MAT148H1 Calculus I with Proofs
The Complete Exam Bible · Fall 2026MAT148H1 Overview
- University of Toronto
- Fall 2026
- Mathematics
- Calculus with proofs
What the course covers
Calculus I with Proofs at the University of Toronto develops differential calculus together with mathematical rigour.
- Definitions drive proofs Every limit, continuity and derivative argument starts from the exact definition.
- Quantifier order matters Swapping for all and there exists changes what must be proved.
- Computation is tested by hand Calculators are not permitted in quizzes written in tutorial, term tests or the final exam.
- Two tests and a final The term tests and the December final exam carry most of the course mark.
How MAT148H1 is assessed
| Component | Weight | Format |
|---|---|---|
| Tutorial quizzes | 6% | Four quizzes written in tutorial; the best three count |
| Pre-class quizzes | 5% | Short quizzes before lectures; the best 25 of up to 35 count |
| Problem sets | 7% | Two problem sets submitted on Gradescope, 3.5% each |
| Term tests | 40% | Two in-person term tests: 25% for the higher mark, 15% for the lower |
| Term test reflections | 2% | One reflection after each term test, 1% each |
| Final exam | 40% | Held in the December 2026 final exam period, scheduled by the registrar |
The Fall 2026 syllabus weights total 100%. Term tests are weighted 25% for the higher mark and 15% for the lower; if a term test is missed for a legitimate reason, that test mark is replaced by the final exam mark. Late submissions are not accepted. The final exam format will be announced during the term; check Quercus for current arrangements.
Current MAT148H1 dates
| Date | Item | Control |
|---|---|---|
| 22 October 2026 | Term Test 1 | In-person term test; confirm room and arrangements on Quercus. |
| 26 November 2026 | Term Test 2 | In-person term test; confirm room and arrangements on Quercus. |
Dates are taken from the Fall 2026 syllabus; all assessment deadlines in its weekly schedule are marked tentative. Confirm exact deadlines and submission settings in the live LMS.
What MAT148H1 covers
Nine chapters follow the syllabus units from logic and sets through limits, derivatives and the Mean Value Theorem to applications.
Sets, Interval Notation and Quantifiers
Set-builder notation, intervals, the universal and existential quantifiers and how their order changes a statementSets as the vocabulary of proof
A set is a collection of well-defined, distinct objects. The symbol ∈ records membership, so 2 ∈ {1, 2, 3}, and A ⊆ B says that every element of A also belongs to B.
- Sets, membership and subsets
- Set-builder notation
- Open, closed and half-open intervals
- Universal and existential quantifiers
- Quantified conditions inside set definitions
- Order of quantifiers
Conditionals, Negation and Proof Strategies
Implications, converses and contrapositives, negating quantified statements, direct proof, counterexamples and inductionConditionals and when they fail
Most theorems take the form if HYPOTHESES, then RESULT, written P ⇒ Q. Such a statement is true when every instance that makes P true also makes Q true.
- Truth conditions of an implication
- Converse versus contrapositive
- Direct proof from definitions
- Proof by contrapositive and by contradiction
- Counterexamples to universal claims
- Mathematical induction
Functions, One-to-One Maps and Inverses
Domain and range, the one-to-one definition and its contrapositive form, and building an inverse functionWhat one-to-one really says
A function f with domain D is one-to-one, also called injective, when different inputs always produce different outputs. Written with quantifiers, the definition reads: for all x₁, x₂ in D, if x₁ ≠ x₂ then f(x₁) ≠ f(x₂).
- Quantified definition of one-to-one
- Contrapositive form used in proofs
- Counterexamples that break injectivity
- Restricting a domain to recover injectivity
- Inverse function, domain and range
- Composition of injective functions
The Epsilon-Delta Definition of a Limit
Reading the formal limit definition, choosing delta from epsilon, writing the proof and negating the definitionReading the definition as a challenge
The statement lim f(x) = L as x → a means: for every ε > 0 there exists δ > 0 such that for all x, if 0 < |x − a| < δ then |f(x) − L| < ε.
