MTH6131 Calculus and Mathematical Modelling
The Complete Exam Bible · MTH6131 Overview
- The Education University of Hong Kong
- Semester 1, 2026/27
- Postgraduate course
- Mathematics
- Final examination and midterm
What this course is actually asking you to do
The opening lecture describes this as a course in mathematical modelling whose main tools are differential equations, and it says plainly that calculus is taught first because most students arrive without strong exposure to it.
- Assessed by A final examination worth sixty per cent, a midterm worth twenty and two assignments worth twenty between them.
- The skill being tested Computing limits, derivatives and integrals correctly, and knowing the condition under which each rule may be used.
- Hardest step Turning an applied problem into a single function of one variable on the right interval before any calculus starts.
- Where to confirm Dates, paper formats and permitted materials are controlled by the course Moodle page.
How MTH6131 is assessed
| Component | Weight | Format |
|---|---|---|
| Two Assignments | 20% | Assignment 1 due 15 October 2026 and Assignment 2 due 2 December 2026, per the course schedule |
| Midterm | 20% | Scheduled for 22 October 2026, after the curve sketching lecture |
| Final examination | 60% | Scheduled for 10 December 2026, after the last lecture |
The course handout lists these three components and they total 100%. The same handout sets out the lecture schedule with the two assignment due dates, the midterm and the final examination, and adds that the schedule may be modified as necessary. For the format of each paper, permitted materials and any pass requirement, check the course Moodle page, which is also where schedule changes are announced.
Assessment structure
Segment widths reproduce the published percentage weights and total 100%.
Current MTH6131 dates
| Date | Item | Control |
|---|---|---|
| 15 October 2026 | Assignment 1 due | Listed in the course schedule beside the curve sketching lecture. |
| 22 October 2026 | Midterm | Listed in the course schedule in the seventh lecture slot. |
| 2 December 2026 | Assignment 2 due | Listed in the course schedule beside the differential equations lecture. |
| 10 December 2026 | Final examination | Listed in the course schedule after the last lecture; the handout notes the schedule may be modified. |
Dates are as published in the course schedule in the course handout for this offering. Confirm exact deadlines and submission settings in the live LMS.
What MTH6131 covers
Sixteen chapters follow the lecture notes from the opening interest model through limits, continuity and differentiation, then the exercise sets on integration and first-order differential equations.
Functions and Continuously Compounded Interest
simple and compound interest · the limit that gives e to the rt · domain, codomain and one to one maps · even, odd and composite functionsWhy a calculus course opens with a bank balance
The first lecture chooses an example that anyone can state and that needs calculus to finish. A loan of principal P at rate r owes P(1 + r) at maturity.
- The simple interest model and its repeated form
- Splitting an annual rate into n credits
- The limit of (1 + r/n) to the power nt
- Domain, codomain and the one-output rule
- One to one, onto and invertible functions
- Composition and why its order matters
Limits of Sequences and Functions
convergent sequences · the epsilon and N definition · one-sided limits that disagree · the limit theorems · sine x over xFrom sequences to functions
The lectures introduce limits with sequences. A sequence is really a function defined on the natural numbers, so 1, 1/2, 1/3 and so on is the function s(n) = 1/n.
- Sequences as functions of a natural number
- The epsilon and N definition of convergence
- Infinity as a direction, not a number
- The epsilon and delta definition for functions
- Left and right limits, and when they disagree
- The six theorems on limits
Continuity and the Value Theorems
the three-condition test · removable, jump and infinite discontinuities · arithmetic of continuous functions · extreme and intermediate valuesContinuity is decided point by point
The lecture warns against the everyday picture of continuity as a graph drawn without lifting the pen.
- Continuity defined at a point, not by a picture
- The three-condition test
- Removable discontinuities and cancelled factors
- Jump and infinite discontinuities
- Absolute values and piecewise joins
- Arithmetic of continuous functions
The Derivative from First Principles
average against instantaneous rate · the difference quotient procedure · differentiable implies continuous · corners, cusps and vertical tangentsDerivatives are about rates of change
The differentiation notes open with a single claim: derivatives are about rates of change.
