Topics in a Calculus I Course

To learn more about a topic listed below, click the topic name to go to the corresponding MathWorld classroom page.

Calculus	The branch of mathematics studying the rate of change of quantities (which can be interpreted as slopes of curves) and the length, area, and volume of objects.
Chain Rule	A formula for the derivative of the composition of two functions in terms of their derivatives.
Continuous Function	A function with no jumps, gaps, or undefined points.
Critical Point	A point of a function's graph where the derivative is either zero or undefined.
Definite Integral	An integral with upper and lower limits.
Derivative	The infinitesimal rate of change in a function with respect to one of its parameters. The derivative is one of the key concepts in calculus.
Discontinuity	A point at which a function jumps suddenly in value, blows up, or is undefined. The opposite of continuity.
Extreme Value Theorem	The theorem that a continuous function on a closed interval has both a maximum and minimum value.
First Derivative Test	A method for determining the maximum and minimum values of a function using its first derivative.
Fundamental Theorems of Calculus	Deep results that express definite integrals of continuous functions in terms of antiderivatives.
Implicit Differentiation	The procedure of differentiating an implicit equation (one which has not been explicitly solved for one of the variables) with respect to the desired variable, treating other variables as unspecified functions of it.
Indefinite Integral	An integral without upper and lower limits.
Inflection Point	A point on a curve at which the concavity changes.
Integral	A mathematical object that can be interpreted as an area or a generalization of area. Integrals and derivatives are the fundamental objects of calculus.
Intermediate Value Theorem	The theorem that if f is continuous on a closed interval [a, b], and c is any number between f(a) and f(b) inclusive, then there is at least one number x in [a, b] such that f(x) = c.
Limit	The value a function approaches as the variable approaches some point. If the function is not continuous, the limit could be different from the value of the function at that point.
Maximum	The largest value of a set, function, etc.
Mean-Value Theorem	The theorem that if f(x) is differentiable on the open interval (a, b) and continuous on the closed interval [a, b], there is at least one point c in (a, b) such that (a - b) f(c) = f(a) - f(b).
Minimum	The smallest value of a set, function, etc.
Newton's Method	An iterative method for numerically finding a root of a function.
Riemann Sum	An estimate, using rectangles, of the area under a curve. An definite integral is defined as a limit of Riemann sums.
Second Derivative Test	A method for determining a function's maxima, minima, and points of inflection by using its first and second derivatives. MathWorld