, functions of two or three variables, partial derivatives, the chain rule, extrema, Lagrange multipliers, differentials and linear approximation, line integrals and multiple integrals
- **Builds on:** MTH 121 Elementary Mathematics II
- **Practice here:** [Calculus 3](/learn/calculus-3) plus [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) from Calculus 2

## Key takeaways

- MTH 242 Mathematical Methods I is a 3-unit, 200-level course at the University of Nigeria, Nsukka.
- It reviews single-variable calculus, adds the mean value theorem and Taylor series, then moves to functions of two or three variables.
- Partial derivatives, the chain rule, extrema and Lagrange multipliers make up the core of the multivariable part.
- It finishes with line integrals and multiple integrals, which are standard Calculus 3 topics.

**UNN MTH 242**, Mathematical Methods I, is the 200-level course where calculus at the University of Nigeria, Nsukka moves into several variables. It is a 3-unit major course in the second semester of the second year for mathematics students, and it builds directly on the single-variable calculus of [MTH 121](/courses/unn-mth-121). This guide maps every MTH 242 topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What UNN MTH 242 covers

The department's description opens with real-valued functions of one real variable: a review of differentiation and integration and their applications, the mean value theorem and Taylor series. It then turns to real-valued functions of two or three variables, covering partial derivatives, the chain rule, extrema, Lagrange's multipliers, increments, differentials and linear approximations. The course closes with the evaluation of line integrals and multiple integrals.

In standard terms, MTH 242 combines the end of **Calculus 2** (the mean value theorem and Taylor series) with most of a **Calculus 3** course. The main Calculus 3 topics it does not list are the big vector calculus theorems, which UNN places in the 300-level course MTH 324 Vector and Tensor Analysis.

## MTH 242 topic map

| MTH 242 topic | Practice | Read |
| --- | --- | --- |
| Review of differentiation | [Mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) | [Chain rule](/rules/chain-rule) |
| Review of integration | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts](/blog/integration-by-parts) |
| Mean value theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean value theorem](/theorems/mean-value-theorem) |
| Taylor series | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |
| Functions of two or three variables | [Functions of several variables](/learn/calculus-3/partial-derivatives/functions-of-several-variables) | [Partial derivatives](/blog/partial-derivatives) |
| Partial derivatives | [Partial derivatives](/learn/calculus-3/partial-derivatives/partial-derivatives) | [Clairaut's theorem](/theorems/clairauts-theorem) |
| The chain rule | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Partial derivatives](/blog/partial-derivatives) |
| Extrema | [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) |
| Increments, differentials, linear approximation | [Tangent planes and linear approximation](/learn/calculus-3/partial-derivatives/tangent-planes) | [Linear approximation](/blog/linear-approximation) |
| Line integrals | [Line integrals of functions](/learn/calculus-3/vector-calculus/line-integrals-of-functions) | [Green's theorem](/theorems/greens-theorem) |
| Multiple integrals | [Double integrals over rectangles](/learn/calculus-3/multiple-integrals/iterated-integrals) | [Double integrals](/blog/double-integrals) |

## MTH 242 in the UNN sequence

| Course | Level | Calculus content |
| --- | --- | --- |
| MTH 121 Elementary Mathematics II | 100 | Limits, derivatives, integrals, areas and volumes |
| MTH 221 Real Analysis I | 200 | Rigorous convergence, continuity, Rolle's and mean value theorems |
| MTH 242 Mathematical Methods I | 200 | Taylor series, several variables, line and multiple integrals |
| MTH 324 Vector and Tensor Analysis | 300 | Gradient, divergence, curl and the Green, Stokes and divergence theorems |

MTH 221 and MTH 242 look at some of the same results from two angles: Real Analysis I proves them, while Mathematical Methods I uses them to compute. If you need to rebuild your foundation, revisit [MTH 121](/courses/unn-mth-121) and [MTH 111](/courses/unn-mth-111).

## Example problems at the MTH 242 level

**1. Partial derivatives.** For $f(x, y) = x^2 y + y^3$,

$$f_x = 2xy, \qquad f_y = x^2 + 3y^2$$

**2. A Lagrange multiplier problem.** Maximize $f = xy$ subject to $x + y = 10$.

Set $\nabla f = \lambda \nabla g$ with $g = x + y$: $y = \lambda$ and $x = \lambda$, so $x = y$. The constraint gives $x = y = 5$, and the maximum is $f = 25$.

**3. A double integral.**

$$\int_0^1 \int_0^2 (x + y)\,dy\,dx = \int_0^1 \left[xy + \frac{y^2}{2}\right]_0^2 dx = \int_0^1 (2x + 2)\,dx = 1 + 2 = 3$$

**4. A line integral.** Evaluate $\int_C x\,dx + y\,dy$ along $\mathbf{r}(t) = (t, t^2)$ for $0 \le t \le 1$.

Here $dx = dt$ and $dy = 2t\,dt$, so

$$\int_0^1 \left(t + t^2 \cdot 2t\right)dt = \int_0^1 (t + 2t^3)\,dt = \frac{1}{2} + \frac{1}{2} = 1$$

## How to do well in UNN MTH 242

- **Refresh single-variable calculus first.** The course starts with a review, and every partial derivative is an ordinary derivative with the other variables held fixed. The [calculus formula sheet](/formulas) keeps the standard results in one place.
- **Draw the region before integrating.** Most errors in multiple integrals come from wrong limits. Sketch the region, then practice [double integrals over general regions](/learn/calculus-3/multiple-integrals/general-regions).
- **Check stationary points with the test.** After solving $f_x = f_y = 0$, classify each point with the second-derivative test rather than guessing.
- **Avoid classic slips.** Read [common calculus mistakes](/blog/common-calculus-mistakes) and see how Calculus 3 differs from earlier courses in [Calculus 1 vs 2 vs 3](/blog/calculus-1-vs-2-vs-3).
- **Rehearse with timed sets** using [practice exams](/practice) before each test.

> **Tip:** Lagrange multiplier problems always end with solving a small system of equations. Write out all the equations, including the constraint, before you start eliminating variables.

## Frequently asked questions

### What is MTH 242 at UNN?

MTH 242 Mathematical Methods I is a 3-unit course in the second semester of the second year. It reviews differentiation and integration, covers the mean value theorem and Taylor series, and then treats functions of several variables, partial derivatives, Lagrange multipliers, line integrals and multiple integrals.

### Is MTH 242 the same as multivariable calculus?

Mostly. Partial derivatives, extrema, Lagrange multipliers, linear approximation, line integrals and multiple integrals are all Calculus 3 topics. The opening review and Taylor series overlap with the end of a Calculus 2 course.

### What should I know before MTH 242?

You need confident differentiation and integration from MTH 121, including the chain rule and definite integrals. Comfort with vectors from MTH 122 also helps with line integrals.

### Is MTH 242 hard?

The new ideas are visualizing functions of two or three variables and setting up the limits of multiple integrals. The calculations themselves reuse single-variable rules, so students who practice setting up problems usually find it manageable.

### How can I prepare for MTH 242?

Refresh single-variable derivatives and integrals first, then preview partial derivatives and double integrals with short practice problems. The Calculus 3 path on PracticeCalculus covers these topics in order.