- Increments and the average rate of change
- The difference quotient and its limit
- The expand, subtract, divide, limit procedure
- Common denominators and conjugates
- Why differentiable implies continuous
- Corners, vertical tangents and endpoints
Rules of Differentiation and Implicit Differentiation
sum, product and quotient rules · the chain rule and its proof · implicit slopes of curves · higher derivatives by two methodsRules proved once, used everywhere
First principles work for any function, but the lectures note that for composite functions they can be a fruitless endeavour.
- The constant, sum and multiple rules
- The product rule and its proof
- The quotient rule and its sign
- The chain rule and the proof with two error terms
- Intermediate variables in Leibniz form
- Implicit differentiation of a curve
Trigonometric and Inverse Trigonometric Derivatives
sine from first principles · the six trigonometric derivatives · restricted domains for inverse functions · arcsine, arctangent and two secant conventionsOne proof, six derivatives
The notes prove the derivative of sine from first principles. Expanding sin(x + Δx) with the addition formula and regrouping gives cos x times sin(Δx)/Δx plus sin x times (cos(Δx) − 1)/Δx.
- Sine differentiated from first principles
- The six trigonometric derivatives and their signs
- The chain factor from the argument
- Identities for tidying answers
- Radians in related rates
- Restricted domains and inverse functions
Logarithmic and Exponential Derivatives
e as a limit and as a series · the logarithm laws · derivatives of log and exponential functions · logarithmic differentiationThe number e, two ways
The notes recall e as the limit of (1 + 1/h) to the power h as h grows, equivalently the limit of (1 + k) to the power 1/k as k tends to zero, and also as the series 1 + 1 + 1/2! + 1/3! and so on.
- The number e as a limit and as a factorial series
- Logarithms and the change of base
- Log laws used before differentiating
- Derivatives of log and exponential functions
- The first-principles proof for the logarithm
- Logarithmic differentiation of products and quotients
Mean Value Theorem and Taylor Approximation
Rolle's Theorem · the Mean Value Theorem and its rearranged form · estimating roots and proving inequalities · Taylor and Maclaurin with remainderRolle's Theorem and the Mean Value Theorem
The notes state Rolle's Theorem first: a function that is unbroken on a closed interval, has a derivative at every interior point and vanishes at both ends must have a horizontal tangent somewhere inside.
- Rolle's Theorem and its hypotheses
- The Mean Value Theorem as a parallel tangent
- The rearranged form f(b) = f(a) + (b − a)f′(x₀)
- Estimating roots by freezing the slope
- Proving inequalities by bounding the derivative
- Taylor's Theorem with remainder
Tangent and Normal Lines and Related Rates
tangent and normal equations · horizontal and vertical tangents of implicit curves · tangents from an outside point · related rates and differentialsTwo lines from one slope
The derivative at a point is the slope of the tangent there, so the tangent line is y − y₀ = f′(x₀)(x − x₀).
- Tangent and normal line equations
- Horizontal and vertical tangents of implicit curves
- Tangents with a prescribed slope
- Tangents through a point off the curve
- The angle between two curves
- Related rates: differentiate, then substitute
Curve Sketching and Extreme Values
critical points and sign charts · the first and second derivative tests · concavity and inflection · asymptotes and absolute extremaThe vocabulary of shape
The notes define each shape word separately. A function is increasing on an interval when larger inputs always give larger outputs, and decreasing when they give smaller ones.
- Increasing, decreasing, stationary and critical
- Relative against absolute extrema
- The first derivative test and sign charts
- The second derivative test and its silent case
- Concavity and inflection points
- Vertical and horizontal asymptotes
Applied Maximum and Minimum Problems
six guidelines from words to one variable · fencing, containers and distances · closed against open intervals · why to square a distanceThe calculus is short; the setting up is not
The applied examples in the lectures are all stories: two ships on crossing courses, a corral fenced against a barn with interior fences, a box-shaped bag under a size limit, a ladder that must clear a fence.
