 TOPICS   # 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 Calculus is 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 The chain rule is a formula for the derivative of the composition of two functions in terms of their derivatives. Continuous Function A continuous function is function with no jumps, gaps, or undefined points. Critical Point A critical point is a point in the graph of a function where the derivative is either zero or undefined. Definite Integral A definite integral is an integral with upper and lower limits. Derivative A derivative is the infinitesimal rate of change in a function with respect to one of its parameters. Discontinuity A discontinuity is a point at which a function jumps suddenly in value, blows up, or is undefined. The opposite of continuity. Extreme Value Theorem The extreme value theorem states that a continuous function on a closed interval has both a maximum and minimum value. First Derivative Test The first derivative test is a method for determining the maximum and minimum values of a function using its first derivative. Fundamental Theorems of Calculus The fundamental theorems of calculus are deep results in analysis that express definite integrals of continuous functions in terms of antiderivatives. Implicit Differentiation Implicit differentiation is 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 indefinite integral, also known as an antiderivative, is an integral without upper and lower limits. Inflection Point An inflection point is a point on a curve at which the concavity changes. Integral An integral is 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 intermediate value theorem states 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 A limit is 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 maximum of a set, function, etc. is the largest value attained by that object. Mean-Value Theorem The mean-value theorem states 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 minimum of a set, function, etc. is the smallest value attained by that object. Newton's Method Newton's method is an iterative method for numerically finding a root of a function. Riemann Sum A Riemann sum is an estimate, using rectangles, of the area under a curve. A definite integral is defined as a limit of Riemann sums. Second Derivative Test The second derivative test is a method for determining a function's maxima, minima, and points of inflection by using its first and second derivatives.