|Lines||Relations and Functions|
Increment: the change in coordinates
Slope: the change in vertical distance divided by the change in horizontal distance; frequently thought of as “Rise over Run”
Parallel Lines: two or more lines lines that never intersect; lines have the same slope
Perpendicular Lines: two lines that intersect at a angle. The product of their slopes will equal -1.
Circle: the set of all points on a plane that are equidistant from the center.
Ellipse: the set of all points on a plane who distances from two fixed points in the plane have a constant sum.
Hyperbola: the set of all points on a plane who distances from two fixed points in the plane have a constant difference.
Parabola: the set of all points in a plane that are equidistant from a given fixed point and a given fixed line in the plane.
Semimajor axis, a: half the distance across an ellipse along the longest of the three principles axes.
Semiminor axis, b: half the distance across an ellipse along the shortest of the three principles axes.
Foci, F: the fixed point related to the construction and properties of a conic section.
Eccentricity, e: the distance from the center of the conic section to the focus point divided by the length of the semimajor axis
Function: a rule that assigns exactly one element y in a set B (called the range) to each element x in a set A (called the domain)
Independent Variable: the number belonging to the set A
Dependent Variable: the number belonging to the set B
Vertical Line Test: If a vertical line intersects a curve at more than one point for any x, then the curve does not represent a function
the amount a function changes over a certain interval x
Vertical asymptote: the line x = a where the function f(x) approaches infinity as x approaches the value a
Horizontal asymptote: the line y = b where the function f(x) approaches the value b as x approaches infinity
Maximum: point on the curve where the function changes from increasing to decreasing
Minimum: point on the curve where the function changes from decreasing to increasing
Algebraic Functions: a function where x is a constant; includes polynomial and rational functions
Polynomial Functions: a function in which x is raised to a power
Rational Functions: a function in which one polynomial is divided by a second (non-zero) polynomial
Absolute Value Functions: a function that takes the absolute value of a variable
Inverse Functions: a function where x and y are switched; noted as
Even Functions: a function is even if f(-x)=f(x)
Odd Functions: a function is odd if f(-x)=-f(x)
Parametric Equations: a set of equations where the functions x and y are dependant on a common variable t
One-To-One Functions: a function in which each value of the domain is mapped to only one value of the range
Domain: the set of numbers for which x is defined
Transformation on x: The curve will shift left or right along the x-axis by the amount of the transformation
Transformation on y: The curve will shift up or down along the y-axis by the amount of the transformation
|Sequences and Series||Exponential and Logarithmic Functions||
Sequence: a function whose domain is a set of integers
Series: the sum of sequences
Summation Notation: the sum of all terms beginning with i and ending with the nth term
Infinite series: a series where the number of terms is infinite
Geometric series: each term is obtained from the previous term by multiplying by a constant, r
Arithmetic series: a series where the difference between terms is a constant; an arithmetic series will be a straight line
P-Series: a sequence in which n is raised to the power of a negative integer, p
Alternating Series: a series in which each term alternates between positive and negative
Convergence: a sequence converges if it has a limit S as n approaches infinity
Divergence: a sequence diverges if it does not have a limit S as n approaches infinity
Recursive sequence: a sequence where each term is related to the preceding term by a formula
Let be a series with positive terms and .
Then, the series converges if , the series diverges if and the test is inconclusive if
Limit Comparison Test
If then and both converge or both diverge.
If and converges then converges.
If and diverges then diverges.
Direct Comparison Test
Let be a series with no negative terms.
converges if there is a convergent series with
diverges if there is a divergent series with
Exponential function: a function defined as a constant b raised to the power x; the most common exponential function is the case where b is the special number e.
Logarithmic function: a function defined as logb of x where b is the base
Natural logarithmic function: a logarithmic function where the base is the number e, written as
Polar Coordinates: system of coordinates defined by a radius r, and an angle .
Complex Number: a number defined as z=x+iy, where x is real component, y is imaginary component and I is defined as