What is a limit in calculus?

A limit describes the value a function approaches as the input approaches a specified point. The function does not need to equal that value at the point itself -- the limit concerns only the behavior during the approach. A two-sided limit exists only when both one-sided limits exist and agree.

What is the relationship between continuity and differentiability?

Continuity means the function has no breaks, jumps, or holes at a point. Differentiability means the function has a well-defined tangent line there. Differentiability implies continuity, but not the reverse -- a function can be continuous at a corner or cusp where no single tangent line exists.

What is the difference between a definite and indefinite integral?

An indefinite integral produces a family of functions (antiderivatives) plus a constant C. A definite integral produces a single number representing the accumulated signed area under the curve between two bounds. The Fundamental Theorem of Calculus connects the two: evaluate the antiderivative at the bounds and subtract.

What is a critical point and how is it used?

A critical point is a value where the derivative equals zero or does not exist. Critical points are the only candidates for local maxima and minima. They are classified using the first derivative test (sign change of the derivative) or the second derivative test (sign of the second derivative).

What does signed area mean in integration?

The definite integral measures signed area: regions where the function is positive contribute positive area, and regions where it is negative contribute negative area. The integral can be zero even when substantial area exists, if positive and negative regions cancel. To find total area, integrate the absolute value of the function.

Calculus Terms and Definitions

About This Glossary

This glossary organizes 29 calculus terms into three categories that follow the natural progression of the subject.

Limits establishes the foundational language with 6 entries: limit, one-sided limit, continuity, discontinuity, indeterminate form, and asymptote. These terms define what it means for a function to approach a value and when that approach breaks down.

Derivatives covers 14 entries on rates of change and curve analysis: the derivative itself, differentiability, differentials, higher-order derivatives, partial derivatives, instantaneous and average rates of change, tangent lines, critical points, local extrema, concavity, inflection points, and monotonic functions. Together these terms describe how functions change and how their graphs bend.

Integrals spans 9 entries on accumulation: antiderivatives, indefinite and definite integrals, the integrand, bounds of integration, Riemann sums, improper integrals, signed area, and the average value of a function. These terms cover both the process of reversing differentiation and the geometric interpretation of area under a curve.

Each definition includes an intuitive explanation, key properties, common errors where applicable, and links to the detailed lesson page. Use the search bar or category filters above to navigate.

DerivativesIntegralsLimits

Derivatives(13)

Integrals(9)

Limits(6)

28 of 28 terms

28 terms

Limits

(6 items)

Limit

The value that

f(x)

approaches as

x

approaches a specified point

a

\lim_{x \to a} f(x) = L

means

f(x)

can be made arbitrarily close to

L

by taking

x

sufficiently close to

a

The Central Idea of a Limit

See details

↑ Back to top

intuitionnotationrelated concepts

A limit describes where a function is headed, not where it is. The function value at

a

may differ from

L

, or may not exist at all — the limit cares only about the approach.

↑ Back to top

One-Sided Limit

The value

f(x)

approaches as

x

approaches

a

from one direction only:

\lim_{x \to a^-} f(x)

(from the left) or

\lim_{x \to a^+} f(x)

(from the right).

Left-Hand Limits

See details

↑ Back to top

intuitionnotationrelated concepts

Sometimes a function behaves differently depending on the direction of approach. One-sided limits isolate each direction. The two-sided limit exists precisely when both one-sided limits exist and are equal.

↑ Back to top

Continuity

A function

f

is continuous at

x = a

if three conditions hold:

f(a)

is defined,

\lim_{x \to a} f(x)

exists, and

\lim_{x \to a} f(x) = f(a)

The Formal Definition

See details

↑ Back to top

intuitionpropertiesrelated concepts

No breaks, no jumps, no holes. The function value matches what the surrounding values predict. Small changes in input produce small changes in output — no surprises.

↑ Back to top

Discontinuity

A point where a function fails to be continuous — at least one of the three continuity conditions is violated.

Types of Discontinuities

See details

↑ Back to top

intuitionspecial casesrelated concepts

Something goes wrong at this point: the function jumps, blows up, oscillates, or has a hole. The type of failure determines whether the discontinuity can be repaired.

↑ Back to top

Indeterminate Form

An expression arising from direct substitution in a limit whose value cannot be determined without further analysis:

\frac{0}{0}

\frac{\infty}{\infty}

0 \cdot \infty

\infty - \infty

0^0

1^\infty

\infty^0

Indeterminate Forms

See details

↑ Back to top

intuitioncommon errorsrelated concepts

A competition between opposing tendencies whose outcome depends on relative rates. In

\frac{0}{0}

, both numerator and denominator vanish — which vanishes faster determines whether the limit is

0

, finite, or infinite.

↑ Back to top

Asymptote

A line that the graph of a function approaches arbitrarily closely as

x

f(x)

tends toward infinity or a boundary point.

