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Covenant University MAT121 Calculus: Study Guide and Practice

Study guide for MAT121 Calculus at Covenant University: limits, derivatives, hyperbolic functions, integration techniques and reduction formulae, with practice.

Covenant University Updated 3 min read

At a glance

Course code
MAT121 (Calculus)
Credit units
3 units
Semester
Omega semester, 100 level (Industrial Mathematics course list)
Covers
Functions and limits, derivatives from first principles and by rule, higher derivatives, small increments and errors, extrema, definite integrals and applications, hyperbolic and inverse functions, logarithmic and implicit differentiation, substitution, integration by parts and reduction formulae
Earlier courses
MAT111 Algebra and MAT112 Trigonometry and Analytical Geometry (Alpha semester, 100 level)
Later calculus
MAT211 Real Analysis I and MAT212 Mathematical Methods I (Alpha semester, 200 level)
Practice here
Calculus 1 and integration techniques in Calculus 2
On this page
  1. What MAT121 covers
  2. MAT121 topic map
  3. MAT121 in the Covenant sequence
  4. Example problems at the MAT121 level
  5. How to do well in MAT121

MAT121 Calculus is the first calculus course in the Covenant University Department of Mathematics' Industrial Mathematics programme. It is a 3-unit course in the Omega semester of 100 level, after MAT111 Algebra and MAT112 Trigonometry and Analytical Geometry in the Alpha semester. It moves quickly from limits to advanced integration, so it covers more than a typical first calculus course. This guide maps every MAT121 topic to free practice with instant feedback.

What MAT121 covers

According to the department's 100 level course list, MAT121 starts with the concept and notation of functions, polynomial and rational functions, and the idea of a limit with techniques for finding limits. Derivatives are defined and computed from first principles, then by rule for constants, powers, sums, products, quotients, composite functions, circular functions and logarithmic and exponential functions. Applications include higher-order derivatives, small increments, approximations and errors, and extrema.

On the integration side, the course covers integrals of sums and scalar multiples, integrals of circular functions, and definite integrals with their general properties and applications in geometry, mechanics, biology and the social sciences. Later topics include derivatives of hyperbolic functions and inverse circular and hyperbolic functions, logarithmic differentiation, successive differentiation of implicit functions and of functions such as e^{at}\sin(bt + c), and "hard integration": irrational functions, substitution, integration by parts and reduction formulae.

That is all of a standard Calculus 1 course plus the integration techniques of early Calculus 2.

MAT121 topic map

MAT121 topic Practice Read
Functions, polynomial and rational functions Functions and notation Polynomial functions, rational functions
Limits and techniques for finding them Factor and cancel How to evaluate limits
Derivatives from first principles The limit definition Limit definition of the derivative
Powers, sums, products and quotients Product rule Power rule, quotient rule
Composite, circular, log and exponential functions Chain rule: trig Chain rule, derivatives of exponential and log functions
Small increments, approximations and errors Tangent lines at work Linear approximation
Extrema Critical points Optimization problems
Integration of sums and circular functions Sine, cosine, eˣ and 1/x Integral linearity rules
Definite integrals and applications Evaluating definite integrals Definite integral properties
Hyperbolic and inverse functions Exponentials and logarithms Hyperbolic functions, arcsine, inverse function rule
Logarithmic and implicit differentiation Implicit differentiation Logarithmic differentiation, implicit differentiation
Integration by substitution U-substitution u-substitution
Integration by parts and reduction formulae Repeated parts and LIATE Integration by parts
Irrational functions Trig substitution Trig substitution

MAT121 in the Covenant sequence

Course Semester Units Focus
MAT111 Algebra Alpha, 100 level 3 Indices, logs, polynomials, partial fractions, series, complex numbers, matrices
MAT112 Trigonometry and Analytical Geometry Alpha, 100 level 2 Trig, exponential, log and hyperbolic functions, lines, planes and conics
MAT121 Calculus Omega, 100 level 3 Single-variable differentiation and integration
MAT122 Intro. to Vector Analysis Omega, 100 level 2 Vectors, vector functions, grad, div and curl
MAT212 Mathematical Methods I Alpha, 200 level 3 Partial differentiation, Lagrange multipliers, Taylor series

Example problems at the MAT121 level

These are original practice problems.

1. Logarithmic differentiation. For y = x^x, take logs: \ln y = x\ln x. Differentiating,

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

2. A product of exponential and trig.

\frac{d}{dt}\left[e^{2t}\sin 3t\right] = e^{2t}\left(2\sin 3t + 3\cos 3t\right)

3. A reduction formula. Let I_n = \int x^n e^x\,dx. Integration by parts gives I_n = x^n e^x - nI_{n-1}, with I_0 = e^x. So

I_2 = x^2e^x - 2(xe^x - e^x) = e^x(x^2 - 2x + 2) + C

4. Small increments. If y = x^3 and x changes from 2 to 2.01, then \delta y \approx 3x^2\,\delta x = 12(0.01) = 0.12.

How to do well in MAT121

  • Lock in the derivative rules first. Everything from extrema to reduction formulae relies on them. Drill the chain rule until it is automatic.
  • Build integration in layers. Master basic antiderivatives, then substitution, then integration by parts, and only then reduction formulae. Our guide to integration by parts explains how to choose u.
  • Keep the hyperbolic identities with your formulas. The calculus formula sheet and the hyperbolic functions page put them in one place.
  • Practise little and often with practice, and review common calculus mistakes before each test.

Practice it now

What is a limit?

Solve real problems on this topic with instant feedback, hints and a step-by-step walkthrough for every answer. Free, no sign-up needed to start.

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Frequently asked questions

What is MAT121 at Covenant University?
MAT121 Calculus is a 3-unit, 100 level course taught in the Omega semester. The department's description covers functions and limits, differentiation from first principles and by rule, applications such as extrema and small errors, definite integrals, hyperbolic and inverse functions, and integration techniques up to reduction formulae.

What should I know before MAT121?
The Alpha semester courses MAT111 Algebra and MAT112 Trigonometry and Analytical Geometry cover the algebra, indices, logarithms, partial fractions and trigonometric, exponential and hyperbolic functions that MAT121 uses throughout.

What comes after MAT121?
In the 200 level Industrial Mathematics list, MAT211 Real Analysis I (sequences, series, L'Hospital's rule, Taylor's theorem) and MAT212 Mathematical Methods I (partial differentiation, Lagrange multipliers, Taylor and Maclaurin series) continue the calculus thread in the Alpha semester.

Is MAT121 hard?
It covers a lot for one course, from first-principles derivatives to reduction formulae. Students who practise each new rule straight away usually cope well; leaving integration techniques until the exam is the most common way to fall behind.

How should I study for MAT121?
Learn the differentiation rules early and practise them daily, then build up integration step by step: basic antiderivatives, substitution, then integration by parts and reduction formulae. Short, frequent practice beats long revision sessions.

Written by the PracticeCalculus team

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Related university courses

Covenant University MAT122 Calculus 3 (vectors) Intro. to Vector Analysis Study guide for MAT122 Introduction to Vector Analysis at Covenant University: vector algebra, products, space curves, grad, div and curl, with practice. Covenant University MAT212 Calculus 3 Mathematical Methods I Study guide for MAT212 Mathematical Methods I at Covenant University: partial derivatives, critical points, Lagrange multipliers and Taylor series.