bythreedu · Math Tracks · Curriculum v1.0 · 2026

Limits, derivatives,
and the start of

the infinite.

A complete Calculus I course from limits through the Fundamental Theorem of Calculus. Built for students who want to understand calculus from the ground up — not just pass the exam. Every concept is derived, every technique is explained, and every application is set up before it is solved.

Duration~26weeks
Total Lessons26at your pace
Units8progressive phases
Assignments8+problem sets
DestinationCalculus II Ready
00Approach
Course Philosophy

The Limit Is the Foundation

Calculus is built on the limit — every definition, every theorem, every technique. This course takes limits seriously. Students who understand limits deeply don't struggle later when integration and series require them again.

Concepts Drive Technique

Most Calc I failures are conceptual, not computational. If you don't know what a derivative means, you can't set up a related rates problem. If you don't know what an integral is, you can't interpret the result. Concept first, mechanics second.

01Curriculum
Learning Path
01
Unit 1 · Weeks 1–2
Limits
3 lessons · Limit concept · Limit laws · Limits at infinity
limit notationindeterminate formslimits at infinity
+
L1
The Limit Concept
Two-sided and one-sided limits. The distinction between the limit at a point and the function value at that point — the most important distinction in calculus.
L2
Evaluating Limits Algebraically
Direct substitution. Indeterminate forms and how to resolve them: factoring, rationalizing, conjugate multiplication. When a limit does not exist.
L3
Limits at Infinity
End behavior formalized. Horizontal asymptotes re-examined through limits. Limits of rational functions as x grows large — the dominant term method.
Assignment 1
Evaluate 15 limits. For each, write which technique you used and why direct substitution failed or worked.
You evaluate any limit and know which technique applies before you start.
02
Unit 2 · Weeks 3–4
Continuity and the Derivative Defined
3 lessons · Continuity · The difference quotient · Differentiability
continuity definitiondifference quotientdifferentiability
+
L4
Continuity
Three conditions for continuity. Types of discontinuity — removable, jump, infinite. The Intermediate Value Theorem stated and applied.
L5
The Derivative as a Limit
The difference quotient as the slope of a secant line. The derivative as the limit of the difference quotient as the secant becomes a tangent. The derivative defined precisely.
L6
Differentiability vs. Continuity
Where derivatives fail to exist: corners, cusps, vertical tangents, discontinuities. Every differentiable function is continuous, but not every continuous function is differentiable.
Assignment 2
Find the derivative of 5 functions using only the limit definition. No rules. Check each with differentiation rules in Unit 3.
You know where the derivative comes from — not just how to compute it.
03
Unit 3 · Weeks 5–7
Differentiation Rules
3 lessons · Power rule · Product and quotient rules · Chain rule
power ruleproduct rulechain rule
+
L7
Power, Constant, Sum, Difference Rules
Mechanics and their limits origins. Differentiating polynomials, constants, and sums with full fluency.
L8
Product and Quotient Rules
Derived, not just stated. Common mistakes unpacked: the quotient rule denominator, distributing when you shouldn't. Simplifying before differentiating when possible.
L9
Chain Rule
Composite function structure — outside-inside. Recognizing the composition before differentiating. Multiple applications of the chain rule.
Assignment 3
Differentiate 20 functions. 10 require the chain rule. Annotate every step on 5 of them.
You differentiate any algebraic combination fluently.
04
Unit 4 · Weeks 8–9
Derivatives of Trig, Exp, and Log
3 lessons · Trig derivatives · e^x and ln(x) · Implicit differentiation
trig derivativese^x derivativeimplicit differentiation
+
L10
Derivatives of Trig Functions
Sin, cos, tan, and their reciprocals. The two special limits that make sin differentiable at 0. Chain rule applications with trig.
L11
Derivatives of e^x and ln(x)
Why e is defined the way it is — the one base whose exponential is its own derivative. Derivatives of a^x and log_a(x) via change of base.
L12
Implicit Differentiation
When y is not isolated. Differentiating both sides with respect to x, treating y as a function of x. Applications to curves like circles and ellipses.
Assignment 4
10 implicit differentiation problems. Find dy/dx and evaluate at a given point. Interpret the slope geometrically for each.
You differentiate any function — explicit or implicit, algebraic or transcendental.
05
Unit 5 · Weeks 10–12
Applications: Rates, Motion, Related Rates
3 lessons · Motion derivatives · Related rates · Linearization
position velocity accelerationrelated rates setuplinearization
+
L13
Position, Velocity, and Acceleration
Derivatives as rates of change. Reading motion from a graph — when is the object moving forward, backward, speeding up, slowing down.
L14
Related Rates
Setting up the implicit differentiation model from a geometric relationship. Diagram and equation before any differentiation. The setup is harder than the calculus.
L15
Linear Approximation and Differentials
The tangent line as a local model of a function. Using linearization to estimate nearby values. The differential as a tool for error analysis.
Assignment 5
Solve 5 related rates problems. Each must include a labeled diagram and a written setup paragraph before any calculus.
You translate a real-world rate problem into calculus without being told what to differentiate.
06
Unit 6 · Weeks 13–15
Curve Sketching and Optimization
4 lessons · MVT · First and second derivative tests · Curve sketching
MVTfirst derivative testconcavity
+
L16
Mean Value Theorem and Rolle's Theorem
What they guarantee and why it matters. Using MVT to prove a function has a root or to bound a function's values.
L17
Increasing, Decreasing, and First Derivative Test
Finding and classifying critical points. Intervals of increase and decrease from the sign of f'.
L18
Concavity, Inflection Points, Second Derivative Test
What the second derivative tells you: direction of bending. Inflection points. Using the second derivative test when the first test is inconclusive.
L19
Curve Sketching
Domain, intercepts, asymptotes, critical points, concavity — all six steps. Synthesizing everything into a complete sketch without a graphing calculator.
Assignment 6
Fully analyze and sketch 3 functions using all six steps. Explain in writing why one local max is a max and not a min.
You sketch any differentiable function from scratch using calculus.
07
Unit 7 · Weeks 16–18
Optimization and L'Hopital's Rule
3 lessons · Optimization · Closed interval method · L'Hopital
closed interval methodobjective functionL'Hopital
+
L20
Optimization: Closed Interval Method
Setting up the objective function before differentiating. Endpoints matter on a closed interval. Real-world optimization problems from geometry, economics, and physics.
L21
Unconstrained Optimization
When endpoints don't bound the problem. Verifying that a critical point is a minimum or maximum using the first or second derivative test.
L22
L'Hopital's Rule
Indeterminate forms 0/0 and ∞/∞. When not to use it: forms that aren't indeterminate. Converting 0·∞ and ∞-∞ forms before applying.
Assignment 7
Solve 5 optimization problems. Each setup must be written out in full before touching calculus.
You solve optimization problems by building models, not by matching to templates.
08
Unit 8 · Weeks 19–26
Integration and Capstone
4 lessons + capstone · Antiderivatives · Riemann sums · FTC
antiderivativesRiemann sumsFTC
+
L23
Antiderivatives and Indefinite Integrals
Reversing differentiation. Power rule for integration. The constant of integration and why it exists.
L24
The Definite Integral
Riemann sums as the foundation — left, right, midpoint. Area under a curve as a limit of sums. The definite integral as a signed area.
L25
The Fundamental Theorem of Calculus
Both parts. Part I: the derivative of an integral. Part II: evaluating definite integrals using antiderivatives. Why the theorem is fundamental.
L26 (cap)
Capstone: Simulated Final Exam
A full-length Calc I final exam simulation. Every missed problem corrected with reasoning — the decision that should have been made, not just the right answer.
Assignment 8
A full-length simulated Calc I final. Every missed problem gets a written correction with reasoning.
You finish Calc I with differentiation mastered and integration grounded.
02Reference
Core Concepts
lim
The Limit
The limit of f(x) as x approaches a is the value f(x) gets arbitrarily close to — regardless of what happens at x=a. The limit and the function value are different things.
One-sided · Two-sided · Limits at infinity
f'(x)
The Derivative
The derivative is the limit of the difference quotient as h approaches 0. It is the slope of the tangent line — the instantaneous rate of change. It is defined as a limit, not a rule.
Difference quotient · Differentiability · Notation
d/dx[f·g]
Product Rule
The derivative of a product is NOT the product of the derivatives. d/dx[fg] = f'g + fg'. The chain rule is the analogous result for compositions.
Quotient rule · Chain rule · Implicit differentiation
f''(x)
Concavity
f'' > 0 means f is concave up (opening upward). f'' < 0 means concave down. An inflection point is where concavity changes — not just where f'' = 0.
First derivative test · Second derivative test · MVT
∫f dx
The Integral
The definite integral is the limit of Riemann sums — the signed area between a curve and the x-axis. It is defined independently of the derivative, then connected to it by the FTC.
Riemann sums · FTC · Antiderivatives
FTC
Fundamental Theorem
Part I: the derivative of an integral with a variable upper limit is the integrand. Part II: a definite integral equals the antiderivative evaluated at the bounds. This connects differentiation and integration.
Part I · Part II · Net change theorem
03Capstone
Simulated Final Exam

