practicecalculus
Log in Start free
Ahmadu Bello University (ABU Zaria)

ABU MATH105 Differential and Integral Calculus: Study Guide

A study guide for ABU Zaria MATH105 (Differential and Integral Calculus): functions, limits, first principles, differentiation and integration, with practice.

Ahmadu Bello University (ABU Zaria) Updated 3 min read

At a glance

Course code
MATH105
Credit units
2
Semester
100 level, first semester
Prerequisites
O level Mathematics
Covers
Odd, even and periodic functions and their graphs, an intuitive treatment of limits and continuity, differentiation from first principles, techniques of differentiation, higher derivatives, techniques of integration and evaluating definite integrals
Next course
MATH104 Conic Sections and Applications of Calculus (second semester); MATH201 Mathematical Methods I at 200 level
Practice here
Calculus 1
On this page
  1. What ABU MATH105 covers
  2. MATH105 topic map
  3. MATH105 and the ABU calculus sequence
  4. Example problems at the MATH105 level
  5. How to do well in ABU MATH105

ABU MATH105, Differential and Integral Calculus, is the first calculus course at Ahmadu Bello University, Zaria. It is a 2-credit-unit course in the first semester of 100 level in the Department of Mathematics, with O level Mathematics as its only prerequisite. This guide maps every MATH105 topic to free practice, so you can learn each idea by solving problems with instant feedback.

What ABU MATH105 covers

The department's course outline has three parts. The first is functions of a real variable: odd, even and periodic functions, their symmetries and graphs, and limits and continuity treated intuitively rather than with formal proofs. The second is differentiation, from first principles and then by the general techniques, including higher derivatives. The third is integration: integration as the inverse of differentiation, the general techniques of integration, and the definite integral, where the outline says the focus is on evaluation.

That is the core of a Calculus 1 course. Applications such as rates of change, curve sketching, areas and volumes are not in MATH105; ABU teaches them in MATH104 in the second semester.

MATH105 topic map

MATH105 topic Practice Read
Functions of a real variable Functions and notation Polynomial functions
Odd, even and periodic functions, graphs Key trig values Sine function, power functions
Limits (intuitive) What is a limit? What is a limit?, how to evaluate limits
Continuity Continuity Continuity
Differentiation from first principles The limit definition Limit definition of the derivative
Techniques of differentiation The product rule Product rule, quotient rule, chain rule
Higher derivatives Concavity and the second derivative Power rule
Integration as the inverse of differentiation Undoing derivatives Antiderivatives
Techniques of integration U-substitution U-substitution, integration by parts
Evaluating definite integrals Evaluating definite integrals Fundamental Theorem of Calculus

MATH105 and the ABU calculus sequence

Course Level and semester Units Calculus content
MATH105 Differential and Integral Calculus 100, first 2 Limits, differentiation, integration
MATH104 Conic Sections and Applications of Calculus 100, second 2 Conics, rates of change, curve sketching, areas, volumes
MATH201 Mathematical Methods I 200, first 3 Taylor series, reduction formulae, partial differentiation

MATH105 runs in the same semester as MATH101 Sets and Number System and MATH103 Trigonometry and Coordinate Geometry, which supply the algebra and trigonometry you will use while differentiating. MATH105 is the listed prerequisite for MATH201, so the techniques you build now carry straight into 200 level.

Example problems at the MATH105 level

1. Odd or even? Show that f(x) = x^3 - x is odd.

f(-x) = (-x)^3 - (-x) = -x^3 + x = -(x^3 - x) = -f(x)

So the graph is symmetric about the origin.

2. First principles. Differentiate f(x) = \dfrac{1}{x} from the definition.

f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h} = \lim_{h \to 0} \frac{-h}{h\,x(x + h)} = \lim_{h \to 0} \frac{-1}{x(x + h)} = -\frac{1}{x^2}

3. A higher derivative. If y = x^4 - 3x^2, then

\frac{dy}{dx} = 4x^3 - 6x, \qquad \frac{d^2y}{dx^2} = 12x^2 - 6

4. Evaluating a definite integral.

\int_1^2 (3x^2 + 1)\,dx = \Big[x^3 + x\Big]_1^2 = (8 + 2) - (1 + 1) = 8

How to do well in ABU MATH105

  • Practice first principles until it is mechanical. Expanding, cancelling h and taking the limit is the same pattern every time. Use the limit definition lesson for repetition.
  • Learn the standard derivatives and integrals cold. With only 2 credit units, the course moves quickly. Keep the calculus formula sheet beside you while practicing.
  • Check symmetry before you sketch. Testing f(-x) for odd or even functions saves time and catches graphing errors.
  • Watch for common slips. Forgetting the chain rule factor and dropping the constant of integration are the classic ones. Read common calculus mistakes.
  • Rehearse under exam conditions with practice exams, and follow our plan for how to prepare for a calculus exam.

Tip: when integrating, differentiate your answer to check it. If you get back the original integrand, the answer is right.

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.

Practice this lesson

Frequently asked questions

What is MATH105 at ABU Zaria?
MATH105 Differential and Integral Calculus is a 2-credit-unit, first-semester, 100-level course in the ABU Department of Mathematics. It covers functions of a real variable and their graphs, limits and continuity, differentiation from first principles and by standard techniques, higher derivatives, integration techniques and definite integrals.

What comes after MATH105?
In the second semester, MATH104 Conic Sections and Applications of Calculus applies differentiation and integration to rates of change, curve sketching, approximations, areas and volumes. At 200 level, MATH201 Mathematical Methods I lists MATH105 as its prerequisite.

Is MATH105 hard?
Limits are treated intuitively and definite integrals are for evaluation only, so the course focuses on technique. Students who practice differentiation and integration regularly, rather than only before exams, usually do well.

What should I revise before MATH105?
Revise O level algebra, indices, logarithms and the trigonometric ratios. Being able to factorize and simplify fractions quickly makes first-principles questions much easier.

How is MATH105 different from MATH104?
MATH105 teaches how to differentiate and integrate. MATH104 uses those skills for conic sections, rates of change, curve sketching, small increments, Newton's approximation, areas and volumes, and first-order differential equations.

Written by the PracticeCalculus team

We build interactive calculus lessons and write every page to be clear and correct. Spotted something off? Tell us and we will fix it.

Related university courses

ABU MATH104 Calculus 1 Conic Sections and Applications of Calculus A study guide for ABU Zaria MATH104: conic sections, rates of change, curve sketching, small increments, Newton's method, areas and volumes, with practice. ABU MATH201 Calculus 2 and 3 Mathematical Methods I A study guide for ABU Zaria MATH201 (Mathematical Methods I): Leibniz's theorem, Taylor series, reduction formulae, improper integrals and partial derivatives.