- From a story to an objective and a constraint
- The six guidelines in order
- Eliminating all but one variable
- The interval of interest
- Comparing endpoints on a closed interval
- Arguing on an open interval
L'Hôpital's Rule and Indeterminate Forms
the seven indeterminate forms · the rule and its hypotheses · converting products, differences and powers · misuses that give clean wrong answersWhen substitution stops working
A limit can be found by substitution whenever the function is continuous at the point.
- What makes a form indeterminate
- The seven indeterminate forms
- L'Hôpital's rule and its hypotheses
- Repeating the rule while the form survives
- Rewriting products and differences as quotients
- The logarithm device for power forms
Integration Formulas and Substitution
the core table of antiderivatives · adjusting the differential · log, exponential and trigonometric forms · arctangent, arcsine and log-quotient formsIntegration reverses differentiation
The schedule opens the second half of the course with an introductory lecture on integration, followed by three weeks of techniques.
- Antiderivatives and the constant of integration
- The core table of standard integrals
- The power rule and its n = −1 exception
- Adjusting the differential by a constant
- Expanding and dividing before integrating
- Log and exponential forms
The Definite Integral and the Fundamental Theorem
Riemann sums and signed area · evaluation with F of b minus F of a · substitution with changed limits · integration by parts and arc lengthFrom sums to a number
The course outline in the opening lecture lists integrability and the Fundamental Theorem of Calculus among its topics.
- Riemann sums and the limit that defines the integral
- Signed area and the properties of the integral
- The Fundamental Theorem and the bracket layout
- Differentiating an integral with a variable limit
- Substitution with changed limits
- Integration by parts in definite form
Trigonometric Substitution and Partial Fractions
sine, tangent and secant substitutions · reference triangles and completing the square · partial fractions for four factor types · half-angle substitutionThree substitutions for three roots
The schedule gives integration techniques three consecutive weeks, and the exercise sets split them into trigonometric substitution, partial fractions and further substitutions.
- Choosing among sine, tangent and secant substitutions
- Reference triangles for returning to x
- Completing the square under a root
- The integral of secant cubed
- Dividing before decomposing
- Partial fractions for linear, repeated and quadratic factors
Separable and Linear First Order Differential Equations
verifying solutions and initial value problems · separating variables · the integrating factor · the interest model with withdrawalsWhere the course has been heading
The opening lecture describes the course as mathematical modelling with differential equations and lists modelling with ordinary differential equations as the last of its topics.
- Differential equations as models of change
- Verifying a proposed solution
- General solutions and initial value problems
- Separating the variables
- Constant solutions lost by division
- Standard form for a linear equation
So the semester runs from limits and continuity through differentiation and its applications to the techniques of integration, and closes on ordinary differential equations.
The final examination carries sixty per cent of the course, the midterm twenty and two assignments twenty, which makes fluent, checked computation the skill that is rewarded most.
The opening model and the language of functions
The first lecture builds interest from a single loan to repeated compounding to the continuous limit, where the balance becomes P times e to the power rt.
Getting there needs a limit, a logarithm and a derivative, which is the argument for learning calculus first.
The same lecture fixes the function vocabulary used for the rest of the course: domain and codomain, one to one and onto maps, composition, inverses, and even and odd functions.
Limits and continuity
Limits arrive through sequences, with the epsilon and N definition stated as a game between a tolerance and a tail.
Function limits follow, with the rule that a two-sided limit exists only when both one-sided limits agree.
Continuity is then tested at a point with three conditions, which also sort failures into removable, jump and infinite discontinuities, and the Extreme Value and Intermediate Value Theorems show what continuity on a closed interval guarantees.
The derivative and its rules
Derivatives are introduced as rates of change, first as an average over an interval and then as the limit of that average.
The lectures prove that differentiable functions are continuous and show, with corners and vertical tangents, that the converse fails.
