Calculus Glossary

A function whose derivative gives you the function you started with. If \( F'(x) = f(x) \), then \( F \) is an antiderivative of \( f \). Because any constant's derivative is zero, there are infinitely many antiderivatives — they differ only by a constant, which is why we write \( +C \).

Example: \( \frac{x^3}{3} + C \) is an antiderivative of \( x^2 \), since \( \frac{d}{dx}\left(\frac{x^3}{3}\right) = x^2 \).

A line that a curve approaches but never quite reaches. A vertical asymptote at \( x = a \) means the function blows up to infinity as \( x \to a \). A horizontal asymptote at \( y = L \) means the function levels off toward \( L \) as \( x \to \pm\infty \).

The rule for differentiating a composite function — a function applied to another function. The derivative is the outer derivative (evaluated at the inner function) multiplied by the derivative of the inner function. \( \frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x) \). Often described as "working from the outside in, one layer at a time."

Whether a function curves upward (like a bowl) or downward (like a hill) at a given point. A function is concave up where \( f''(x) > 0 \) — the slope is increasing. It is concave down where \( f''(x) < 0 \) — the slope is decreasing. Concavity tells you how the rate of change is itself changing.

The unknown constant added to every indefinite integral result. Because differentiating any constant gives zero, the antiderivative of a function is not unique — \( x^2 + 5 \) and \( x^2 - 100 \) both have derivative \( 2x \). Writing \( +C \) acknowledges this entire family of solutions. Forgetting it is one of the most common AP exam mistakes.

A function is continuous at a point if three conditions hold: it exists at that point, its limit exists at that point, and the limit equals the function value. Intuitively: you can draw the graph through that point without lifting your pen. Differentiability implies continuity, but continuity does not imply differentiability — a sharp corner is continuous but not differentiable.

A point where \( f'(x) = 0 \) or \( f'(x) \) is undefined. Critical points are candidates for local maxima and minima — but not every critical point is an extremum (a flat inflection point, for example, has \( f' = 0 \) but is neither a max nor a min). Always test with the first or second derivative test to classify them.

The integral of a function over a specific interval \([a, b]\), producing a number rather than a function. Geometrically, it represents the signed area between the curve and the x-axis on that interval — positive where the curve is above the axis, negative where it's below. Notation: \( \int_a^b f(x)\,dx \).

The instantaneous rate of change of a function at a point. Geometrically, it's the slope of the tangent line to the curve at that point. Defined as the limit: \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). In context, if \( f(t) \) is position, then \( f'(t) \) is velocity.

An equation that relates a function to one or more of its derivatives. Instead of solving for a number, you solve for a function. \( \frac{dy}{dx} = ky \) is a simple differential equation whose solution is \( y = Ce^{kx} \) — the model for exponential growth and decay.

The process of finding a derivative. Synonymous with "taking the derivative." When you apply the power rule, chain rule, or any other differentiation rule, you are performing differentiation.

The mathematical constant \( e \approx 2.71828 \ldots \), the base of the natural logarithm. Its defining property is that \( \frac{d}{dx}e^x = e^x \) — it is its own derivative. This makes it the natural choice for modeling continuous growth and decay in science, finance, and engineering.

A maximum or minimum value of a function. A local (relative) extremum is a peak or valley in a neighborhood — higher (or lower) than nearby points. A global (absolute) extremum is the highest or lowest value on the entire domain or interval being considered. The Extreme Value Theorem guarantees at least one of each on a closed, continuous interval.

The theorem that links differentiation and integration — the two branches of calculus. It has two parts:

Part 1: If \( F(x) = \int_a^x f(t)\,dt \), then \( F'(x) = f(x) \). Differentiating an integral gives back the integrand.

Part 2: \( \int_a^b f(x)\,dx = F(b) - F(a) \), where \( F \) is any antiderivative of \( f \). This is how you evaluate definite integrals without summing infinite rectangles. It is the most important theorem in all of calculus.

A technique for finding \( \frac{dy}{dx} \) when the equation relates \( x \) and \( y \) in a form where \( y \) is not isolated (e.g. \( x^2 + y^2 = 25 \)). Differentiate both sides with respect to \( x \), treating \( y \) as a function of \( x \) and applying the chain rule whenever \( y \) appears, then solve for \( \frac{dy}{dx} \).

