# FUTA MTS 102 Introductory Mathematics II: Study Guide, Practice

> A study guide for FUTA MTS 102 (Introductory Mathematics II): limits, differentiation, curve sketching, integration methods, areas and volumes, with practice.

Source: https://practicecalculus.com/courses/futa-mts-102
Updated: 2026-11-03

## At a glance

- **Course code:** MTS 102
- **Semester:** 100 level, second semester
- **Department:** Mathematical Sciences
- **Covers:** Functions of a real variable and their graphs, limits and continuity, differentiation from first principles, differentiation rules including inverse, implicit and parametric functions, kinematics, tangents and normals, maxima and minima, curve sketching, integration by substitution and partial fractions, trigonometric integrals, areas and volumes
- **Practice here:** [Calculus 1](/learn/calculus-1)

## Key takeaways

- MTS 102 Introductory Mathematics II is the second-semester, 100-level calculus course in the FUTA Department of Mathematical Sciences.
- It covers functions, limits and continuity, and differentiation from first principles through implicit and parametric differentiation.
- Applications include kinematics, tangents and normals, maxima and minima, and curve sketching.
- The integration part covers substitution, partial fractions and trigonometric integrals, with applications to areas and volumes.

**FUTA MTS 102**, Introductory Mathematics II, is the 100-level calculus course in the Department of Mathematical Sciences at the Federal University of Technology, Akure. It runs in the second semester and takes students from functions and limits to differentiation, integration and their applications. This guide maps every MTS 102 topic to free practice, so you can learn each idea by solving problems with instant feedback.

## What FUTA MTS 102 covers

The department's course synopsis groups MTS 102 into seven parts:

- **Functions of a real variable:** definitions, types of function and their graphs.
- **Limits and continuity:** the idea of a limit, rates of change, differentiation from first principles and continuity.
- **Techniques of differentiation:** sums and differences, products, quotients, second and higher derivatives, function of a function (the chain rule), inverse functions, implicit functions and parametric equations.
- **Applications of differentiation:** kinematics, tangents and normals to a curve, and maximum and minimum values.
- **Curve sketching:** turning points and maximum and minimum values of a curve.
- **Integration:** methods of integration, rational algebraic fractions, substitution, partial fractions and trigonometric functions.
- **Applications of integration:** geometry and mechanics, areas and volumes.

That is a complete **Calculus 1** course, with integration by partial fractions added from what is usually early Calculus 2.

## MTS 102 topic map

| MTS 102 topic | Practice | Read |
| --- | --- | --- |
| Functions and their graphs | [Functions and notation](/learn/calculus-1/functions-warmup/function-notation) | [Polynomial functions](/functions/polynomial-functions), [piecewise functions](/functions/piecewise-functions) |
| Limits | [What is a limit?](/learn/calculus-1/limits/what-is-a-limit) | [What is a limit?](/blog/what-is-a-limit) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity](/blog/continuity) |
| Differentiation from first principles | [The limit definition](/learn/calculus-1/the-derivative/limit-definition) | [Limit definition of the derivative](/blog/limit-definition-of-the-derivative) |
| Sum, product and quotient rules | [The quotient rule](/learn/calculus-1/derivative-rules/quotient-rule) | [Product and quotient rules](/blog/product-and-quotient-rules) |
| Function of a function | [Chain rule: any function](/learn/calculus-1/derivative-rules/chain-rule-general) | [Chain rule](/rules/chain-rule) |
| Inverse functions | [Exponentials and logarithms](/learn/calculus-1/derivative-rules/exponential-and-log-derivatives) | [Inverse function rule](/rules/inverse-function-rule) |
| Implicit functions | [Implicit differentiation](/learn/calculus-1/derivative-rules/implicit-differentiation) | [Implicit differentiation](/blog/implicit-differentiation) |
| Parametric equations | [Parametric curves](/learn/calculus-2/power-series-parametric-polar/parametric-curves) | [Chain rule](/blog/chain-rule) |
| Kinematics | [Related rates](/learn/calculus-1/applications-of-derivatives/related-rates) | [Related rates](/blog/related-rates) |
| Tangents and normals | [Tangent lines](/learn/calculus-1/the-derivative/tangent-lines) | [Linear approximation](/blog/linear-approximation) |
| Maxima, minima and turning points | [Critical points](/learn/calculus-1/applications-of-derivatives/critical-points) | [Curve sketching with derivatives](/blog/curve-sketching-with-derivatives) |
| Integration by substitution | [U-substitution](/learn/calculus-1/integrals/u-substitution) | [U-substitution](/blog/u-substitution) |
| Partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions](/blog/partial-fractions) |
| Trigonometric integrals | [Sine, cosine, eˣ and 1/x](/learn/calculus-1/integrals/trig-exponential-integrals) | [Antiderivatives](/blog/antiderivatives) |
| Areas | [Area between curves](/learn/calculus-1/applications-of-integrals/area-between-curves) | [Area between curves](/blog/area-between-curves) |
| Volumes | [Volumes with disks](/learn/calculus-1/applications-of-integrals/volumes-disks) | [Volumes of revolution](/blog/volumes-of-revolution) |