- Quantifier order in the ε-δ definition
- Epsilon bands and delta windows on a graph
- Factor and bound routine for polynomials
- Choosing δ = min(1, ε/k)
- Negation of the limit definition
- Witness points for limits that fail
Infinite Limits, Continuity and Value Theorems
Limits equal to infinity, continuity at a point, the Squeeze Theorem, the Intermediate and Extreme Value TheoremsWhat an infinite limit actually claims
Saying lim x→a f(x) = ∞ is not a statement that the function equals infinity anywhere.
- Infinite limits with M and δ
- Limit does not exist versus limit equals infinity
- Continuity at a point in ε-δ form
- Removable, jump and essential discontinuities
- Squeeze Theorem for oscillating limits
- Intermediate Value Theorem root proofs
Derivatives from First Principles and Rules
The derivative as a limit, differentiability failures, and proofs of the product, quotient and chain rulesThe derivative as a limit of secant slopes
Join the graph points above x = a and x = b by a straight line, the secant; its slope (f(b) − f(a))/(b − a) is the average rate of change of f between the two points.
- Secant slope and the limit definition
- First principles with reciprocals and roots
- Product, quotient and chain rules
- Quotient rule proof from the definition
- Differentiable implies continuous
- Corners, discontinuities and vertical tangents
Trigonometric, Exponential and Log Derivatives
Trigonometric review, inverse trigonometric functions, exponentials, logarithms and logarithmic differentiationWhere these derivatives sit in the course
Unit 4 of the syllabus introduces transcendental functions: sine, cosine and tangent, their inverses, the exponential eˣ and the natural logarithm.
- Sine and cosine derivatives from the special limits
- Tangent and secant via the quotient rule
- Unit circle coordinates and reference angles
- Inverse function rule and the arcsine derivative
- Arctangent and its domain
- Exponential and natural log derivatives
The Mean Value Theorem and Its Consequences
Rolle's Theorem, the Mean Value Theorem, and using them to prove inequalities and monotonicityWhat the theorem says
Unit 5 of the syllabus centres on the Mean Value Theorem. Take f continuous on the closed interval [a, b] and differentiable at every point of the open interval (a, b).
- Statement of the Mean Value Theorem with exact hypotheses
- Rolle's Theorem and interior extrema
- Proof of the MVT by tilting to Rolle
- Counterexamples when a hypothesis fails
- Zero derivative implies constant
- Counting roots with Rolle and the IVT
Extreme Values, Indeterminate Forms and Curve Sketching
Critical points, the first and second derivative tests, L'Hôpital's Rule and sketching a curve from its derivativesWhere maxima and minima can hide
An extreme value of f is a largest or smallest output, either across a whole domain (absolute or global) or only among nearby inputs (local).
- Critical points and the closed-interval method for absolute extrema
- First Derivative Test with sign charts of f′
- Second Derivative Test and higher-order checks when f″(c) = 0
- L'Hôpital's Rule for 0/0 and ∞/∞, including repeated use
- Rewriting 0 · ∞, ∞ − ∞ and exponential forms before applying the rule
- Concavity, inflection points and the curve sketching checklist
The Fall 2026 syllabus lists six units: logic, sets, notation, definitions and proofs; limits and continuity; derivatives; transcendental functions; the Mean Value Theorem and its applications; and applications of limits and derivatives.
This guide follows that order in nine chapters, splitting the first unit into sets and quantifiers, proof strategies, and functions, and separating the formal limit definition from infinite limits and continuity.
The syllabus sets four learning outcomes: fluency with calculus concepts and their computation, mathematical rigour in reading and writing proofs, problem solving on unfamiliar questions, and academic integrity.
Because the course asks you to understand why methods work, each chapter pairs a definition or theorem with a fully written argument, then asks you to adapt it to a new function.
