# MTH 201 Mathematical Methods I: Topics, Study Guide, Practice

> MTH 201 Mathematical Methods I study guide: mean value theorem, Taylor series, partial derivatives, Lagrange multipliers and multiple integrals, with practice.

Source: https://practicecalculus.com/courses/nuc-mth-201
Updated: 2026-11-03

## At a glance

- **Course code:** MTH 201 (Mathematical Methods I)
- **Credit units:** 2 units, compulsory (30 lecture hours)
- **Level:** 200 level, core in the CCMAS B.Sc. Mathematics programme and in computing programmes
- **Covers:** Review of differentiation and integration, the mean value theorem, Taylor series, functions of two and three variables, partial derivatives and the chain rule, extrema, Lagrange multipliers, differentials and linear approximation, line integrals and multiple integrals
- **Comes after:** MTH 102 Elementary Mathematics II (Calculus)
- **Practice here:** [Calculus 2](/learn/calculus-2) series units and [Calculus 3](/learn/calculus-3)

## Key takeaways

- MTH 201 Mathematical Methods I is the 200 level calculus course in the NUC CCMAS, worth 2 units.
- It reviews MTH 102, adds the mean value theorem and Taylor series, then moves to functions of several variables.
- Partial derivatives, Lagrange multipliers, line integrals and multiple integrals make up the multivariable half, the core of a Calculus 3 course.
- PracticeCalculus has a lesson for almost every MTH 201 topic in its Calculus 2 and Calculus 3 paths.

**MTH 201 Mathematical Methods I** is the main 200 level calculus course in the National Universities Commission's Core Curriculum and Minimum Academic Standards (CCMAS). It is a compulsory 2-unit course in the B.Sc. Mathematics programme, and it also appears in the CCMAS computing programmes. It takes the single-variable calculus of MTH 102 and extends it to functions of several variables. This guide maps every MTH 201 topic to practice and reading on PracticeCalculus.

## What MTH 201 covers

The CCMAS outline starts with real-valued functions of a real variable and a review of differentiation, integration and their applications. It then covers the mean value theorem and Taylor series. The second half moves to real-valued functions of two and three variables: partial derivatives and the chain rule, extrema, Lagrange multipliers, increments, differentials and linear approximation, the evaluation of line integrals, and multiple integrals.

The learning outcomes ask students to explain real-valued functions, solve problems with the mean value theorem and Taylor expansions, and evaluate line, surface and volume integrals.

Compared with a standard sequence, MTH 201 takes the series and approximation ideas of **Calculus 2** and most of the differential and integral calculus of **Calculus 3**.

## MTH 201 topic map

| MTH 201 topic | Practice | Read |
| --- | --- | --- |
| Review of differentiation and integration | [Mixed rules practice](/learn/calculus-1/derivative-rules/derivative-rules-practice) | [Chain rule](/rules/chain-rule), [u-substitution](/blog/u-substitution) |
| Mean value theorem | [Tangent lines at work](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Mean value theorem](/theorems/mean-value-theorem), [Rolle's theorem](/theorems/rolles-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 and 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) |
| Chain rule for several variables | [The multivariable chain rule](/learn/calculus-3/partial-derivatives/multivariable-chain-rule) | [Gradient and directional derivatives](/blog/gradient-and-directional-derivatives) |
| Extrema | [Critical points and the second-derivative test](/learn/calculus-3/partial-derivatives/critical-points-and-second-derivative-test) | [Partial derivatives](/blog/partial-derivatives) |
| 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 201 and the courses around it

| Course | Level | Focus |
| --- | --- | --- |
| [MTH 102](/courses/nuc-mth-102) Elementary Mathematics II | 100 level | Single-variable calculus |
| MTH 201 Mathematical Methods I | 200 level | Taylor series and calculus of several variables |
| MTH 202 Elementary Differential Equations | 200 level | First and second order differential equations |
| [MTH 210](/courses/nuc-mth-210) Vector Analysis | 200 level | Vector algebra, space curves, gradient, divergence and curl |

MTH 201 and MTH 210 complement each other: MTH 201 supplies partial derivatives and multiple integrals, and MTH 210 supplies the vector language used for gradients and line integrals.

## Example problems at the MTH 201 level

These are original practice problems.

**1. A Maclaurin series.** Expanding $\ln(1 + x)$ about $x = 0$:

$$\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots, \qquad -1 < x \le 1$$

**2. Partial derivatives.** For $f(x, y) = x^2 y + \sin(xy)$,

$$f_x = 2xy + y\cos(xy), \qquad f_y = x^2 + x\cos(xy)$$

**3. Lagrange multipliers.** Maximise $f = xy$ subject to $x + y = 10$. Setting $\nabla f = \lambda \nabla g$ gives $y = \lambda$ and $x = \lambda$, so $x = y = 5$ and the maximum is $f = 25$.

**4. A double integral.**

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

## How to do well in MTH 201

- **Refresh MTH 102 first.** A partial derivative is an ordinary derivative with the other variables held constant, so weak differentiation rules show up immediately. The [formula sheet](/formulas) is a quick refresher.
- **Sketch regions before integrating.** Most mistakes in multiple integrals come from wrong limits. Draw the region, then decide the order of integration.
- **Use a checklist for Lagrange problems.** Write $\nabla f = \lambda \nabla g$, list every equation including the constraint, then solve. The [Lagrange multipliers guide](/blog/lagrange-multipliers) walks through the method.
- **Practice in short, regular sessions.** Use [practice](/practice) to mix topics, and read the [Calculus 1 vs 2 vs 3 guide](/blog/calculus-1-vs-2-vs-3) to see how MTH 201 fits the wider calculus picture.

## Frequently asked questions

### What is MTH 201?

MTH 201, Mathematical Methods I, is a 200 level course in the National Universities Commission core curriculum (CCMAS). It reviews single-variable calculus, covers the mean value theorem and Taylor series, and then studies functions of two and three variables: partial derivatives, extrema, Lagrange multipliers, linear approximation, line integrals and multiple integrals.

### What should I know before MTH 201?

You need the MTH 102 skills: limits, all the differentiation rules, and integration including substitution. Being comfortable with vectors from MTH 103 also helps with line integrals.

### Is MTH 201 multivariable calculus?

Largely, yes. The first part extends single-variable calculus with the mean value theorem and Taylor series, and the rest is multivariable calculus: partial derivatives, constrained optimisation with Lagrange multipliers, line integrals and multiple integrals.

### Is MTH 201 hard?

It covers a lot of ground in 2 units, so the pace is the main challenge. Students who are fluent in MTH 102 differentiation usually find partial derivatives straightforward; Lagrange multipliers and setting up limits for multiple integrals need the most practice.

### What is the difference between MTH 201 and MTH 202?

Both are 200 level CCMAS courses. MTH 201 Mathematical Methods I extends calculus to several variables, while MTH 202 Elementary Differential Equations covers first and second order differential equations and their applications.