The product, quotient and chain rules are proved from the definition, and implicit differentiation extends slopes to curves such as circles that are not functions.
Special functions and the value theorems
Trigonometric derivatives start from a first-principles proof for sine; inverse trigonometric derivatives come from implicit differentiation on restricted domains.
The number e is defined as a limit, logarithm laws are used to simplify before differentiating, and logarithmic differentiation handles variable exponents.
Rolle's Theorem, the Mean Value Theorem and Taylor's Theorem with remainder turn derivatives into estimates with a known error.
Applications of the derivative
The applied topics are tangent and normal lines, related rates, curve sketching with sign charts and the second derivative, applied maximum and minimum problems, and limits of indeterminate form evaluated with L'Hôpital's rule.
Each has its own procedure, and each has a characteristic way of producing a clean-looking wrong answer when a hypothesis is ignored, which this guide makes explicit chapter by chapter.
Integration and differential equations
Integration is taught through a table of standard forms and the habit of adjusting the differential to match one, then the definite integral and the Fundamental Theorem, then trigonometric substitution, partial fractions and further substitutions.
The course ends with first-order differential equations, separable and linear, solved by separating variables or with an integrating factor, and verified by substitution.
How to use this guide
Read the front matter first, then work one chapter at a time in order, because each chapter's methods are built from the previous chapter's definitions.
Every chapter contains a full model solution and practice items with answers; write your own answer before reading the one given.
The practice chapter is a mixed paper in the order the course is taught, and the closing chapter is a final revision pass for the morning of a paper.
Evidence and assessment control
Assessment components, weights and dates follow the course handout for this offering, which also notes that the schedule may be modified as necessary.
Teaching explanations, worked examples and practice items are independently written and are not University assessment material. Confirm paper formats, permitted materials, deadlines and any change to the schedule on the course Moodle page.
Worked example · free
A tangent line used as an estimate
- 2Differentiate and find the slope and the point of contact at x = 9.
- 2Write the tangent line in point-slope form.
- 2Evaluate the line at 9.3 and compare with the curve's concavity.
Key terms
- Limit
- The number that the values of a function or sequence approach as the input approaches a point or grows without bound. It describes where the values are heading and does not depend on the value at the point itself.
- One-sided Limit
- A limit taken as the input approaches from the left only or from the right only. A two-sided limit exists exactly when both one-sided limits exist and agree.
- Continuity
- The property of a function at a point where the value is defined, the limit exists, and the two are equal. A function is continuous on a set when this holds at every point of it.
- Removable Discontinuity
- A point where the limit exists but the value is missing or different, so that defining or redefining one value makes the function continuous there.
- Derivative
- The limit of the average rate of change as the increment tends to zero, which gives the instantaneous rate of change and the slope of the tangent line.
- Chain Rule
- The rule that the derivative of a composite function is the derivative of the outer function at the inner one, multiplied by the derivative of the inner function.
- Implicit Differentiation
- A method for finding a slope from an equation in x and y: differentiate every term with respect to x, regarding y as depending on x, and solve for the derivative.
- Critical Point
- An interior point of the domain where the derivative is zero or does not exist. Every relative extremum of a differentiable function occurs at one.
- Inflection Point
- A point where the direction of bending of a curve switches, which requires the second derivative to change sign there.
- Indeterminate Form
- An outcome such as zero over zero produced by substitution that does not by itself determine a limit, and signals that the expression must be rewritten or L'Hôpital's rule applied.
- Antiderivative
- A function whose derivative is a given function. Any two antiderivatives of the same function differ by a constant, which is why indefinite integrals carry plus C.
- Integrating Factor
- The multiplier e raised to the integral of P that turns a linear first-order equation in standard form into the derivative of a product, so it can be integrated directly.
MTH6131 FAQ
How is this course assessed, and how much is the final worth?
