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GRE Math Subject Test
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This is for the GRE Math Subject test. Not the regular GRE
Terms in this set (70)
Function (pg 5)
A function is a rule that assigns to each element of one set(domain), exactly one element of another set(range)
Map (pg 5)
If f assigns to the element x of A the element y of B, we say that f maps x to y.
The image (pg 5)
If f maps x to y, then y is the image of x.
Fixed Point(pg 6)
A point x₀ is called of f is called a fixed point if f(x₀) = x₀
Composite Function (pg 7)
You have two functions f and g. If you put the range of f into the domain of g then you have a composite function.
Injective (One to One) (pg 7)
A function from A to B is said to be injective if no two elements in A are mapped by f to the same element in B.
Surjective (Onto) (pg 7)
A function from A to B is said to be onto (or surjective) if every element in B is the image of some element in A.
Bijective (pg 7)
A function is bijective if it is both injective and surjective
f⁻¹(pg 8)
f⁻¹ is the inverse of f
How do you determine the inverse of f in one dimension(pg 8)
in the function f replace x with y and vice versa.
Graph (pg 9)
The Graph fo f consists of all points (x,y) in the plane such that y = f(x).
Symmetric with respect to the Y-axis (pg 9)
A graph is said to be symmetric with respect to the y axis when ever (x,y) is on the graph then (-x,y) is also on the graph.
Symmetric with respect to the origin (pg 9)
A graph is said to be symmetric to the origin whenever (x,y) is on the graph, (-x, y) is also on the graph.
The function f(x) and its inverse (pg 9)
Are symmetric with respect to the linfe y = x
The Vertical Line Test (pg 9)
The vertical-line test says that a given graph is not the graph of a function if there are two or more points that line on the same vertical line.
Symmetric with respect to the x axis (pg 9)
A graph is said to be symmetric with respect to the x-axis whenever (x,y) is on the graph then (x, -y) is on the graph.
Abscissa (pg 9)
the value of a coordinate on the horizontal axis
Ordinate (pg 9)
the value of a coordinate on the vertical axis
[x] (Pg 10)
This is an example of a step function. It should take a different value for each n-1≤x<N
Examples of Conic Sections (pg 11)
Examples include lines, parabolas, circles, ellipses, and hyperbolas
The Universal Equation for Conic Sections (pg 11)
Ax²+Bxy+Cy²+Dx+Ey+F = 0
Equation for a straight linge (pg 11)
ax+by + c =0, in which a and b are not both zero.
Point Slope form (pg 11)
y-y₁ = m(x-x₁)
Slope (pg 11)
m = (y₂ -y₁)/(x₂-x₁)
Parrallel (pg 11)
If two lines have the same slope then they are parallel
Perpendicular (pg 11)
If the slope of one line is the negative reciprocal of the slope of a different line, then those two lines are perpendicular
Slope of a vertical line (pg 11)
Undefined
Parabola (pg 11)
Let F be a fixed point and D be a given fixed line that doesn't contain f. By definition, a parabola is the set of points in the plane containing F and D that are equidistant from the point F(the focus) and the line D(the Directrix)
Axis of a parabola (pg 11)
The axis of a parabola is the line through the focus and perpendicular to the directrix
Vertex of a parabola (pg 11)
The vertex of a paraboa is the turning point, the point on the parabola's axis that's midway between the focus and the directrix.
The standard equation for a parabola with vertex at the origin and the y axis is axis of the parabola(pg 11)
y = ±x²/(4p)
The standard equation for a parabola with vertex at the origin and the x axis is the axis of the parbola(pg 11)
y = ±y²/(4p)
Latus Rectum (pg 12)
The line segment with endpoints on the parabola that passes through the focus.
The Directrix has the opposite sign (pg 12)
of the direction of the parabola
Circle (pg 13)
A circle is the set of points in the plane that are all at a constant, positive distance from a given fixed point.
The standard equation for a circle with radius a centered at the origin is
x²+y²=a²
The standard equation for a circle with radius a centered at (h,k) is
(x-h)²+(y-k)²=a²
Ellipse
An ellipse is the set of points in the plane such that the sum of the distances from every point on the ellipse to two give fixed (the foci) is a constant.
Standard equation for an Ellipse
x²/a² + y²/b² = 1
Eccentricity of an elipse
The measure of flatness of an elipse from 0 to 1.
Characteristic of an Ellipse where a > b
The foci is on the x axis (±c,0) where c = √a²-b²
Vertices are on the x axis at (±a,0)
Major axis length is of course 2a
Minor axis length is of course 2b
eccentrity = c/a
Characteristic of an Ellipse where b > a
The foci is on the y axis (0,±c) where c = √b²-a²
Vertices are on the y axis at (0,±b)
Major axis length is of course 2b
Minor axis length is of course 2a
eccentrity = c/b
Hyperbola
A hyperbola is the set of points in the plane such that the difference between the distances from every point on the hyperbola to two fixed points is a constant.
Focal Axis
Line in a hyperbola that contains the foci.
The standard equation for a hyperbola
x²/a² - y²/b² = 1 or -x²/a² + y²/b² = 1
When x²/a² - y²/b² = 1
The foci are on the x axis at (±c, 0) where c = c = √b²+a²
Vertices are at (+-a,0)
Asymptoes y=±(b/a)x
When -x²/a² + y²/b² = 1
The foci are on the y axis (0,±c) where c=√b²+a²
Vertices are on the y axis at (0,±b)
asymptoes are still at ±(b/a)x±
Polynomial
it would be confusing to type some stuff
Discriminant
The part under the square root in the quadratic formula. The discriminant determines the nature of the roots inside a quatdratic polynomial
Division Algorithm
Let p(x) and d(x) be two different polynomials. If you divide p(x) by d(x) and d(x) does not equal zero then you get q(x) and r(x) which are both polynomials such that p(x) = q(x)d(x)+r(x)
Fundamental Theorem of Algebra and Roots of polynomials equatios
This fundamental theorem states that every polynomial equation of degree >= 1can be writeen as the product of unique inear degree 1 factors.
The root location theorem
if p(x) is a real polynomial, and real numbers a and b can be found such that p(a) < 0 and p(b) > 0, then there must be at least one value of x between a and b such that p(x) = 0.
The rational roots theorem pg 18
if the coefficient of a polynomial are integers the RRT says that if p(x) = o has any rational roots, then when expressed in the lowerst terms, they must be of the form x = s/t where s is a factor of the constant term (a0) and t is a factor of the leading term an.
The conjugate Radical roots theorem
If the coefficients of are rational then the CRRT states that if p(x) = 0 has a root of the form x = s + t√u where √u is irrational then the equation must also have conjugate radical x=s-t√u as a root.
The Complex Conjugate Roots Theorem
If the coefficients of a polynomial are real then the CCRT states that if p(x) = 0 has a complex root of the form x = s+ti then it must also have the root x=s-ti.
The sum of roots in a polynomial
go to pg19
d(k)=
0
d(uⁿ)=
nuⁿ⁻¹du
d(e^u)=
e^u du
d(a^u)=
(log a) a^u du (Remember log for the test is actually ln)
d(log u)=
(1/u)du
d(log base a u)=
1/(u log a) du as long as a≠1
d(sin u)=
cos(u)du
d(cos u)=
-sin(u)du
d(tan u)=
sec²(u)du
d(cot(u))=
-csc²(u)du
d(sec u)=
sec u tan u du
d(csc u)=
-csc u cot u du
d(arcsin u)=
du/(√1-u²)
d(arctan u)=
du/(1+u²)
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