Glossary

The single largest or smallest value a function attains over its entire domain (or a given interval).

How fast velocity itself is changing.

A function built by running a "total so far" tally of another function's area underneath it.

A function whose derivative is the given function — differentiation run in reverse.

The region trapped between two graphs, found by integrating the difference of the top and bottom functions.

A line a curve gets closer and closer to without ever quite touching it.

How much a quantity changes per unit input, averaged over an entire interval.

The height of the flat rectangle whose area over an interval matches the actual area under the curve.

The rule for differentiating a composite function — multiply the outer derivative by the inner derivative.

Which way a curve bends — up like a cup (f''>0) or down like a frown (f''<0).

A function has no breaks, jumps, or holes at a point — its limit there equals its actual value.

A function is continuous at every point of an interval, with one-sided continuity required at closed endpoints.

A point in a function's domain where its derivative is zero or undefined — the only places extrema can occur.

The signed area between a curve and the x-axis over an interval, written \int_a^b f(x)\,dx.

The instantaneous rate of change of a function — the slope of its tangent line at a point.

The expression \dfrac{f(x+h)-f(x)}{h} — the slope of a secant line through two nearby points on a curve.

A function has a well-defined derivative at a point — its graph has a single, unambiguous tangent line there.

The linear-approximation notation dy = f'(x)\,dx — the predicted change in y for a small nudge dx in x.

Finding the volume of a solid of revolution by slicing it into circular disks perpendicular to the axis of rotation.

The precise, quantitative statement of what a limit means, using tolerances \varepsilon and \delta.

The small raised number that says how many times to multiply a base by itself.

A function that multiplies by the same fixed factor every time its input goes up by one.

Classify a critical point as a local max, local min, or neither by watching whether f' changes sign there.

The theorem linking differentiation and integration as inverse operations, in two parts.

The derivative of a derivative — how the rate of change is itself changing.

Differentiating both sides of an equation that isn't already solved for y, treating y as a function of x.

The notation \int f(x)\,dx = F(x)+C for the entire family of antiderivatives of f.

A break where the function shoots off toward positive or negative infinity instead of settling on a value.

A limit that grows without bound — the function's outputs increase or decrease without settling on a number.

A point where a curve switches from bending one way to bending the other.

A "find the antiderivative" question that also pins down the free constant using one known starting value.

The rate of change at a single instant — the limit of average rates over shrinking intervals.

The function written right after the integral sign — the thing actually being integrated.

A point strictly between the two ends of an interval, not sitting on either edge.

A continuous function on [a,b] hits every value between f(a) and f(b) at least once.

A break where the function's value suddenly steps from one level to another — the one- and two-sided limits disagree.

For a 0/0 or \infty/\infty limit of a quotient, replace numerator and denominator with their derivatives and try again.

The value a function's outputs approach as the input approaches some point, without necessarily reaching it.

The value a function's outputs approach as the input itself grows without bound.

The algebraic rules — sum, product, quotient, power — that let limits of combined functions be computed from limits of their pieces.

Using a function's tangent line at a nearby point to estimate its value, since a smooth curve looks almost straight up close.

A point where a function's value is largest or smallest compared only to its immediate neighbors.

A continuous, differentiable function's instantaneous rate of change matches its average rate of change somewhere on the interval.

The total change in a quantity over an interval equals the definite integral of its rate of change.

An iterative procedure that uses tangent lines to home in on a root of a function.

The value a function approaches from only one direction — the left or the right.

Finding the input that makes some quantity (area, cost, time, ...) as large or as small as possible, subject to constraints.

An expression built by adding and subtracting whole-number powers of a variable, each multiplied by a fixed number.

The shortcut \dfrac{d}{dx}[x^n] = nx^{n-1} for differentiating a power of x.

The rule for differentiating a product of two functions — it is not just the product of the derivatives.

A phrase that says how many values a statement covers, "for every" (\forall) or "there exists" (\exists).

The rule for differentiating a ratio of two functions.

Problems where several quantities are linked by an equation, and one rate of change is found from another.

A single-point hole in a graph that could be patched by defining (or redefining) one value.

An approximation of the area under a curve by adding up areas of thin rectangles.

If a continuous, differentiable function returns to the same value at both ends of an interval, its derivative is zero somewhere in between.

The backward reasoning, from the goal |f(x)-L|<\varepsilon, used to discover a working \delta; it is not part of the written proof.

Classify a critical point as a local max or min by checking the sign of the second derivative there instead of watching for a sign change.

The compact \sum symbol for writing a sum of many similar terms without writing every one out.

If a function is trapped between two others that share the same limit, it shares that limit too.

The line that touches a curve at one point and matches its exact direction there.

An area-approximation method that uses trapezoids instead of rectangles, usually more accurate for the same number of pieces.

The integration technique that reverses the chain rule by renaming an inner expression u and rewriting the integral in terms of it.

How fast something's position is changing, including which direction it's moving.

Finding the volume of a solid of revolution with a hole through it, by slicing into ring-shaped washers.