The capstone is a full-length Calc I final exam simulation. Every missed problem becomes a lesson — not just the right answer, but the reasoning that should have driven each step. The goal is to walk into Calc II without a single gap.

○
Gap Review
Two weakest units revisited before the exam simulation.
○
Timed Final Exam
Full-length, independent, same conditions as a real exam.
○
Written Corrections
Every missed problem corrected with reasoning — not just the right answer.
○
Calculus II Readiness Review
A final session on what Calc II will demand from day one.
04Investment
Pricing Options
per Hour
$85
1-hr lesson
Flexible · No commitment
Online only
  • Flexible scheduling
  • 1-on-1 focused attention
  • No long-term commitment

Best for trying the curriculum before committing.

Book a session
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per Month
$650
4 lessons · 1.5 hours/lesson
1 lesson/week · book more to move faster
Online + In-Person available
  • Session notes after every lesson
  • Async DM support between sessions
  • Priority scheduling

Booking 2x/week draws down lessons faster — no extra charge.

Get started
Semester
$3,775
26 lessons · 1.5 hours/lesson
~6 months · save vs. monthly
Online + In-Person available
  • Full course arc
  • Session notes after every lesson
  • Capstone review + corrections
  • Next track readiness session

The full arc in one commitment.

Enroll now
In-person sessions available on Monthly and Semester plans at $800/mo and $4,625/semester. NYC only. Per-hour sessions are online only.
05Contact
Let's Talk

Starting Calc I for the first time or need a second pass? One session is enough to figure out where you are and what the path forward looks like.

What do I need to know coming in? +
Solid precalculus: function behavior, trig functions and identities, logarithms and exponentials. Shaky on any of those? The Precalculus track is the right starting point.
Is this AP Calc AB or BC? +
This track covers AB completely. BC topics (integration techniques, series, parametric and polar) are in the Calculus II track.
Are lessons online or in-person? +
Per-hour sessions are online only. Monthly and Semester plans include the option for in-person sessions in NYC.
What if I need to reschedule? +
Give 24 hours notice and we'll find another time. Monthly and Semester students get priority rebooking.