## How MTS 102 compares with other Nigerian first calculus courses

Many Nigerian universities teach a similar first calculus course at 100 level, with different codes and emphases.

| Course | Notable extras beyond core Calculus 1 |
| --- | --- |
| FUTA MTS 102 Introductory Mathematics II | Kinematics, parametric differentiation, partial fractions |
| [UNIBEN MTH 112 Calculus](/courses/uniben-mth-112) | Logarithmic differentiation, integration by parts, surface area of revolution |
| [UNN MTH 121 Elementary Mathematics II](/courses/unn-mth-121) | Close to a standard Calculus 1 course |

If you are studying from another university's materials, the core ideas match, so the practice lessons on this page serve all of them.

## Example problems at the MTS 102 level

**1. The quotient rule.** Differentiate $y = \dfrac{x}{x + 1}$.

$$\frac{dy}{dx} = \frac{(x + 1)(1) - x(1)}{(x + 1)^2} = \frac{1}{(x + 1)^2}$$

**2. Implicit differentiation.** Find the gradient of $x^2 + y^2 = 25$ at $(3, 4)$.

$$2x + 2y\frac{dy}{dx} = 0 \quad\Rightarrow\quad \frac{dy}{dx} = -\frac{x}{y} = -\frac{3}{4}$$

**3. Kinematics.** A particle moves with $s = t^3 - 6t^2 + 9t$. Then

$$v = \frac{ds}{dt} = 3t^2 - 12t + 9 = 3(t - 1)(t - 3), \qquad a = \frac{dv}{dt} = 6t - 12$$

so the particle is momentarily at rest at $t = 1$ and $t = 3$.

**4. Partial fractions.** Since $\dfrac{1}{x(x + 1)} = \dfrac{1}{x} - \dfrac{1}{x + 1}$,

$$\int \frac{dx}{x(x + 1)} = \ln|x| - \ln|x + 1| + C = \ln\left|\frac{x}{x + 1}\right| + C$$

**5. An area.** The area between $y = 4 - x^2$ and the $x$-axis for $0 \le x \le 2$ is

$$\int_0^2 (4 - x^2)\,dx = \left[4x - \frac{x^3}{3}\right]_0^2 = 8 - \frac{8}{3} = \frac{16}{3}$$

## How to do well in FUTA MTS 102

- **Build the differentiation rules one at a time.** Master the [power rule](/learn/calculus-1/derivative-rules/power-rule), then products, quotients and the chain rule, before moving on to implicit and parametric differentiation.
- **Connect calculus to motion.** In kinematics questions, velocity is $\frac{ds}{dt}$ and acceleration is $\frac{dv}{dt}$. Writing these down first prevents most errors.
- **Practice partial fractions separately.** Decomposing the fraction is algebra, not calculus, and it is where most marks are lost. Use the [partial fractions](/blog/partial-fractions) guide.
- **Keep the standard results nearby.** The [calculus formula sheet](/formulas) has every derivative and integral you need for MTS 102.
- **Rehearse with timed problems** from [practice exams](/practice), and read [common calculus mistakes](/blog/common-calculus-mistakes) before each test.

> **Tip:** after finding a turning point, always check which kind it is with the second derivative or a sign change in $\frac{dy}{dx}$. Questions on curve sketching almost always ask you to classify them.

## Frequently asked questions

### What is MTS 102 at FUTA?

MTS 102 Introductory Mathematics II is a second-semester, 100-level course in the Department of Mathematical Sciences at the Federal University of Technology, Akure. Its synopsis covers functions and graphs, limits and continuity, differentiation and its applications, integration methods, and areas and volumes.

### Is MTS 102 a calculus course?

Yes. Apart from the opening review of functions and graphs, every topic in the official synopsis is differential or integral calculus, so it lines up with a standard Calculus 1 course.

### Is MTS 102 hard?

It covers a lot in one semester: all the differentiation rules, curve sketching and several integration methods. Students who practice a few problems every day usually cope much better than those who rely on reading notes.

### How should I prepare for MTS 102?

Revise algebra, indices, logarithms, factorization and trigonometric identities, since they appear in almost every derivative and integral. Then work through limits and first-principles derivatives before the semester starts.

### What topics in MTS 102 carry over to later courses?

Differentiation rules, integration by substitution and partial fractions, and applications to areas and volumes are used constantly in later mathematics, physics and engineering courses, so it pays to master them now.