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University of Benin (UNIBEN)

UNIBEN MTH 112 Calculus: Topics, Study Guide and Practice

A study guide for UNIBEN MTH 112 (Calculus): limits, differentiation rules, stationary points, integration by substitution and parts, volumes and surfaces.

University of Benin (UNIBEN) Updated 4 min read

At a glance

Course code
MTH 112
Credits
3
Semester
100 level, first semester (core course)
Covers
Functions and graphs, limits and continuity, rates of change, tangents and normals, differentiation rules including implicit, logarithmic and parametric differentiation, the binomial expansion, stationary points, integration by substitution and by parts, definite integrals, volumes and surface areas of revolution
Taken with
MTH 110 Algebra and Trigonometry (first semester)
Later calculus courses
MTH 218 Mathematical Methods I and MTH 213 Vector Analysis (200 level)
Practice here
Calculus 1
On this page
  1. What UNIBEN MTH 112 covers
  2. MTH 112 topic map
  3. MTH 112 in the UNIBEN sequence
  4. Example problems at the MTH 112 level
  5. How to do well in UNIBEN MTH 112

UNIBEN MTH 112, Calculus, is the first calculus course at the University of Benin. It is a 3-credit core course in the first semester of 100 level for students in the Department of Mathematics, and it also appears in the first year of the combined Mathematics with Chemistry programme. This guide maps every MTH 112 topic to free practice, so you can learn each idea by solving problems with instant feedback.

What UNIBEN MTH 112 covers

The department's course description begins with elementary functions of one real variable and their graphs (polynomial, rational and trigonometric), limits and the idea of continuity, rates of change, and tangents and normals to a curve. Differentiation is introduced as the limit of a rate of change, followed by the product, quotient and function of a function (chain) rules, implicit differentiation, derivatives of trigonometric, inverse trigonometric and exponential functions, and logarithmic and parametric differentiation. The binomial expansion for any index and stationary values (maxima, minima and points of inflexion) complete the differential part.

The integral part treats integration as the inverse of differentiation, integration by substitution and by parts, definite integrals, and volumes and surface areas of revolution.

That is a full Calculus 1 course with a little extra: logarithmic and parametric differentiation, integration by parts and surface area of revolution usually appear early in Calculus 2 elsewhere.

MTH 112 topic map

MTH 112 topic Practice Read
Functions and their graphs Functions and notation Polynomial functions, rational functions
Limits and continuity What is a limit? What is a limit?, continuity
Rate of change, tangents and normals Tangent lines Limit definition of the derivative
Product, quotient and chain rules Chain rule: any function Product rule, quotient rule, chain rule
Implicit differentiation Implicit differentiation Implicit differentiation
Trig and inverse trig derivatives Sine, cosine and tangent Derivatives of trig functions, arcsine
Exponential and logarithmic differentiation Exponentials and logarithms Logarithmic differentiation
Parametric differentiation Parametric curves Chain rule
Stationary values and points of inflexion Critical points Curve sketching with derivatives
Integration as the inverse of differentiation Undoing derivatives Antiderivatives
Integration by substitution U-substitution U-substitution
Integration by parts Integration by parts Integration by parts
Definite integrals Evaluating definite integrals Fundamental Theorem of Calculus
Volume and surface area of revolution Volumes with disks Volumes of revolution

MTH 112 in the UNIBEN sequence

Course Level and semester Credits Calculus content
MTH 110 Algebra and Trigonometry 100, first 3 Foundations: series, trigonometry, complex numbers
MTH 112 Calculus 100, first 3 Limits, differentiation, integration
MTH 125 Differential Equations and Dynamics 100, second 3 First- and second-order differential equations, particle dynamics
MTH 218 Mathematical Methods I 200, first 3 Integration techniques, partial differentiation, Taylor series
MTH 213 Vector Analysis 200, first 3 Vector products, space curves, grad, div and curl

MTH 112 is where every later calculus course at UNIBEN starts. Continue with the guides to MTH 218 Mathematical Methods I and MTH 213 Vector Analysis.

Example problems at the MTH 112 level

1. Logarithmic differentiation. Differentiate y = x^x for x > 0.

Take logs: \ln y = x \ln x. Differentiate both sides:

\frac{1}{y}\frac{dy}{dx} = \ln x + 1 \quad\Rightarrow\quad \frac{dy}{dx} = x^x(\ln x + 1)

2. Parametric differentiation. If x = t^2 and y = t^3, then

\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{3t^2}{2t} = \frac{3t}{2}, \quad t \ne 0

3. Stationary points. For y = x^3 - 3x, y' = 3x^2 - 3 = 0 gives x = \pm 1. Since y'' = 6x:

  • at x = 1, y'' = 6 > 0, a minimum at (1, -2);
  • at x = -1, y'' = -6 < 0, a maximum at (-1, 2).

4. Integration by parts. With u = x and dv = \cos x\,dx,

\int x\cos x\,dx = x\sin x - \int \sin x\,dx = x\sin x + \cos x + C

How to do well in UNIBEN MTH 112

  • Learn the differentiation rules in order. Get the power rule automatic, then the product, quotient and chain rules, then implicit and logarithmic differentiation. Each builds on the last.
  • Use the formula sheet while practicing. The calculus formula sheet lists derivatives of inverse trigonometric functions, which are easy to mix up.
  • Classify every stationary point. Exam questions usually ask for maxima, minima and points of inflexion, so always finish with the second derivative test.
  • Avoid the classic mistakes. Missing chain rule factors and sign errors with \cos x are common. Read common calculus mistakes.
  • Practice under exam conditions with practice exams, using our guide on how to prepare for a calculus exam.

Tip: for integration by parts, choose u as the factor that gets simpler when differentiated. For x\cos x, x becomes 1, which is why the integral collapses so quickly.

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What is a limit?

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Frequently asked questions

What is MTH 112 at UNIBEN?
MTH 112 Calculus is a 3-credit, first-semester, 100-level core course in the University of Benin Department of Mathematics. It covers functions and graphs, limits and continuity, differentiation and its applications, and integration with applications to volumes and surface areas of revolution.

What is the difference between MTH 110 and MTH 112?
Both are first-semester 100-level courses. MTH 110 Algebra and Trigonometry covers the real number system, induction, sequences and series, quadratic equations, sets, trigonometry and complex numbers. MTH 112 is the calculus course.

Is MTH 112 hard?
MTH 112 covers a full first calculus course in one semester, including logarithmic and parametric differentiation and integration by parts. Students who keep up with weekly practice find it far easier than those who cram before exams.

What comes after MTH 112?
At 200 level, mathematics students take MTH 218 Mathematical Methods I, which covers integration techniques, partial differentiation, Lagrange multipliers and Taylor series, and MTH 213 Vector Analysis.

How should I prepare for MTH 112?
Revise algebra, indices, logarithms and trigonometric identities from secondary school, then practice limits and basic derivatives. The free Calculus 1 path on PracticeCalculus follows a similar order.

Written by the PracticeCalculus team

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Related university courses

UNIBEN MTH 218 Calculus 2 and 3 Mathematical Methods I A study guide for UNIBEN MTH 218 (Mathematical Methods I): integration techniques, partial derivatives, Lagrange multipliers, Taylor series and complex numbers. UNIBEN MTH 213 Calculus 3 Vector Analysis A study guide for UNIBEN MTH 213 (Vector Analysis): vector and triple products, vector equations, space curves, gradient, divergence and curl, with practice.