🎯Calculus I

The toolkit calculus assumes you already own: what a function is, how its graph moves, and the trig, exponential, and log functions in every later chapter.

1.1 What a Function Actually Is · 1.1 Domain and Range from a Formula and a Graph · 1.1 Piecewise-Defined Functions · 1.1 Shifting, Stretching, and Reflecting Graphs · +11 more

Where calculus starts: the tangent and area problems, limits pinned down numerically and with epsilon and delta, continuity proved rather than assumed.

2.1 The Two Problems That Invented Calculus · 2.2 Estimating a Limit from a Table of Values · 2.3 Estimating a Limit from a Graph · 2.4 One-Sided Limits · +14 more

The derivative born from a limit, the shortcuts that make it fast to compute, and what it means for a function to fail to have one.

3.1 The Derivative as the Slope of a Tangent Line · 3.1 The Difference Quotient and the Limit Definition of the Derivative · 3.1 Computing a Derivative Directly from the Definition · 3.2 The Derivative as a Function · +9 more

Trig, exponential, log, inverse, and hyperbolic functions get their own derivative rules, and the chain rule links every composed function back to them.

3.5 Derivatives of Sine and Cosine · 3.5 Derivatives of the Other Four Trigonometric Functions · 3.6 The Chain Rule · 3.6 The Chain Rule with More Than Two Layers · +10 more

Two quantities linked through a shared equation, a tangent line standing in for a curve, and the theorem guaranteeing a derivative matches an average rate.

4.1 Setting Up a Related Rates Problem · 4.1 Related Rates with a Sliding Ladder · 4.1 Related Rates with a Shrinking or Growing Shape · 4.2 Linear Approximation and the Tangent Line · +4 more

Where a function peaks and dips, how its graph bends, the theorems guaranteeing those peaks exist, and turning a real constraint into the best answer.

4.3 Critical Points · 4.3 The Extreme Value Theorem · 4.3 The Candidates Test for Absolute Extrema · 4.5 Increasing and Decreasing Intervals from the First Derivative · +13 more

Running differentiation backward, then building the definite integral from scratch out of thinner and thinner rectangles.

4.10 Reversing Differentiation: Antiderivatives · 4.10 Antiderivatives of Trigonometric and Exponential Functions · 4.10 Solving an Initial Value Problem · 5.1 Approximating Area with Left, Right, and Midpoint Sums · +6 more

The theorem tying every rectangle-counting limit back to antiderivatives, proved in full, plus the substitution toolkit for evaluating integrals fast.

5.3 The Fundamental Theorem of Calculus, Part 1: The Accumulation Function · 5.3 Proving the Fundamental Theorem of Calculus, Part 1 · 5.3 The Fundamental Theorem of Calculus, Part 2: Evaluating Definite Integrals · 5.4 The Net Change Theorem · +5 more

The definite integral at work: area between curves, solids built by slicing and revolving, motion from velocity, work against a force, exponential growth.

6.1 The Area Between Two Curves · 6.1 The Area Between Curves That Cross More Than Once · 6.1 Finding Area by Integrating with Respect to y · 6.2 Volumes by Slicing with General Cross-Sections · +17 more