A limit describes the behavior of a function as the input approaches a certain value. It represents the value that the function gets arbitrarily close to without actually reaching it.

The derivative is the instantaneous rate of change of a function at a given point. It measures the sensitivity of a function to changes in its input value.

The Fundamental Theorem of Calculus relates the concepts of differentiation and integration. It states that the derivative of an integral of a function is the original function, and that the definite integral of a function's derivative over an interval equals the change in the function's values over that interval.

A definite integral has specified limits of integration and gives the exact area under the curve. An indefinite integral does not have limits specified, and its result is a family of functions whose derivatives are the original function.

The Mean Value Theorem states that for a continuous function on a closed interval, there exists at least one point in the interval where the instantaneous rate of change (derivative) equals the average rate of change over the interval.

The antiderivative, or indefinite integral, of a function is found by reversing the process of differentiation using integration techniques like substitution, integration by parts, etc.

The main integration techniques include substitution, integration by parts, partial fractions, trigonometric substitutions, and integration of rational functions.

A Taylor polynomial is a representation of a function as an infinite sum of terms calculated from the values of the function's derivatives at a single point.

A sequence is an ordered list of numbers or terms. A series is the sum of the terms in a sequence.

Euler's method is a numerical technique used to approximate solutions to first-order differential equations given initial conditions.

A polar equation describes a relation between a point's distance from the origin (r) and its angle from the positive x-axis (theta) in the polar coordinate system.

A non-parametric equation directly relates x and y values. A parametric equation expresses both x and y in terms of a third variable (parameter t).

Applications include optimization problems, related rates, motion/velocity analysis, geometric modeling, differential equations modeling growth/decay, and more.

Find critical points by taking the derivative and setting it equal to 0. Evaluate the function at critical points and endpoints of the interval. The absolute max/min will be among those values.

A local extreme is maximum or minimum value in a local region around a point. A global extreme is the overall maximum or minimum of the function on its entire domain.

A relative extremum is a point where the function changes from increasing to decreasing or vice versa. An absolute extremum is the overall maximum or minimum value of the function.

A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

An even function satisfies f(-x) = f(x). An odd function satisfies f(-x) = -f(x).

A function is continuous if its graph can be drawn without lifting the pencil from the paper. Formally, it has no breaks, holes, or discontinuities.

An increasing function has a positive rate of change (derivative) over an interval. A decreasing function has a negative rate of change over an interval.

A critical point of a function is a point where the derivative is zero or undefined. These may correspond to relative maximum, minimum or inflection points.

L'Hôpital's Rule provides a technique for evaluating limits of indeterminate forms like 0/0 or infinity/infinity by taking the limit of the derivative ratio.

A position function describes the position of an object in terms of time. Its derivative is the velocity function, and the derivative of velocity is acceleration.

The area between two curves f(x) and g(x) from a to b is given by the integral from a to b of (f(x) - g(x)) dx.

The Second Derivative Test is used to determine whether a critical point is a relative maximum, minimum or neither by analyzing the sign of the second derivative.

A series converges if the sum of its terms approaches a finite value as more terms are added. If the sum increases without bound, the series diverges.

Integrate the area under/around one revolution of the curve being revolved around an axis, multiplied by the circumference of the revolution at each step.

The chain rule allows you to find the derivative of a composite function by taking the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Mathematical induction is a proof technique used to prove statements that hold for all positive integers greater than a certain value. It uses a basis step and inductive step.

Differentiation finds the rate of change or derivative of a function. Integration finds the accumulated area under the curve or antiderivative of a function.

The Intermediate Value Theorem states that if a continuous function takes values f(a) and f(b) of opposite signs on a closed interval [a,b], then it takes on every value between f(a) and f(b) for some point in the interval.

The power rule states that d/dx (x^n) = nx^(n-1), where n is any real number except -1.

A cusp is a point on a curve where the derivative or tangent line is not defined and the curve makes a sharp turn back on itself.

An inflection point is a point on the graph of a function where the concavity changes from positive to negative or vice versa.

Exact methods like the Fundamental Theorem give precise integral values. Approximation methods like numerical integration estimate areas under curves.

A separable differential equation is one that can be rewritten with separable variables, so that one side is a function of x and the other side is a function of y.

A slope field is a visual representation showing slopes of tangent lines to a differential equation at various points, which can help visualize solution curves.

If the limit of a function as x goes to infinity is infinite, it means the function grows without bound as x increases without limit.

Common antiderivatives include integrals of polynomial, exponential, logarithmic, trigonometric and inverse trigonometric functions.

A positive second derivative indicates the function is concave up, while a negative second derivative means it is concave down.

Some techniques for improper integrals include taking limits of definite integrals as the interval extends to infinity, and evaluating convergence using comparison tests.

Arc length can be found by evaluating the integral of the square root of 1 + (dy/dx)^2 dx over the interval of interest.

Differential equations are used to model rates of change in systems like population growth/decay, motion, mixing problems, and many other applied areas.

A Riemann sum provides an approximation of a definite integral using a finite number of rectangular areas under the curve.