MATH1851 Calculus and ordinary differential equations
The Complete Exam Bible · Semester 1, 2026-27MATH1851 Overview
- The University of Hong Kong
- First Semester 2026-27
- Level 1 mathematics
- Calculus and differential equations
- Final exam published
What the course asks you to do
MATH1851 is organised into Part I Calculus and Part II Ordinary Differential Equations. The available Moodle files publish the complete assessment split and a detailed calculus schedule. They also contain lecture slides, tutorial sheets and revision drills for the opening calculus block.
- Assessment shape The final exam carries the largest published share, alongside two tests and assignment work in both course parts.
- Core habit State the limiting direction and the rule conditions before beginning algebra.
- Coverage boundary This edition teaches the complete available calculus lectures and routes later material back to Moodle.
- Revision move Work an original problem, audit its domain and sign, then explain the result in one sentence.
How MATH1851 is assessed
| Component | Weight | Format |
|---|---|---|
| Part I: 3 Assignments | 5% | Three assignments |
| Part I: Test | 15% | In class on 2 October 2026 |
| Part II: 3 Assignments | 5% | Three assignments |
| Part II: Test | 15% | Date shown as TBD on the course page |
| Final Exam | 60% | Date and paper details shown as TBD on the course page |
The five published numeric rows total 100%. The course page separately offers three bonus percentage points for class participation. It does not publish the final-exam duration, section split, question count, permitted-materials rule or a hurdle flag, so confirm those operational details on the live Moodle page and official timetable.
Assessment structure
Segment widths reproduce the five published weights and total 100%.
Current MATH1851 dates
| Date | Item | Control |
|---|---|---|
| 2 October 2026 | Part I Test | Published as an in-class test on the Moodle course page. |
Dates are as published in the Moodle course information for this offering. Confirm exact deadlines and submission settings in the live LMS.
What MATH1851 covers
Six chapters follow the complete available lecture sequence from one-sided limits and bounding arguments through continuity, tangent slopes, differentiation rules and implicit curves.
Limits and One-Sided Behaviour
nearby behaviour, one-sided agreement, limit laws, factorisation and rationalisationApproach is not value
A limit asks what outputs do when inputs move close to a target while remaining different from it. That is why a missing point, or a point plotted at an unexpected height, can leave the limit unchanged.
- Average and instantaneous rates of change
- Nearby behaviour versus the function value
- Left-hand and right-hand limits
- Three ways a finite limit can fail
- Limit laws and direct substitution
- Factorisation and conjugate rationalisation
Sandwich Theorem and Oscillatory Limits
bounding inequalities, damped oscillation, the fundamental trigonometric limit and failure conditionsBound the difficult function
The Sandwich Theorem is a comparison argument.
- The lower and upper inequality
- Equal outer limits as the forcing condition
- Oscillation with a shrinking envelope
- Absolute-value bounding across both sides
- Fundamental trigonometric limit reductions
- Unequal and unbounded bounds as failure cases
Limits at Infinity and Asymptotes
infinite limits, degree comparison, sign-aware end behaviour and three asymptote testsSeparate two meanings of infinity
An infinite limit keeps the input near a finite point and records output growing without bound. A limit at infinity sends the input far along the number line and asks about end behaviour.
- Infinite limits near a finite input
- Limits at positive and negative infinity
- Rational degree comparison
- One-sided sign charts at denominator zeros
- Horizontal and vertical asymptotes
- Oblique asymptotes through polynomial division
Continuity and the Intermediate Value Theorem
the three-condition test, discontinuity types, continuous extensions, composites and root existenceContinuity is a three-condition test
At an interior point, continuity requires a defined function value, an existing two-sided limit and equality between the two. The order matters because each condition answers a different question.
