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 as published in 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 statement02Conditionals, Negation and Proof Strategies
Implications, converses and contrapositives, negating quantified statements, direct proof, counterexamples and induction03Functions, One-to-One Maps and Inverses
Domain and range, the one-to-one definition and its contrapositive form, and building an inverse function04The Epsilon-Delta Definition of a Limit
Reading the formal limit definition, choosing delta from epsilon, writing the proof and negating the definition05Infinite Limits, Continuity and Value Theorems
Limits equal to infinity, continuity at a point, the Squeeze Theorem, the Intermediate and Extreme Value Theorems06Derivatives from First Principles and Rules
The derivative as a limit, differentiability failures, and proofs of the product, quotient and chain rules07Trigonometric, Exponential and Log Derivatives
Trigonometric review, inverse trigonometric functions, exponentials, logarithms and logarithmic differentiation08The Mean Value Theorem and Its Consequences
Rolle's Theorem, the Mean Value Theorem, and using them to prove inequalities and monotonicity09Extreme Values, Indeterminate Forms and Curve Sketching
Critical points, the first and second derivative tests, L'Hôpital's Rule and sketching a curve from its derivativesThe 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.
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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