|Limits and Continuity||Differential Calculus|
Limit: the value of the function as x approaches a certain value
Right-Hand Limit: the limit as x approaches from the positive direction; is noted as
Left-Hand Limit: the limit as x approaches from the negative direction; is noted as
Two-Sided 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 right-handed and left-handed 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 non-zero 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 anti-derivative.
Anti-Derivative: the function F(x) such that the derivative is the original function f(x)
Indefinite Integral: the set of all anti-derivatives of f(x); the process of finding the anti-derivative of a function over no boundary
Definite Integral: the process of finding the anti-derivative 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 anti-derivative 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 x-axis 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 x-axis 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 x-axis 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 x-axis 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 x-axis 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
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 anti-derivative
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.