MATH1011 University Mathematics I

How MATH1011 is assessed

The four graded components published for this course total 100%. The course materials also list a 5% bonus through GEAR on top of that total; they do not say what GEAR involves, how the bonus is earned or whether it is capped, so treat the size as published and check the mechanism on the course Moodle site. The weighting table names the two in-semester components Test 1 and Test 2, while the lecture calendar marks the same two sittings Quiz 1 and Quiz 2; no course file maps one set of labels onto the other.

Current MATH1011 dates

Dates are as published in the course lecture calendar and the tutorial notice on the course Moodle site. Confirm exact deadlines and submission settings in the live LMS.

What MATH1011 covers

Twelve chapters follow the course notes from sets and counting through functions and limits to differentiation and integration.

Most early marks are lost when an absolute-value condition such as |x − 1| < 2 is converted into an interval carelessly, or when a set that contains another set as a single element is counted as though it contained that set's members.

Counting Methods and Basic Probability

Counting is the part of probability where the answer is decided before any arithmetic happens.

The question is whether order matters and whether repetition is allowed, and the four answers to that pair of questions give the four standard counting formulas.

Once a count is correct, a probability on a finite equally likely sample space is the count of the favourable outcomes divided by the count of all of them.

Sigma Notation and the Binomial Theorem

Summation and product notation exist so that a long expression can be manipulated by its index rather than rewritten in full.

Shifting an index, splitting a sum and pulling a constant out are the three moves used constantly later in integration.

The binomial theorem then gives every term of a power of a sum in closed form, which is what makes a specific coefficient obtainable without expanding anything.

Mathematical Induction in Two Forms

Induction is the course's first formal proof technique and the first place where presentation is assessed alongside the result.

The first principle verifies one starting value and derives each case from the one immediately before it. The second principle, in the form this course states, verifies two starting values and derives each case from the two before it, which is what a relation linking three consecutive terms needs.

A proof that omits its base cases establishes nothing at all.

Functions, Domains and Composition

A function is the pairing of each input with exactly one output, and almost every error in this chapter is a domain error rather than an algebra error. Composition inherits a restriction from the inner function that the simplified formula no longer shows.

The standard shapes are worth memorising outright, because recognising one saves the work of deriving its behaviour from scratch later in the calculus chapters.

Exponential and Logarithmic Functions

Exponential and logarithmic functions invert each other, so an equation in one is solved by moving to the other.

The log laws are what make that practical, but each application of them narrows or widens the set of admissible values.

Solving a logarithmic equation therefore has two stages: find the candidate roots, then test every candidate against the original argument conditions and discard those that fail.

Trigonometric Functions and Identities

Trigonometry arrives twice over: as ratios in a right triangle, and as coordinates on the unit circle that extend those ratios to any angle.

Radian measure is the version calculus needs, because the derivative formulas later in the course are only valid in radians. The identities are the working tools, and the skill being tested is picking the identity that removes the obstacle rather than reciting all of them.

Limits, Asymptotes and Continuity

A limit describes the value a function approaches, which is a different question from the value it takes.

Three cases must stay distinct: the two one-sided limits disagree, the function grows without bound, or the function oscillates. Only the first two have standard notation.

Continuity at a point then requires three separate facts to hold together, and an asymptote is read off the limits rather than guessed from the formula.

Differentiation Rules and Techniques

The derivative is defined as a limit of difference quotients, and the rules are labour-saving consequences of that definition.

Product, quotient and chain rules cover the ordinary cases; implicit differentiation handles a relation that cannot be solved for one variable; logarithmic differentiation handles a variable exponent or a long product.

Choosing between them is fast once the structure of the expression is read correctly.

Extrema, Concavity and Curve Sketching

A stationary point is where the derivative vanishes, which is not the same claim as a maximum or a minimum. The first derivative test reads the sign change around the point; the second derivative test reads the concavity at it and is silent when the second derivative is zero.