- Value, limit and agreement at a point
- Endpoint continuity on a closed interval
- Four discontinuity mechanisms
- Continuous extensions at removable holes
- Closure rules for combinations and composites
- Intermediate values and root existence
Derivatives, Tangents and Differentiability
difference quotients, tangent equations, derivative graphs and four differentiability failuresA derivative is a limit of slopes
Average rate of change is the slope of a secant through two curve points. Move the second point toward the first and the secant slopes may approach a finite limit.
- Secant slopes converging to a tangent
- Difference quotient definitions
- Point-slope equations for tangents
- Positive, negative and zero derivative regions
- Differentiability on intervals and endpoints
- Discontinuity, corners, cusps and vertical tangents
Differentiation Rules and Implicit Curves
product and quotient structure, nested chain factors, implicit differentiation and higher derivativesStructure chooses the rule
Before differentiating, identify the outermost operation.
- Constant, power, sum and trigonometric rules
- Product-rule structure and verification
- Quotient-rule signs and domain restrictions
- Nested layers in the chain rule
- Implicit differentiation workflow
- Higher derivatives and repeating trig cycles
Those materials emphasise theory, worked examples, group problem-solving and discussion, so this guide follows the same rhythm: define the object, choose a legal theorem or rule, work an original case and then diagnose a near-miss.
The reasoning chain across the six chapters
Limits begin by separating nearby behaviour from the value at a point.
One-sided agreement decides existence, while limit laws, factorisation and rationalisation supply algebraic routes. The Sandwich Theorem handles expressions whose oscillation is difficult but whose magnitude is easy to bound. Limits involving infinity then convert local blow-up and global end behaviour into vertical, horizontal and oblique asymptotes. Continuity adds a three-condition test and the Intermediate Value Theorem.
Derivatives reinterpret a limit as tangent slope and instantaneous rate, before product, quotient, chain and implicit rules make repeated calculations efficient.
How to use the available material
The Moodle schedule names later calculus topics and the ODE half, but their detailed teaching bodies are not present in the available files. This guide does not recreate those methods from memory.
Use the live lecture resources and the assigned text for the remaining syllabus, and use these chapters to master the limit and differentiation reasoning on which those later topics depend.
Evidence and originality
Assessment labels and weights follow the course page. Every worked problem and practice item in this guide is newly authored with fresh functions and values.
The tutorial and revision files are used only to identify the kinds of reasoning the course practises. No past examination paper, practice final or question-level university mark scheme is available here.
Worked example · free
Distinguish a removable hole from a vertical asymptote
- 2Factor and simplify the first expression for nearby inputs.
- 2Test the one-sided signs of the second expression.
- 2State the different graph features and the evidence for each.
Key terms
- One-Sided Limit
- A limit taken using inputs from only the left or only the right of a target point.
- Punctured Neighbourhood
- Inputs close to a point while excluding the point itself, which is the region a finite limit examines.
- Sandwich Theorem
- A theorem forcing a middle function to a shared limit when it remains between two functions with that limit.
- Infinite Limit
- Unbounded function behaviour as the input approaches a finite point from a specified direction.
- Horizontal Asymptote
- A line y equals L approached by a function along at least one infinite input direction.
- Oblique Asymptote
- A non-horizontal line whose difference from the function tends to zero at infinity.
- Continuity
- The condition that a function value exists, its limit exists and the two are equal at the point.
- Continuous Extension
- A definition at a missing point that assigns the existing finite limit and repairs a removable discontinuity.
- Intermediate Value Theorem
- A result guaranteeing that a continuous function on a closed interval takes every value between its endpoint outputs.
- Difference Quotient
- The ratio of output change to input change whose limit defines a derivative.
- Differentiability
- Existence of a finite derivative, which guarantees continuity but is not guaranteed by continuity.
- Implicit Differentiation
- Differentiating a relation in x and y while attaching dy over dx whenever a y term changes.
MATH1851 FAQ
How is the course assessed?
The course page lists assignment work in both parts, a test in each part and a final exam. The five numeric weights total one hundred per cent, and the page separately offers bonus percentage points for class participation. Check Moodle for live deadlines and settings.
