Limits and Continuity  Differential Calculus  

Limit: the value of the function as x approaches a certain value RightHand Limit: the limit as x approaches from the positive direction; is noted as LeftHand Limit: the limit as x approaches from the negative direction; is noted as TwoSided Limit: the limit is the same whether x approaches from the right or left Indeterminate Limits: Undefined limits such as 0 divided by 0, infinity divided by infinity, infinity times zero, infinity minus infinity, one to the power infinity, zero to the power zero and infinity to the power zero Continuity: a function, f(x), is continuous over an interval if there exists a value of the function for every value of x on the interval Discontinuity: a function, f(x), is discontinuous at a point if the function is not defined at that point Removable Discontinuity: a limit of the function exists at the point of discontinuity Jump Discontinuity: the righthanded and lefthanded limits are different Infinite Discontinuity: the limit of the function as x approaches the discontinuity is infinity By Substitution: limits for polynomial and rational functions with nonzero denominators can be found by substituting the value x into the function By Factoring: limits for rational functions where the denominator is zero can be found by factoring the function and then simplifying By Estimating: limits can be found for any function by using a graph or by calculating the value of f(x) for different values of that approach the limit Vertical Asymptote: a line, x=a that the graph of a function approaches but never touches Horizontal Asymptote: a line, y=a that the graph of a function approaches but never touches Intermediate Value Theorem: if the function is continuous between two points a and b, then there exists a value c between a and b where f(c) will be between f(a) and f(b) Extreme Value Theorem: if a function is continuous over a closed interval then the function has a maximum and minimum on the interval Sandwich or Squeezed Theorem: if a function f is between two other functions g and h with a common limit, then the function f has the same limit as g and h 
Derivative: the function that describes the rate of change of another function Geometrically, the derivative is the slope of a curve Analytically, the derivative is the rate of change of a curve Numerically, the derivative is the difference in y values on a very small interval of x Higher Order Derivatives: taking the derivative of a derivative; the order of the derivative represents the number of times the derivative was taken Partial Derivative: the derivative of only one variable in the function; other variables are treated as constants Derivative Notation or Second Order Derivative or Mean Value Theorem: if a function is continuous and differentiable on an interval [a,b], then there exists at least one point on that interval where the derivative is the slope through a and b. If a function is differentiable at a point then it is continuous at that point; the opposite is not always true

Integral Calculus  Application of Derivatives  

Integral: the process of finding the antiderivative. AntiDerivative: the function F(x) such that the derivative is the original function f(x) Indefinite Integral: the set of all antiderivatives of f(x); the process of finding the antiderivative of a function over no boundary Definite Integral: the process of finding the antiderivative of a function over a fixed boundary of x Fundamental Theorem of Calculus: a function that is continuous on a given interval will have an antiderivative that is continuous on that interval and the integral of the derivative will be the function itself Riemann Sum: the sum of all the rectangles that approximate the area under the curve Midpoint Evaluation: the area between the curve and the xaxis is divided into equal rectangles; the function passes through the top midpoint of each rectangle Left or Right Evaluation: the area between the curve and the xaxis is divided into equal rectangles; the function passesf through the top left or right corner of each rectangle Trapezoidal Rule: the area between the curve and the xaxis is divided into trapezoids; the sum of the area of all trapezoids approximates the area under the curve. Simpson’s Rule: the area between the curve and the xaxis is found by creating a parabola that intersects the curve at an even number of intervals of equal width;the area between the parabola and the xaxis will approximate the area of the curve.
If f(x) and g(x) have an integral on [a,b] and k is a constant then the functions , and have an integral on [a,b].

Local Maximum or Minimum: a point on the graph where derivative changes from positive to negative or negative to positive Absolute Maximum: the highest point on the graph Absolute Minimum: the lowest point on the graph Monotonicity: a function that only increases or only decreases Concavity: the rate of change of the slope of a curve Velocity: the first derivative of a function describing position Acceleration: the first derivative of a function describing velocity; the second derivative of function describing position
Slope of a Curve: the value of the derivative at a specific point
Average Value of a Function: the average of the value of a function over a given interval Differential Equation: an equation that relates a function to its antiderivative Separable Differential Equation: a differential equation where x and y variables can be separated Volume: If f(x,y) has an integral over the plane region, R, and f(x,y)0 for all (x,y) in R then the volume, V, that is between the graph of f(x,y) and R is defined as Areas and Integrals: the integral over the interval [a,b] is the area under the curve of the function between the points a and b Volumes and Integrals: the volume of a shape can be found by solving the double or triple integral of the function describing the volume. Separable Differential Equations: combine like terms; evaluate the integral of both sides of the equation; solve for y. 