An absolute extremum on a closed interval also requires the endpoints to be compared, which is the step most often skipped.

Antiderivatives and Integration Methods

Integration is recognition before computation. Substitution undoes a chain rule and needs the derivative of the inner function present as a factor.

Integration by parts trades one integral for another and is chosen by which factor simplifies on differentiation.

Long division comes first whenever the numerator's degree is at least the denominator's, and partial fractions splits what remains into pieces with standard antiderivatives.

The Definite Integral and Area

The definite integral is a signed area, so a region below the axis contributes negatively and a geometric area question needs the integral split at every crossing.

The fundamental theorem connects it to antiderivatives and makes evaluation possible. The trapezoidal rule approximates an integral that cannot be evaluated exactly, and an improper integral is defined as a limit, so it converges or diverges rather than simply having a value.

How to use this guide

Work one chapter at a time and in the order above, because each chapter's notation is assumed by the next.

For each topic, state the method in one sentence before touching a problem, then work the example in full, then attempt the practice without looking back.

When an attempt fails, find the first line that is wrong rather than restarting: the first wrong line is the thing you did not know, and everything after it is a consequence.

Evidence and assessment control

The component names and weights below are the ones published in the course materials available here.

The explanations, worked reasoning and practice questions are written independently for this guide and are not the University's questions or marking scheme. Confirm dates, submission settings and any operational change on the course Moodle site, which is the surface that controls them.

Worked example · free

Turn an absolute-value condition into an interval and intersect

Key terms

Rule Method

The Rule Method specifies a set by a property rather than a list, written as the elements x for which a stated condition holds. It is the only one of the three specification methods that handles an infinite or uncountable set.

Cardinality

Cardinality is the number of distinct elements in a set. Repeated entries are counted once, and a set that appears inside another set contributes one element regardless of how many members it has.

Combination

A combination is a selection in which order carries no information, counted by the binomial coefficient. It is the right count whenever rearranging the chosen items would not produce a different outcome.

Sigma Notation

Sigma notation compresses a sum into an index, a lower limit and an upper limit. Its value is that a sum can then be shifted, split or scaled by manipulating the index instead of writing every term.

Binomial Coefficient

The binomial coefficient counts the ways of choosing r items from n without regard to order, and supplies the coefficient of each term when a binomial is raised to a power.

Composite Function

A composite function applies one function to the output of another. Its domain is restricted by the inner function as well as the outer one, which a simplified formula for the composite no longer displays.

Natural Logarithm

The natural logarithm is the inverse of the exponential function with base e. It converts a product into a sum and an exponent into a factor, which is why it linearises multiplicative expressions.

Radian

A radian measures an angle as the arc length it subtends on a unit circle. Calculus uses radians because the standard derivative formulas for the trigonometric functions are false in degrees.

One-Sided Limit

A one-sided limit describes the behaviour of a function as the input approaches a point from one direction only. A two-sided limit exists precisely when both one-sided limits exist and agree.

Vertical Asymptote

A vertical asymptote is a line the graph approaches as the function grows without bound near a finite input. It is established by a one-sided limit, not by the denominator vanishing on its own.

Chain Rule

The chain rule differentiates a composite by multiplying the outer derivative, evaluated at the inner function, by the inner derivative. Nested compositions require one factor for every layer present.

Stationary Point

A stationary point is an input where the first derivative is zero. It may be a maximum, a minimum or neither, so a separate test is always needed before it is classified.

Improper Integral

An improper integral has an infinite limit or an unbounded integrand, and is defined as a limit of ordinary definite integrals. It either converges to a number or diverges.

MATH1011 FAQ

Questions students ask in each MATH1011 chapter

Does the set {2, {3, 4}, 5} have three members or four?

