The derivative as a linear approximation
Linked to companion
f(a+h) ≈ f(a) + f′(a)h
Turn your textbook into a connected learning system.
Upload a PDF, read it beside a source-linked AI companion, build a concept progression, and practice each idea without losing the context of the book. Generalize is opening access gradually while the product is in private beta.
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✓Source-linked explanations✓Skill trees from your textbook✓Verified practice in sets of 5
Start with your own textbook
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Generalize does not start from a generic course. Your PDF is the source. Choose the material you want, then turn the same textbook into a linked companion, a Skill Tree, and focused practice.
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PDF
Numerical Linear Algebra.pdf40 lectures detected
ReadyPDF
Probability and Random Processes.pdf18 chapters detected
Ready01
Bring the sourceYour PDF stays at the center of the workflow.
Numerical Linear AlgebraChoose what to generate
3 selected✓
Lecture 3 · Normspp. 16–23
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Lecture 4 · The SVDpp. 24–31
Lecture 5 · More on the SVDpp. 32–38
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Lecture 6 · Projectorspp. 39–46
Selected chapters count as one generation
02
Choose the materialGenerate only the chapters or lectures you want.
AI CompanionRead beside source-linked explanations
Highlights · intuition · examplesSkill TreeTurn the same material into concepts
Progression · lessons · saved progressPracticeWork through focused question sets
Verified answers · more on demand03
Learn from the same sourceEach feature stays anchored to your textbook.
What this is really saying
Concept breakdownSource pages 42–43
Intuition
Close to a point, a differentiable function behaves almost like its tangent line. The derivative supplies the slope of that local linear model.
Example
If f(x)=x² near x=3, then f′(3)=6, so a small change h gives f(3+h)≈9+6h.
From PDF to mastery
More than a chatbot beside a textbook.
Generalize turns the structure of the book into an actual learning workflow: understand the source, see the concepts, then practice them.
01
See the subject as a skill tree.
Generalize breaks the textbook into compact concepts and lays them out as a progression. Open any generated lesson, see what comes next, and keep your initial-practice progress visible on every node.
1Vector spacesInitial 5
100%
2Vector normsInitial 5
60%
3Induced normsInitial 5
0%
4ConditioningLesson readyOpen anytime →
Practice · Vector normsQuestion 3 of 5
60%Application
If ‖x‖₂ = 5, what is ‖3x‖₂, and which norm property justifies the result?
Type your answer…
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Answers are checked independentlySolutions are generated when you need them.
02
Practice in focused sets of 5.
Each concept starts with 5 questions. Complete them, then generate 5 more whenever you want. Answers are verified separately, while the node keeps its progress anchored to that first set so your baseline mastery stays easy to read.
03
Use the textbook's real structure.
Upload a PDF and Generalize detects the book's instructional units—chapters, lectures, sections, or modules—then lets you build the companion and Skill Tree from the material you actually want to study.
PDF
Numerical Linear AlgebraStructure detected · 40 lectures
ReadyPart I · Fundamentals
01Matrix-Vector Multiplicationpp. 1–7
02Orthogonal Vectors and Matricespp. 8–15
03Normspp. 16–23
04The Singular Value Decompositionpp. 24–31
Stay tied to the source
Highlights and companion blocks move together, so explanations stay grounded in the original pages.
Build around intuition
Concept lessons include compact explanations, dedicated intuition, examples, and counterexamples when they help.
Generation keeps going
Skill Tree work is saved durably and can keep processing in the background while you move around the app.
Plans
One shared allowance. Two ways to use it.
A generation can build a textbook analysis or a Skill Tree. Multiple selected chapters in one textbook submission still use only one generation.
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