Horizontal Asymptotes

See details

↑ Back to top

intuitionspecial casesrelated concepts

A boundary the curve chases but never settles on. Horizontal asymptotes describe long-run behavior as

x \to \pm\infty

. Vertical asymptotes mark points where the function blows up.

↑ Back to top

Derivatives

(13 items)

Derivative

The instantaneous rate of change of

f

x = a

, defined as

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

, when this limit exists.

The Difference Quotient and Its Limit

See details

↑ Back to top

intuitionnotationrelated concepts

The slope of the tangent line at a single point. The derivative captures how fast the function changes at that exact instant — not over an interval, but at a point.

↑ Back to top

Differentiability

A function

f

is differentiable at

x = a

\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

exists and is finite.

The Definition of Differentiability

See details

↑ Back to top

intuitionspecial casesrelated concepts

The graph has a well-defined, non-vertical tangent line at the point. The secant lines from both sides converge to the same slope. Differentiability implies continuity, but not the reverse.

↑ Back to top

Differential

The independent differential

dx

is a freely chosen increment in

x

; the dependent differential is

dy = f'(x) \cdot dx

, the change predicted by the tangent line.

The Differential dy

See details

↑ Back to top

intuitionpropertiesrelated concepts

Differentials separate the derivative

\frac{dy}{dx}

into two independent quantities. The differential

dy

approximates the actual change

\Delta y

— exact for linear functions, increasingly approximate for curved ones as

dx

grows.

↑ Back to top

Higher-Order Derivative

The

n

th derivative

f^{(n)}(x)

, obtained by differentiating

f

a total of

n

times:

f''(x) = \frac{d^2y}{dx^2}

f'''(x) = \frac{d^3y}{dx^3}

, and so on.

Definition and Notation

See details

↑ Back to top

intuitionpropertiesrelated concepts

Each derivative measures how the previous one changes. The second derivative captures concavity — how the slope itself varies. The third and beyond capture progressively finer aspects of the curve's shape.

↑ Back to top

Partial Derivative

The derivative of a multivariable function with respect to one variable while all others are held constant:

\frac{\partial f}{\partial x}

Differentials of Multiple Variables

See details

↑ Back to top

intuitionnotationrelated concepts

Freeze every variable except one, then differentiate as usual. The partial derivative measures the function's sensitivity to changes in that single variable, treating the others as constants.

↑ Back to top

Instantaneous Rate of Change

The rate of change of

f

at a single point

x = a

, equal to the derivative

f'(a)

: the limit of average rates of change as the interval shrinks to zero.

The Derivative as Rate of Change

See details

↑ Back to top

intuitioncommon errorsrelated concepts

How fast the function is changing right now — not over a span, but at one frozen instant. Velocity is the instantaneous rate of change of position; marginal cost is the instantaneous rate of change of total cost.

↑ Back to top

Average Rate of Change

The ratio

\frac{f(b) - f(a)}{b - a}

, measuring the overall change in

f

per unit change in input over the interval

[a, b]

The Difference Quotient and Its Limit

See details

↑ Back to top

intuitionpropertiesrelated concepts

The slope of the secant line connecting

(a, f(a))

and

(b, f(b))

. It summarizes the function's total change over the interval without revealing what happens in between.

↑ Back to top

Tangent Line

The line through

(a, f(a))

with slope

f'(a)

y - f(a) = f'(a)(x - a)

Tangent Lines

See details

↑ Back to top

intuitionpropertiesrelated concepts

The best linear approximation to the curve at a single point. Near the point of tangency, the tangent line and the curve are nearly indistinguishable. This is the geometric foundation of differentials and linearization.

↑ Back to top

Critical Point

A value

x = c

in the domain of

f

where

f'(c) = 0

f'(c)

does not exist.

Critical Points

See details

↑ Back to top

intuitionpropertiesrelated concepts

The only candidates for local extrema. If

f

has a local maximum or minimum at

c

, then

c

must be a critical point. But not every critical point is an extremum —

f(x) = x^3

has a critical point at

x = 0

with no extremum.

↑ Back to top

Local Extremum

A point where

f

achieves a value greater than (local maximum) or less than (local minimum) all nearby values:

f(c) \geq f(x)

f(c) \leq f(x)

for all

x

in some open interval around

c

The First Derivative Test

See details

↑ Back to top

intuitionpropertiesrelated concepts

A peak or valley in the immediate neighborhood. The function rises to a local max then falls, or falls to a local min then rises. Farther away, higher peaks or deeper valleys may exist.

↑ Back to top

Concavity

A property describing how the slope of

f

changes: concave up where

f''(x) > 0

(slope increasing), concave down where

f''(x) < 0

(slope decreasing).