What is published about the final exam?
The final exam is a published component worth sixty per cent. The available files do not state its duration, sections, question count, permitted materials or hurdle status. Use the live Moodle page and official timetable as the authority for those details.
What should I practise for the Part I test?
The schedule places the test after the calculus sequence through integration techniques, while the tutorial and revision files available here provide detailed practice for the opening limit, continuity and derivative topics. Confirm the current coverage announcement before allocating revision time.
How do I decide whether a limit exists?
Compute or read the left-hand and right-hand limits separately. A finite two-sided limit exists only when those values agree. The function value at the target is a separate fact, so a missing or different point value can break continuity without destroying the limit.
What is the fastest continuity check?
Use the three conditions in order: the value at the point is defined, the two-sided limit exists, and the limit equals the value. For a closed-interval endpoint, use the one-sided limit from inside the interval. Name the failed condition when continuity breaks.
How should I choose a differentiation rule?
Read the outermost structure first. A product needs two changing-factor terms, a quotient needs two complete products over a squared denominator, and a composite needs the outer derivative evaluated at the unchanged inside multiplied by the inner derivative.
Does this guide cover the complete course?
It covers the complete mathematics available for the first six calculus lectures and the published assessment structure. Later calculus topics and the ordinary differential equations teaching body are named on Moodle but not present in the available files, so use the live materials for that content.
Questions students ask in each MATH1851 chapter
- Can a limit exist when the function is undefined at the point?
- Yes. The limit uses nearby inputs and excludes the target itself. If the left-hand and right-hand outputs approach the same finite value, the limit exists even when no point value is defined. Assigning that limiting value would create a continuous extension. Chapter 01 · Limits and One-Sided Behaviour
- Does oscillation always mean a limit fails?
- No. Fixed-amplitude oscillation prevents settling, but a vanishing multiplier can force the whole expression toward one value. Bound the absolute magnitude. If that bound tends to zero, the Sandwich Theorem can establish a zero limit despite infinitely rapid oscillation. Chapter 02 · Sandwich Theorem and Oscillatory Limits
- Does every denominator zero give a vertical asymptote?
- No. Factor the numerator and denominator first. If the zero comes from a common factor that cancels, the graph has a removable hole when the simplified limit is finite. An uncancelled zero still needs a one-sided sign check to describe the unbounded behaviour. Chapter 03 · Limits at Infinity and Asymptotes
- How do I test continuity at a piecewise join?
- Find the left-hand limit, the right-hand limit and the defined value at the join. Continuity requires all three to be equal. When a parameter is unknown, this equality becomes the equation that determines it. Label the common value after solving. Chapter 04 · Continuity and the Intermediate Value Theorem
- Why does substituting the increment as zero fail?
- The raw difference quotient has the increment in its denominator, so immediate substitution creates zero over zero. Simplify the numerator for nonzero nearby increments, cancel the common factor legally and only then take the limit as the increment approaches zero. Chapter 05 · Derivatives, Tangents and Differentiability
- How do I know which differentiation rule to use first?
- Look at the outermost operation. A product of two large expressions uses the product rule even if each factor contains a composite. A power or trigonometric function applied to another expression uses the chain rule. Work from the outside inward. Chapter 06 · Differentiation Rules and Implicit Curves
How to study for the exam
Use a four-pass cycle. First, write the theorem or rule from memory with every condition. Second, work one original problem without notes and keep algebra exact. Third, audit the answer for excluded inputs, one-sided directions, signs and inner derivative factors. Fourth, explain the result in one sentence that connects the symbolic value to the graph or theorem.
Alternate short mixed sets with focused repair sessions from your error log. Before the final assessment, verify the live format and permitted materials on Moodle because the available files publish the weight but not the paper structure. For later calculus and the ordinary differential equations half, study from the current lecture resources rather than extrapolating from this guide.
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