Calculus
Analysis
Single-variable calculus at the AP Calculus BC level: limits, derivatives and their applications, integrals and the fundamental theorem, techniques and applications of integration, differential equations, series, and parametric and polar calculus.
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Approaching Without Arriving
Unit 1 · Limits and Continuity
0 of 12 units complete
- 1Unit 1Limits and Continuity5 parts · about 11 min
Approaching Without Arriving
Limits describe where a function is heading. Evaluate them with algebra, squeeze a wild oscillation to a single value, classify the ways a graph can break, and prove a root exists without finding it.
- 2Unit 2The Derivative5 parts · about 11 min
Slope at a Single Point
The derivative is the slope of a curve at one point. Drag a tangent line along a graph, discover the power rule, find the one exponential whose slope equals its height, and learn the product and quotient rules.
- 3Unit 3Chain, Implicit and Inverse Differentiation5 parts · about 13 min
Gears, Circles and Backwards Functions
The chain rule multiplies rates like gears, implicit differentiation finds slopes on curves that are not functions, and inverse functions get their derivatives for free.
- 4Unit 4Contextual Applications of Differentiation4 parts · about 11 min
Rates in the Real World
Derivatives as velocity and acceleration, a ladder sliding down a wall, linear approximations your calculator would be proud of, and L'Hôpital's trick for 0/0.
- 5Unit 5Analytical Applications of Differentiation5 parts · about 10 min
Shapes from Slopes
The sign of the derivative says up or down, the second derivative says which way the curve bends. Find peaks and valleys, the mean value theorem's promise, and the best box you can fold.
- 6Unit 6Integration and the Fundamental Theorem5 parts · about 12 min
Adding Up Infinitely Thin Slices
Approximate an area with rectangles, watch the error vanish as they get thinner, discover that areas undo derivatives (the Fundamental Theorem), and integrate by substitution.
- 7Unit 7Techniques of Integration5 parts · about 12 min
The Integration Toolbox
Integration by parts, partial fractions, trig integrals, and integrals that run off to infinity — including a horn you can fill with paint but never coat with it.
- 8Unit 8Applications of Integration5 parts · about 10 min
What Integrals Measure
Area between curves, volumes built from stacks of disks, the volume of a cone proved by slicing, arc length, and why a hanging chain is not a parabola.
- 9Unit 9Differential Equations5 parts · about 12 min
Equations About Change
A differential equation describes how something changes. Read solutions straight off a slope field, solve separable equations, model growth, cooling and spreading rumours, and step forward with Euler's method.
- 10Unit 10Parametric, Polar and Vector-Valued Functions5 parts · about 10 min
Calculus on Curves
Slopes and lengths of parametric curves, velocity and speed as vectors, and areas swept out in polar coordinates — including the heart-shaped cardioid and Kepler's law of equal areas.
- 11Unit 11Infinite Sequences and Series5 parts · about 12 min
Adding Forever
When does an infinite sum settle down? The harmonic series that sneaks off to infinity, the tests that decide convergence, and Taylor series that rebuild sine out of nothing but powers of x.
- 12Unit 12Non-Elementary Integration5 parts · about 10 min
Integrals Without Formulas
Some integrals have no formula at all, yet their exact values over special intervals are known. Meet the bell curve's area, the symmetry tricks King's rule and Frullani, and the Gamma function that gives factorials of fractions.