Three. The inner braces wrap 3 and 4 into one object, so the set holds the number 2, a two-member collection, and the number 5. That inner collection is neither the number 3 nor the number 4, which is why an intersection with a set of integers throws it away. Chapter 01 · Sets, Intervals and Set Operations

How do I tell a permutation question from a combination question?

Picture the result already chosen, swap two of the chosen items, and ask whether anything has changed. If the swap produces a different outcome, such as a different seating plan or a different passcode, you are counting permutations. If the same people or cards have merely been written down in another order, it is a combination and the orderings have to be divided out. Chapter 02 · Counting Methods and Basic Probability

How do I move the starting index of a sum without changing its value?

Decide what the new starting value should be, note how far the counter has to travel, then move both limits and the rule by that same amount in that same direction. Confirm with three numbers afterwards: the first term, the last term and the count of terms must all come out as they did before the move. Chapter 03 · Sigma Notation and the Binomial Theorem

Why do several checked examples not prove an induction claim?

A finite list cannot rule out a later counterexample. Induction needs a base case to start the chain and a general implication that transports the statement from an arbitrary index to the next required index. The checked values may suggest the formula, but the implication is what reaches every positive integer. Chapter 04 · Mathematical Induction in Two Forms

Why is a formula not enough to identify a function?

The same formula can define different functions when its allowed inputs or declared output set changes. A rule such as the square root on all nonnegative real numbers is not the same function as the identical rule restricted to one interval, even where their numerical outputs agree. Chapter 05 · Functions, Domains and Composition

Why must logarithmic roots be checked after solving?

Combining logs and solving the resulting algebraic equation can introduce candidates for which an original logarithmic argument is zero or negative. Those candidates satisfy the transformed equation but never belonged to the original domain, so they must be rejected explicitly. Chapter 06 · Exponential and Logarithmic Functions

Why do the calculus formulas insist on radians rather than degrees?

Because the derivative rules for sine and cosine come from a limit that only behaves simply when the angle is measured in radians. Work in degrees and a factor of π/180 appears in front of every one of those derivatives, so a gradient read off a calculator left in degree mode is about 57 times too small. Chapter 07 · Trigonometric Functions and Identities

Can a function have a limit somewhere it is not even defined?

Yes, and that case turns up constantly. A limit inspects inputs near the target and never the target itself, so a fraction that reduces to something harmless once a shared factor cancels has a perfectly good limit at the very input it skips. The graph shows a hole at that spot. Chapter 08 · Limits, Asymptotes and Continuity

When is it worth taking logarithms before differentiating?

Take them when the variable sits up in an exponent, so that neither the power rule nor the exponential rule applies, or when four or more moving factors are piled into one fraction. Logarithms turn that exponent into a coefficient and the product into a sum, which the ordinary rules can then handle comfortably. Chapter 09 · Differentiation Rules and Techniques

Does f prime equal zero always give a maximum or minimum?

No. It gives a stationary candidate. Classify it by checking how the first derivative changes sign, or use the second derivative when that test is conclusive. A horizontal tangent can occur without either kind of extremum if the function keeps increasing or keeps decreasing through the point. Chapter 10 · Extrema, Concavity and Curve Sketching

Why must an indefinite integral include a constant?

Differentiation removes every constant, so infinitely many functions share the same derivative. The arbitrary constant records that entire family. Omitting it turns a general antiderivative question into one unexplained representative and loses information the original integrand cannot recover. Chapter 11 · Antiderivatives and Integration Methods

Why can a definite integral be smaller than the enclosed area?

A definite integral is signed: regions below the axis subtract from regions above it. Geometric area keeps every region positive, so it requires splitting at each sign change and reversing the sign of any below-axis contribution before adding the pieces. Chapter 12 · The Definite Integral and Area

How to study for the exam

Treat each chapter as a method plus a decision. Write the method down in one sentence from memory, work the chapter example without notes, then do the matching question-bank section and compare against the published answers. Keep a single page of the errors you actually made, sorted by first wrong line rather than by topic, and reread it before each test.

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