How the material builds
The opening chapters supply the language every later proof needs: set-builder notation, intervals, quantifiers, implications and their negations.
The epsilon-delta definition then uses all of that language at once, which is why so many students find it the first real hurdle in the course. Continuity, the Intermediate Value Theorem and the Extreme Value Theorem follow from the limit definition, and the derivative is introduced as a limit before the differentiation rules are proved.
The second half moves from rules to consequences.
Transcendental functions extend differentiation to trigonometric, inverse trigonometric, exponential and logarithmic functions.
The Mean Value Theorem connects the derivative to the behaviour of a function on an interval, and the final chapter uses derivatives to locate extreme values, evaluate indeterminate limits with L'Hôpital's Rule and sketch curves.
Working with the weekly rhythm
The syllabus describes short pre-class videos and readings, one pre-class quiz per lecture, three lecture hours spent on problems rather than note taking, one tutorial hour, and an expected eight to ten hours of study each week.
The recommended text is the set of lecture notes Calculus in One Dimension, and the syllabus assigns some pre-class readings from it. Practice problems for each unit are posted but not submitted, and routine computation is left off the problem sets on purpose, so computational speed has to come from your own practice before the term tests.
Worked example · free
Choose delta for a linear limit
- 2Scratch work: |(4x − 1) − 7| = |4x − 8| = 4|x − 2|, so 4|x − 2| < ε whenever |x − 2| < ε/4.
- 2Proof: let ε > 0 and set δ = ε/4, which is positive.
- 2If 0 < |x − 2| < δ, then |(4x − 1) − 7| = 4|x − 2| < 4δ = ε, as required.
Key terms
- Universal Quantifier
- The symbol ∀, read "for all". A statement ∀x ∈ S, P(x) is true only when P(x) holds for every element of S, so one counterexample is enough to make it false.
- Existential Quantifier
- The symbol ∃, read "there exists". A statement ∃x ∈ S, P(x) is true when at least one element of S satisfies P(x); proving it usually means exhibiting that element.
- Contrapositive
- The implication "if not Q, then not P" formed from "if P, then Q". It is logically equivalent to the original, which is why proving it proves the original statement.
- One-to-One Function
- A function f for which f(a) = f(b) forces a = b. Equivalently, distinct inputs give distinct outputs, and only such functions have an inverse on their range.
- Epsilon-Delta Definition
- The formal definition of a limit: for every ε > 0 there is δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε.
- Continuity at a Point
- A function f is continuous at a when f is defined at a and the limit of f(x) as x approaches a equals f(a).
- Derivative
- The limit of the difference quotient (f(a + h) − f(a))/h as h approaches 0, when that limit exists. It gives the slope of the tangent line at a.
- Mean Value Theorem
- A theorem stating that a function continuous on [a, b] and differentiable on (a, b) has some c in (a, b) where f′(c) equals the average rate (f(b) − f(a))/(b − a).
- Critical Point
- An input c at which f is defined and its derivative is either zero or undefined. Local extrema of a function on an open interval can occur only at critical points.
MAT148H1 FAQ
Is calculus with proofs harder than a regular first calculus course?
It asks for a different skill as well as computation. Alongside limits and derivatives, students read precise definitions, critique arguments and write rigorous proofs, so practice must include explaining why a step is valid and not only producing the right number.
How do I start an epsilon-delta proof when I have no idea what delta should be?
Work backwards on scratch paper first. Bound |f(x) − L| by a constant times |x − a|, restrict δ ≤ 1 to control the extra factor, then take δ as the minimum of the two restrictions and write the proof forwards.
What is the difference between a limit that does not exist and a limit equal to infinity?
Saying a limit equals infinity is a precise claim: for every bound M there is a δ making f(x) larger than M. It is one particular way of failing to have a real limit, while oscillation is another.
Can I use a calculator on the term tests and final exam?