The general antiderivative of a function — a function plus \( +C \). Unlike the definite integral, it has no bounds and produces a family of functions, not a number. \( \int f(x)\,dx = F(x) + C \) where \( F'(x) = f(x) \).

A point where a function's concavity changes — from concave up to concave down, or vice versa. At an inflection point, \( f''(x) = 0 \) or \( f''(x) \) is undefined, but the sign of \( f'' \) must actually change. Just having \( f''(x) = 0 \) is not enough on its own.

An integration technique for products of two unrelated function types. Based on reversing the product rule: \( \int u\,dv = uv - \int v\,du \). Choose \( u \) using the LIATE hierarchy (Logarithmic, Inverse trig, Algebraic, Trig, Exponential).

A rule for evaluating limits of the form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). If \( \lim_{x \to a} \frac{f(x)}{g(x)} \) is indeterminate, then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \) — differentiate the numerator and denominator separately (not using the quotient rule) and take the limit again. Only applies to the specific indeterminate forms.

The value a function approaches as the input approaches some value — regardless of what the function actually equals at that point. Written \( \lim_{x \to a} f(x) = L \). Limits are the foundation of all of calculus: derivatives and integrals are both defined as limits. A limit can exist even if the function is undefined at that point.

A local maximum is a point where the function value is higher than all nearby points; a local minimum is lower than all nearby. "Local" means within some neighborhood — there may be higher or lower values elsewhere. A function can have many local maxima and minima. These occur at critical points where \( f' \) changes sign.

If a function is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists at least one point \( c \) where the instantaneous rate of change equals the average rate of change over the interval: \( f'(c) = \frac{f(b)-f(a)}{b-a} \). Intuitively: if your average speed on a road trip was 60 mph, at some moment you were travelling exactly 60 mph.

Using calculus to find the maximum or minimum value of a function, typically subject to some constraint. The standard process: write the quantity to optimize as a function of one variable, find critical points by setting the derivative to zero, then verify with the second derivative test or by evaluating the function at endpoints and critical points.

The most-used differentiation rule: \( \frac{d}{dx}x^n = nx^{n-1} \). Multiply the coefficient by the exponent, then reduce the exponent by 1. Works for any real exponent — positive, negative, fractional. The entry point to all derivative computation.

The rule for differentiating a product of two functions: \( \frac{d}{dx}[f \cdot g] = f'g + fg' \). You cannot distribute differentiation across multiplication — this rule is why. Often remembered as "first times derivative of second plus second times derivative of first."

The rule for differentiating a ratio: \( \frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2} \). The denominator must be squared. Remember: "Low D-High minus High D-Low, over Low squared."

How quickly a quantity changes relative to another quantity. The average rate of change over an interval is \( \frac{f(b)-f(a)}{b-a} \) (slope of the secant line). The instantaneous rate of change at a point is the derivative \( f'(x) \) (slope of the tangent line). All of differential calculus is about rates of change.

A class of problems where two or more quantities are changing with respect to time, and their rates of change are related through an equation. The technique: write an equation relating the quantities, differentiate both sides implicitly with respect to time using \( \frac{d}{dt} \), substitute known values, and solve for the unknown rate.

An approximation of a definite integral by dividing the area under a curve into rectangles and summing their areas. As the number of rectangles approaches infinity and their widths approach zero, a Riemann sum approaches the exact value of the definite integral — this is how the integral is formally defined.

A test for classifying critical points. If \( f'(c) = 0 \) and \( f''(c) > 0 \), then \( c \) is a local minimum (concave up). If \( f''(c) < 0 \), then \( c \) is a local maximum (concave down). If \( f''(c) = 0 \), the test is inconclusive — use the first derivative test instead.

The instantaneous slope of a curve at a specific point, equal to the value of the derivative at that point: \( m = f'(a) \). The equation of the tangent line at \( (a, f(a)) \) is \( y - f(a) = f'(a)(x - a) \). This is one of the most common AP exam question types.

A way of representing a function as an infinite sum of polynomial terms, each involving a derivative of the function at a specific point. A Maclaurin series is a Taylor series centered at zero. They allow functions like \( e^x \), \( \sin x \), and \( \cos x \) to be approximated by polynomials — which is how calculators and computers compute them.

\( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

An integration technique that reverses the chain rule. If an integral contains a composite function and the derivative of the inner function also appears, let \( u \) equal the inner function, replace \( du \) for the corresponding \( dx \) expression, and simplify. After integrating in terms of \( u \), substitute back the original expression.