The course handout lists three components totalling one hundred per cent: two assignments worth twenty per cent between them, a midterm worth twenty per cent and a final examination worth sixty. Because the final carries more than the other two together, fluent computation under time pressure is the skill that matters most. Confirm paper formats and permitted materials on the course Moodle page.
When are the midterm and the final examination?
The schedule in the course handout places the midterm on 22 October 2026, straight after the curve sketching lecture, and the final examination on 10 December 2026, after the last lecture. Assignment 1 is listed as due on 15 October and Assignment 2 on 2 December. The handout says the schedule may be modified, so check Moodle before each date.
Do I need a calculus background before taking this course?
The opening lecture says the course assumes a good grasp of single-variable and multi-variable calculus in principle, but that most students enrolled do not have strong exposure to it. For that reason calculus topics are covered first, from limits upward, and the modelling with differential equations comes later in the semester.
When should I differentiate from first principles rather than use the rules?
Use first principles when a question asks for it, or when you need to prove a rule, as the lectures do for the product rule and for sine. Otherwise use the rules, which were proved from the definition once so that they could be used everywhere. First principles also settle whether a derivative exists at a corner or a vertical tangent, where the rules say nothing.
When am I allowed to use L'Hôpital's rule?
Only when substitution gives zero over zero or infinity over infinity and both functions are differentiable near the point with a non-zero denominator derivative. The other indeterminate forms must first be rewritten as a quotient. Applying the rule to an expression that is not indeterminate produces a clean-looking wrong answer, so always substitute and name the form first.
How do I choose an integration technique?
Read the shape of the integrand. If the derivative of an inner expression is present up to a constant, adjust the differential and use the standard table. A root of a sum or difference of squares calls for a trigonometric substitution; a rational function with a factorable denominator calls for partial fractions, after division if the numerator's degree is too high.
What kinds of differential equation does the course cover?
The exercise sets cover first-order equations of two types: separable equations, solved by collecting each variable on its own side and integrating, and linear equations, solved with an integrating factor after rewriting in standard form. They also ask you to verify that a given function solves an equation and to fix the constant from an initial condition.
Questions students ask in each MTH6131 chapter
- Why does the course start with interest rather than with limits?
- Because the interest example shows why limits are needed. Compounding more and more often leads to a quantity with the variable in both the base and the exponent, and settling what it approaches takes a logarithm and a derivative. The example therefore motivates the order of the whole semester: calculus first, then modelling. Chapter 01 · Functions and Continuously Compounded Interest
- Does a sequence have to reach its limit?
- No. The terms only have to get arbitrarily close to it. The sequence 1/n never equals zero, yet zero is its limit, because for any tolerance the terms are eventually all within it. The lecture makes this point directly: the limit of a convergent sequence need not appear among its terms. Chapter 02 · Limits of Sequences and Functions
- Is a function with a hole in its graph discontinuous?
- It depends on whether the hole is in the domain. If the formula is undefined at the point, the function is simply not defined there; the useful question is whether it can be extended continuously, which it can when the limit exists. If the function is defined at the point with a different value, it is discontinuous there, and redefining that one value removes the discontinuity. Chapter 03 · Continuity and the Value Theorems
- Why can I not just put Δx equal to zero in the difference quotient?
- Because the result is zero over zero, which carries no information. The derivative is a limit precisely because the quotient is undefined at Δx = 0. The procedure simplifies the quotient for non-zero Δx until the division disappears, and only then takes the limit, which is now a simple substitution. Chapter 04 · The Derivative from First Principles
- What is the difference between the chain rule and the product rule?
- The product rule handles two functions multiplied together, each depending on x. The chain rule handles one function applied to the output of another. Many expressions need both: (x² + 1)(3x − 2) to the fourth power needs the product rule for the multiplication and the chain rule for the power of 3x − 2. Chapter 05 · Rules of Differentiation and Implicit Differentiation
- Do I need to memorise all six trigonometric derivatives?