The course policies page says calculators are not permitted in tutorial quizzes, term tests or the final exam, and that questions are written so every computation can be done by hand. Build fluency with exact values and algebra early.
How are the two term tests weighted?
Together they carry 40% of the course mark, split so the higher of the two marks counts for 25% and the lower for 15%. A reflection after each test adds 1% each, separately from the tests themselves.
How should I practise for proof questions on tests?
Write complete proofs from a blank page, then compare them line by line with a model. Check that every variable is introduced, every quantifier is in the right order, and the final sentence states exactly what was claimed.
Questions students ask in each MAT148H1 chapter
- What is the difference between (a, b) and [a, b] in interval notation?
- Round brackets exclude the endpoint and square brackets include it. So (2, 5) contains 2.5 but neither 2 nor 5, while [2, 5] contains both. On a number line this matches hollow versus filled dots, and infinity always takes a round bracket. Chapter 01 · Sets, Interval Notation and Quantifiers
- Is the contrapositive the same as the converse?
- No. For if P then Q, the converse is if Q then P and may be false, while the contrapositive is if not Q then not P and is always equivalent to the original. Only the contrapositive can replace the original in a proof. Chapter 02 · Conditionals, Negation and Proof Strategies
- How do you prove a function is one-to-one?
- Take two arbitrary points u and v from the domain and assume their images agree. Then use algebra such as cross-multiplication or factoring out (u − v) to force u = v. Because a and b were arbitrary, this covers every pair simultaneously and completes the proof. Chapter 03 · Functions, One-to-One Maps and Inverses
- How do you find delta in an epsilon delta proof?
- Start from |f(x) − L| and factor out |x − a|. Restrict |x − a| less than 1 to bound the leftover factor by a constant k, then take δ = min(1, ε/k). That choice makes the bound and the smallness work together in the final inequality. Chapter 04 · The Epsilon-Delta Definition of a Limit
- Is a limit that equals infinity the same as a limit that does not exist?
- Every infinite limit is a nonexistent real limit, but the converse fails. Writing equals infinity records extra information: the values grow past every bound. A function like sin(1/x) near 0 has no limit and no infinite limit either. In a proof the two claims need different arguments. Chapter 05 · Infinite Limits, Continuity and Value Theorems
- Why is |x| not differentiable at 0?
- At 0 the difference quotient |h|/h equals −1 for negative h and 1 for positive h. Those one-sided limits disagree, so the two-sided limit defining the derivative fails, even though the graph is continuous there. The graph shows this as a sharp corner. Chapter 06 · Derivatives from First Principles and Rules
- What is the derivative of arcsin x and where does it exist?
- It equals 1/√(1 − x²). Arcsine accepts inputs in [−1, 1], but the slope formula only works on the open interval (−1, 1), because the denominator is zero at ±1 and the graph has vertical tangents at those two endpoints. Chapter 07 · Trigonometric, Exponential and Log Derivatives
- How do you prove an equation has exactly one real solution?
- Split the claim in two. Existence comes from the Intermediate Value Theorem: find inputs where the function changes sign. Uniqueness comes from Rolle: two roots would force a zero of the derivative between them, so show the derivative never vanishes. Chapter 08 · The Mean Value Theorem and Its Consequences
- Can I use L'Hôpital's Rule on any limit that looks hard?
- No. The rule only applies when the quotient is of type 0/0 or ∞/∞, and f and g must be differentiable near the point. Applying it to a determinate quotient such as 0/3 produces wrong answers, so substitute first and confirm the type on every line. Chapter 09 · Extreme Values, Indeterminate Forms and Curve Sketching
How to study for the exam
Before each lecture, watch the assigned videos and attempt the pre-class quiz so class time goes to hard problems. After class, rewrite one definition and one proof from memory, then check every quantifier. Each week, do the posted practice problems by hand without a calculator, because routine computation is tested on the term tests rather than the problem sets.
Use office hours, the Math Learning Centre and Piazza for questions as soon as a step stops making sense.
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