Concavity

See details

↑ Back to top

intuitionpropertiesrelated concepts

Concavity describes bending, not direction. A function can rise while bending downward (decelerating) or fall while bending upward (decelerating in the negative direction). Concave up means the curve lies above its tangent lines; concave down means below.

↑ Back to top

Inflection Point

A point on the graph of

f

where the concavity changes — from concave up to concave down, or the reverse.

Inflection Points

See details

↑ Back to top

intuitioncommon errorsrelated concepts

Where the bending reverses direction. The curve transitions from cupping upward to cupping downward (or vice versa). On the graph of

f'

, inflection points of

f

appear as local extrema.

↑ Back to top

Monotonic Function

A function that is entirely non-decreasing or entirely non-increasing on an interval. Strictly monotonic: strictly increasing (

a < b \implies f(a) < f(b)

) or strictly decreasing (

a < b \implies f(a) > f(b)

Increasing and Decreasing Functions

See details

↑ Back to top

intuitionpropertiesrelated concepts

A function that moves in only one direction — it never reverses. Strictly increasing means the graph only rises; strictly decreasing means it only falls. No turning points, no plateaus (in the strict case).

↑ Back to top

Integrals

(9 items)

Antiderivative

A function

F

whose derivative equals the given function:

F'(x) = f(x)

. Also called a primitive.

Antiderivatives

See details

↑ Back to top

intuitionpropertiesrelated concepts

The reverse of differentiation. Given a rate of change, the antiderivative recovers the original quantity. Since the derivative of any constant is zero, antiderivatives are never unique — they form a family

F(x) + C

↑ Back to top

Indefinite Integral

The general antiderivative of

f

, written

\int f(x)\,dx = F(x) + C

, representing the entire family of functions whose derivative is

f

Notation

See details

↑ Back to top

intuitioncommon errorsrelated concepts

The notation

\int f(x)\,dx

asks: what function, when differentiated, gives back

f(x)

? The answer is a family, not a single function — the constant

C

captures the vertical ambiguity that differentiation erases.

↑ Back to top

Definite Integral

The limit of Riemann sums over

[a, b]

\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x

, yielding a number that represents accumulated signed area.

Notation and Meaning

See details

↑ Back to top

intuitionpropertiesrelated concepts

Continuous summation over an interval. The definite integral accumulates the values of

f

from

a

b

, producing a single number — not a function. Regions above the

x

-axis contribute positively; regions below contribute negatively.

↑ Back to top

Integrand

The function

f(x)

appearing inside an integral expression

\int f(x)\,dx

— the function being integrated.

Notation and Terminology

See details

↑ Back to top

intuitionrelated concepts

The integrand specifies what is being accumulated at each point. In a definite integral,

f(x)\,dx

represents an infinitesimal contribution to the total; the integral sums all such contributions across the interval.

↑ Back to top

Bounds of Integration

The values

a

(lower bound) and

b

(upper bound) in a definite integral

\int_a^b f(x)\,dx

, specifying where accumulation begins and ends.

Notation and Meaning

See details

↑ Back to top

intuitioncommon errorsrelated concepts

The boundaries of the interval over which the function is accumulated. The lower bound sits at the bottom of the integral sign, the upper bound at the top. Swapping them negates the result.

↑ Back to top

Riemann Sum

An approximation to the definite integral formed by partitioning

[a, b]

into subintervals and summing rectangular areas:

S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x

The Riemann Sum Construction

See details

↑ Back to top

intuitionpropertiesrelated concepts

Approximate the area under a curve with rectangles. Each rectangle has width

\Delta x

and height

f(x_i^*)

at some sample point. As the rectangles become infinitely thin (

n \to \infty

), the sum converges to the exact integral.

↑ Back to top

Improper Integral

A definite integral where the interval is infinite or the integrand is unbounded within the interval, evaluated as a limit of proper integrals.

What Makes an Integral Improper?

See details

↑ Back to top

intuitionspecial casesrelated concepts

Standard integration requires a finite interval and a bounded function. When either condition fails, the integral is improper — it must be computed as a limit. The integral converges if the limit is finite; it diverges otherwise.

↑ Back to top

Signed Area

The value of a definite integral interpreted geometrically: area above the

x

-axis counts as positive, area below counts as negative.

Signed Area Interpretation

See details

↑ Back to top

intuitionpropertiesrelated concepts

The integral does not simply measure area — it measures directed area. Regions where

f(x) > 0

add to the total; regions where

f(x) < 0

subtract. The integral can be zero even when substantial area exists, if positive and negative regions cancel.

↑ Back to top

Average Value of a Function

The mean output of

f

over

[a, b]

f_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x)\,dx

Average Value of a Function

See details

↑ Back to top

intuitionpropertiesrelated concepts

The height of a rectangle with base

[a, b]

whose area equals the area under the curve. It generalizes the discrete average to continuous functions: sum everything up (integrate), then divide by the length of the interval.

↑ Back to top