- It is worth it, but only two need real memorising: sine differentiates to cosine and cosine to minus sine. The other four follow from the quotient rule in a line each. The pattern that every function whose name begins with co has a minus sign in its derivative catches most sign errors. Chapter 06 · Trigonometric and Inverse Trigonometric Derivatives
- What is the difference between the derivative of x squared and the derivative of 2 to the power x?
- The first has a constant exponent and variable base, so the power rule gives 2x. The second has a constant base and variable exponent, so the exponential rule gives 2 to the power x times ln 2. Confusing the two is common; ask which part is constant before choosing a rule. Chapter 07 · Logarithmic and Exponential Derivatives
- What does the Mean Value Theorem actually tell me?
- It guarantees that somewhere between a and b the instantaneous rate of change equals the average rate over the interval, provided the function is continuous on the closed interval and differentiable inside it. It does not say where, which is why it is used for estimates and inequalities rather than for finding a specific point. Chapter 08 · Mean Value Theorem and Taylor Approximation
- What is the difference between a tangent and a normal?
- The tangent at a point has the curve's slope there; the normal passes through the same point at right angles to the tangent. If the tangent's slope is m, the normal's is minus one over m. When the tangent is horizontal, the normal is vertical and is written as x equals a constant. Chapter 09 · Tangent and Normal Lines and Related Rates
- What should I do when the second derivative test gives zero?
- Go back to the first derivative test. A zero second derivative at a stationary point means the test is silent: the point could be a maximum, a minimum or neither. The function x to the fourth has a minimum at zero, while x cubed has neither, and both have zero second derivative there. Chapter 10 · Curve Sketching and Extreme Values
- How do I know whether an optimisation answer is a maximum or a minimum?
- On a closed interval, compare the value at each critical point with the values at the endpoints. On an open interval, use the sign change of the derivative at the critical point, or the behaviour near the ends of the interval. If there is only one critical point and the derivative changes sign there, it is the absolute extremum. Chapter 11 · Applied Maximum and Minimum Problems
- Can I use L'Hôpital's rule on any limit of a quotient?
- No. The quotient must have the form zero over zero or infinity over infinity at the point, and both functions must be differentiable near it. If substitution gives an ordinary number, that number is the limit, and applying the rule would replace it with something wrong. Substitute first, every time. Chapter 12 · L'Hôpital's Rule and Indeterminate Forms
- Why do I have to add plus C to an indefinite integral?
- Because the derivative of a constant is zero, so if F is an antiderivative of f, so is F plus any constant. The indefinite integral names that whole family. Leaving out plus C gives only one member and loses the information needed later, for example to fit an initial condition in a differential equation. Chapter 13 · Integration Formulas and Substitution
- Why is there no plus C in a definite integral?
- Because the constant cancels. The Fundamental Theorem subtracts the antiderivative at the lower limit from its value at the upper limit, and any constant added to the antiderivative appears in both terms and disappears. Any antiderivative gives the same answer. Chapter 14 · The Definite Integral and the Fundamental Theorem
- How do I choose between sine, tangent and secant substitutions?
- Read the root. A square root of a squared minus x squared takes x equals a sine theta, a root of a squared plus x squared takes a tangent, and a root of x squared minus a squared takes a secant. Each choice is the one whose Pythagorean identity turns the expression under the root into a perfect square. Chapter 15 · Trigonometric Substitution and Partial Fractions
- How do I tell whether a first-order equation is separable or linear?
- Write the derivative on its own. If the right-hand side factors as a function of x times a function of y, the equation is separable. If it can be rearranged as y′ plus a function of x times y equals a function of x, it is linear. Some equations, such as y′ equals k y, are both, and either method works. Chapter 16 · Separable and Linear First Order Differential Equations
How to study for the exam
Keep one error log for the whole semester and add to it every time a practice answer goes wrong, naming the rule and the slip: a missing chain factor, a quotient-rule sign, a limit not changed after substitution. Before each paper, reread only that log and work one practice item per entry. Calculus marks are lost to a short, personal list of repeated slips far more than to unfamiliar topics.
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