# Calculus

Calculus is the mathematics of change and accumulation. Its whole
edifice rests on a single idea — the **limit** — which makes
precise what it means to approach a value without ever arriving.


_Figure 001 — The derivative: a secant line sweeps to the tangent as Δx → 0, giving the
instantaneous slope f'(x).
_

_Figure 002 — The integral: midpoint rectangles refine into the exact area under a
curve.
_

Everything begins with the tangent-line problem. Fix a point on a curve,
take a second nearby, and draw the line through both. As the second point
slides in, that secant pivots toward a single limiting line — the
**tangent**, whose slope is the derivative.


The derivative is an _instantaneous rate of change_: velocity from
position, marginal cost from cost, slope from height. Differentiation
turns a function into its rate function, and a small kit of rules — the
product, quotient, and chain rules — differentiates almost anything you
can write down.


_Figure 003 — The ε–δ definition: for every tolerance ε there is a neighbourhood δ whose
graph stays inside the band.
_

Integration runs the other way. Instead of measuring how fast a quantity
changes, it **accumulates** — summing infinitely many
infinitesimal contributions into an area, a volume, a total distance.
The definite integral is the limit of Riemann sums as the rectangles
thin.


The two operations are inverses. The Fundamental Theorem of
Calculus says that differentiation and integration undo each
other, which is why an area problem can be solved by finding an
antiderivative rather than summing rectangles by hand.


_Figure 004 — Taylor series: polynomials of rising degree hug a function over a widening
interval.
_

Push the limit further and functions become infinite sums. A
**Taylor series** rebuilds a function from its derivatives at
a point, so a few polynomial terms approximate sine, the exponential, or a
logarithm as closely as you please.


_Figure 005 — Optimization: at a maximum the tangent goes flat and f'(x) = 0.
_

The payoff is everywhere. Setting the derivative to zero locates maxima
and minima — the heart of _optimization_ — while related rates,
arc length, and accumulated change turn geometry and physics into
solvable equations.


The same three ideas — limit, derivative, integral — then extend to
curves and surfaces and to fields in space, ending with the great
theorems of Green, Stokes, and Gauss that unify them all.


---

## Contents

### 1. Limits and Continuity

1. [Functions and Mathematical Models](/calculus/limits-and-continuity/functions-and-models)
2. [The Limit of a Function](/calculus/limits-and-continuity/the-limit-of-a-function)
3. [Limit Laws and the ε–δ Definition](/calculus/limits-and-continuity/limit-laws-and-the-precise-definition)
4. [Continuity](/calculus/limits-and-continuity/continuity)

### 2. Derivatives

1. [The Derivative and Rates of Change](/calculus/derivatives/the-derivative-and-rates-of-change)
2. [Differentiation Rules and the Chain Rule](/calculus/derivatives/differentiation-rules-and-the-chain-rule)
3. [Implicit Differentiation and Related Rates](/calculus/derivatives/implicit-differentiation-and-related-rates)
4. [Linear Approximations and Differentials](/calculus/derivatives/linear-approximations-and-differentials)

### 3. Applications of Derivatives

1. [Extrema and the Mean Value Theorem](/calculus/applications-of-derivatives/extrema-and-the-mean-value-theorem)
2. [How Derivatives Shape a Graph](/calculus/applications-of-derivatives/how-derivatives-shape-a-graph)
3. [Curve Sketching and Optimization](/calculus/applications-of-derivatives/curve-sketching-and-optimization)
4. [Newton's Method and Antiderivatives](/calculus/applications-of-derivatives/newtons-method-and-antiderivatives)

### 4. Integrals

1. [Area and the Definite Integral](/calculus/integrals/area-and-the-definite-integral)
2. [The Fundamental Theorem of Calculus](/calculus/integrals/the-fundamental-theorem-of-calculus)
3. [The Substitution Rule](/calculus/integrals/the-substitution-rule)

### 5. Applications of Integration

1. [Areas Between Curves and Volumes](/calculus/applications-of-integration/areas-and-volumes)
2. [Work, Average Value, Arc Length, and Surface Area](/calculus/applications-of-integration/work-average-value-and-arc-length)
3. [Applications to Physics, Economics, and Probability](/calculus/applications-of-integration/physics-economics-and-probability)

