U of T · Calculus with Proofs

MAT137, unit by unit, in full.

MAT137 has no textbook — it is taught through Alfonso Gracia-Saz's video playlists and problem sets. This guide expands all 14 units, each its own page with definitions, theorems, worked examples, pitfalls, and practice.

Why this course for summer

The proof on-ramp, free and complete.

MAT137 (now offered as MAT148 + MAT149) is built for the high-school → proof jump. The same 14 topics underlie the harder analysis stream, MAT158 + MAT159, so finishing MAT137 over the summer turns September's analysis course into review-then-extend. Watch the unit video, read the chapter page, then prove the problems by hand. MAT137 video library ↗

01

The 14 Units

In official order. Term 1 is differential calculus (matches MAT148 / MAT158); Term 2 is integral calculus and series (matches MAT149 / MAT159). Each unit opens its own detailed page.

MAT148H1Calculus I · Fall MAT149H1Calculus II · Winter MAT158H1Analysis 1A MAT159H1Analysis 1B
Start here

Prologue — Why Calculus Exists

The 2,000-year story: how calculus was discovered, the feud over it, and how it built the modern world.

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Term 1 — differential calculus

  1. Logic, sets, definitions & proofs

    Quantifiers, implication, and the proof methods everything is written in.

    Open →

  2. Limits & continuity

    The ε–δ definition, limit laws, and continuity.

    Open →

  3. Derivatives

    The derivative as a limit and the differentiation rules, proved.

    Open →

  4. Transcendental functions

    Exponential, logarithm, trig functions and their inverses.

    Open →

  5. The MVT & inverse functions

    Rolle, the MVT, inverse function theorem, monotonicity, and inequalities.

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  6. Applications of derivatives & limits

    Optimization, curve sketching, L'Hôpital.

    Open →
Term 2 — integral calculus & series

  1. The definition of the integral

    Riemann sums and integrability.

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  2. The Fundamental Theorem of Calculus

    Linking the derivative and the integral.

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  3. Integration methods

    Substitution, by parts, partial fractions.

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  4. Applications of the integral

    Area, volume, average value.

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  5. Sequences

    Convergence and the monotone convergence theorem.

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  6. Improper integrals

    Unbounded intervals and convergence tests.

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  7. Series

    Convergence tests and absolute convergence.

    Open →

  8. Power series & Taylor series

    Radius of convergence and Taylor's theorem.

    Open →
After the 14 units

Calculus at Work

Where these ideas run today: AI, finance, engineering, medicine, graphics, space, and data science — each tied back to a unit.

Read →
02

Source Spine

MAT137 is not a textbook course in the usual sense. The course spine is Alfonso Gracia-Saz's inverted-classroom material: videos, unit guides, classroom questions, problem sets, and past assessments. This guide follows that order closely and expands each unit into a self-contained reading chapter.

How this guide follows the course materials
Primary source

Use the MAT137 video playlist and the Alfonso calculus question bank as the source of truth. The local pages do not replace them; they explain the same topics in a slower written format and add worked proof models.

Unit alignment

Units 1-6 match the differential-calculus half: proof language, limits, derivatives, transcendental functions, MVT, and applications. Units 7-14 match the integral and series half: Riemann sums, FTC, integration methods, applications, sequences, improper integrals, series, and Taylor series.

What to produce

For every unit, write a one-page definition sheet, a theorem sheet with proof sketches, a mistake log, and at least two fully written solutions from the official question bank before moving to old tests.

VIDEO

Watch before reading

Pause whenever a definition appears. Rewrite it in quantifier form and predict the first line of a proof using that definition.

GUIDE

Read the local chapter

Use the local page to fill gaps: theorem statements, proof templates, worked examples, and common traps.

CLASS

Do classroom questions

Use Alfonso's question bank immediately after each unit. Treat these as active recall, not examples to skim.

TEST

Finish with past tests

After every two or three units, attempt old MAT137 problems under time pressure and rewrite weak solutions cleanly.

03

How to Study

MAT137 is a flipped course: watch first, then do.

01

Watch, then prove

Watch the unit's videos, then work the problems by hand before checking. Understanding lives in the doing.

02

Do the real problem sets

The linked CC-licensed banks and past tests are the actual course problems. Use them after each unit page.

03

Master limits early

The ε–δ definition (Unit 2) is the wall. Everything downstream leans on it.

04

Re-prove from memory

Weekly, re-derive key theorems — MVT, FTC, the chain rule — with the page closed.

04

Practice Plan

The unit pages already include local practice. Use this ladder to turn that practice into course-level mastery.

01

Definition drills

Before solving, write the exact definition needed: limit, continuity, derivative, integrability, convergence, or uniform convergence.

02

One proof, three versions

Redo important proofs from memory, then write a shorter exam version, then write a version explaining every quantifier choice.

03

Mixed-topic sets

Every weekend, mix one problem from each previous unit so limits, derivatives, integrals, and series do not live in separate boxes.

04

Reflection log

Record the exact line where each failed proof broke. MAT137 rewards fixing proof habits more than memorizing more formulas.

Course-wide challenge bank

Use these after the relevant units. They are meant to force proof choices across the whole MAT137 sequence.

Challenge 1 · Quantifiers

Negate the statement: "$f$ is continuous at $a$." Then explain in one sentence what the negation says geometrically.

Solution sketch
Continuity is $\forall\varepsilon>0\ \exists\delta>0\ \forall x\, (|x-a|<\delta\Rightarrow |f(x)-f(a)|<\varepsilon)$. Its negation is $\exists\varepsilon>0\ \forall\delta>0\ \exists x$ with $|x-a|<\delta$ but $|f(x)-f(a)|\ge\varepsilon$. No matter how close one zooms near $a$, some point stays visibly away from $f(a)$.
Challenge 2 · Epsilon-delta

Prove from the definition that $\lim_{x\to2}(3x+1)=7$.

Solution sketch
$|(3x+1)-7|=3|x-2|$. Given $\varepsilon>0$, choose $\delta=\varepsilon/3$. If $0<|x-2|<\delta$, then $|(3x+1)-7|<3\delta=\varepsilon$.
Challenge 3 · MVT application

Let $0<a<b$. Prove that $\dfrac{b-a}{b}<\ln b-\ln a<\dfrac{b-a}{a}$.

Solution sketch
Apply the Mean Value Theorem to $\ln x$ on $[a,b]$. There is $c\in(a,b)$ with $\ln b-\ln a=(b-a)/c$. Since $a<c<b$, we have $1/b<1/c<1/a$, and multiplying by $b-a>0$ gives the result.
Challenge 4 · Series decision

Determine whether $\sum_{n=1}^{\infty}\dfrac{n}{n^3+1}$ converges.

Solution sketch
For large $n$, $\dfrac{n}{n^3+1}\sim\dfrac1{n^2}$. Use limit comparison with $\sum 1/n^2$: the ratio tends to $1$, so the series converges.
05

Resources

MAT137 video library — 178 videos, Alfonso Gracia-Saz
MAT137 question banks & classroom problems (by unit)
MAT137 course materials archive — past tests & problem sets
MAT148H1 — current calendar listing
MAT149H1 — current calendar listing

The practice problems in each unit are adapted from the MAT137 question bank by Alfonso Gracia-Saz, released under a Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) license. Solutions are written for this guide. The official problem sets, classroom questions, and past tests carry the same licence — see the links above to use them directly.