### 6. Exponential, Logarithmic, and Inverse Functions

1. [Inverse Functions, Logarithms, and Exponentials](/calculus/exponential-logarithmic-and-inverse-functions/inverse-functions-logarithms-and-exponentials)
2. [Growth, Decay, Inverse Trigonometric, and Hyperbolic Functions](/calculus/exponential-logarithmic-and-inverse-functions/growth-decay-inverse-trig-and-hyperbolic-functions)
3. [Indeterminate Forms and l'Hospital's Rule](/calculus/exponential-logarithmic-and-inverse-functions/lhospitals-rule)

### 7. Techniques of Integration

1. [Integration by Parts](/calculus/techniques-of-integration/integration-by-parts)
2. [Trigonometric Integrals and Substitution](/calculus/techniques-of-integration/trigonometric-integrals-and-substitution)
3. [Partial Fractions and Integration Strategy](/calculus/techniques-of-integration/partial-fractions-and-integration-strategy)
4. [Approximate and Improper Integrals](/calculus/techniques-of-integration/approximate-and-improper-integrals)

### 8. Parametric Equations and Polar Coordinates

1. [Parametric Curves and Their Calculus](/calculus/parametric-and-polar/parametric-curves-and-their-calculus)
2. [Polar Coordinates](/calculus/parametric-and-polar/polar-coordinates)
3. [Conic Sections](/calculus/parametric-and-polar/conic-sections)

### 9. Infinite Sequences and Series

1. [Sequences](/calculus/sequences-and-series/sequences)
2. [Series and the Integral Test](/calculus/sequences-and-series/series-and-the-integral-test)
3. [The Convergence Tests](/calculus/sequences-and-series/the-convergence-tests)
4. [Power Series](/calculus/sequences-and-series/power-series)
5. [Taylor and Maclaurin Series](/calculus/sequences-and-series/taylor-and-maclaurin-series)

### 10. Vectors and the Geometry of Space

1. [Three-Dimensional Coordinates, Vectors, and the Dot Product](/calculus/vectors-and-space-curves/vectors-and-the-dot-product)
2. [The Cross Product, Lines, and Planes](/calculus/vectors-and-space-curves/the-cross-product-lines-and-planes)
3. [Cylinders and Quadric Surfaces](/calculus/vectors-and-space-curves/cylinders-and-quadric-surfaces)
4. [Vector Functions and Space Curves](/calculus/vectors-and-space-curves/vector-functions-and-space-curves)
5. [Arc Length, Curvature, and Motion in Space](/calculus/vectors-and-space-curves/arc-length-curvature-and-motion)

### 11. Partial Derivatives

1. [Functions of Several Variables, Limits, and Continuity](/calculus/partial-derivatives/functions-of-several-variables)
2. [Partial Derivatives](/calculus/partial-derivatives/partial-derivatives)
3. [Tangent Planes, Linear Approximation, and the Chain Rule](/calculus/partial-derivatives/tangent-planes-and-the-chain-rule)
4. [Directional Derivatives and the Gradient](/calculus/partial-derivatives/directional-derivatives-and-the-gradient)
5. [Optimization and Lagrange Multipliers](/calculus/partial-derivatives/optimization-and-lagrange-multipliers)

### 12. Multiple Integrals and Vector Calculus

1. [Double Integrals](/calculus/multiple-integrals-and-vector-calculus/double-integrals)
2. [Triple Integrals and Coordinate Systems](/calculus/multiple-integrals-and-vector-calculus/triple-integrals-and-coordinate-systems)
3. [Vector Fields and Line Integrals](/calculus/multiple-integrals-and-vector-calculus/vector-fields-and-line-integrals)
4. [Green's Theorem, Curl, and Divergence](/calculus/multiple-integrals-and-vector-calculus/greens-theorem-curl-and-divergence)
5. [Parametric Surfaces and Surface Integrals](/calculus/multiple-integrals-and-vector-calculus/surface-integrals)
6. [Stokes' Theorem and the Divergence Theorem](/calculus/multiple-integrals-and-vector-calculus/stokes-and-the-divergence-theorem)