Section 1
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What is the discriminant in the quadratic equation?
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Cards (246)
Section 1
(50 cards)
What is the discriminant in the quadratic equation?
The terms under the radical
What is the vertex?
The "extremity" of the closed end of the parabola. It is where the parabola intercepts the parabolic axis
What is the minor axis of an ellipse?
The short axis of the ellipse perpendicular to the major axis
What does (h,k) in the conic section denote?
The coordinates of the center of the conic section
What are the latera recta of an ellipse?
The lines passing through the foci and perpendicular to the major axis
What is the latus rectum for a parabola?
The distance between the focus and the directrix. The vertex is halfway between the focus and the directrix
Perpendicular Slopes Equation
Straight Line Equation
If A=C but not equal to 0, then...
The conic section is a circle
What is the semi-minor distance of an ellipse?
The distance along the minor axis from the center point to where the ellipse intersects the minor axis
What is the perpendicular distance from the vertex to the asymptote?
b as designated by the standard form of hyperbola
What is an reflex angle?
An angle more than 180 degrees
Distance between two points on a plane
Standard form equation for horizontal parabola
What is the directrix of a parabola?
A line that is outside the closed end of the parabola by the distance of half the latus rectum. The slope of the directrix is the slope of the parabola through the vertex
What is the distance between the center of the hyperbola and either vertex?
a as designated by the standard form of hyperbola
Normal Equation of conic section
Distance between points in space
General form equation of conic section
What is an acute angle?
An angle less than 90 degrees
Equation for angle between lines
If A does not equal C, then...
B^2-4AC <0 is an ellipse B^2-4AC >0 is a hyperbola B^2-4AC =0 is a parabola
What is an obtuse angle?
An angle more than 90 degrees
What is an related angle?
An angle that differs from some other by a multiple of 90 degrees
What is the parabolic axis?
The line across which the parabola is symmetric
Features of ellipse with center at origin
Standard form equation of a hyperbola
If A=C=0 in a conic section, then...
The conic section is a line
What is the semi-major distance of a an ellipse?
The distance along the major axis from the center point to a vertex
Point-Slope form for equation of a line
Features of parabola with vertex at origin
What are the vertices of an ellipse?
The points where the ellipse intersects the major axis
Equation for eccentricity of cutting plane through conic section
Features of an hyperbola with center at the origin
What is the transverse axis of a hyperbola?
The line that passes through both foci of the hyperbola
if A=C in a conic section, then...
B of the conic section must be 0
What are the conjugate axis of a hyperbola?
The lines that pass through the foci of a hyperbola but are perpendicular to the transverse axis
What is an straight sngle?
An angle that is 180 degrees
Quadratic Equations
Standard form equation for ellipse
What is the major axis of an ellipse?
The long axis of the ellipse on which both Foci reside
General form for equation of a line
Standard form equation of a circle
Length of tangent of circle to a point
What are complementary angles?
2 angles whose sum is 90 degrees
Standard form equation for a sphere
What does the r denote?
A negative r is not a conic section A r value of 0 denotes a point A r value greater than 0 denotes the size of a conic section such as the radius of a circle
What is a reciprocal?
A pair of numbers that can be multiplied to produce 1
What are the asymptotes of a hyperbola?
The lines to which the hyperbola approaches to infinity. The lines intersect at the center of the hyperbola
Standard form for equation of a line
Section 2
(50 cards)
Area of parallelogram
What is a right circular cylinder?
a cylinder whose base is a circle and whose axis is perpendicular to the base
Reciprocal Identities
What is cosecant?
hypotenuse over opposite
Area of circular sector
Law of Sines
Law of Cosines
Perimeter of a parallelogram
What is the apothem of a polygon?
a line segment drawn from the center of the polygon to the midpoint of one of the sides
Lengths of diagonals for parallelogram
What is secant?
hypotenuse over adjacent
What is sine?
opposite over hypotenuse
Two Angle Identities
Surface Area of Sphere
When are adjacent angles supplementary?
When their exterior sides form a straight line
What are supplementary angles?
2 angles whose sum is 180 degrees
Precise Perimeter of Ellipse
Volume of sphere
Approximate Perimeter of Ellipse
Operation Identities
What is a right circular cone?
a cone whose base is a circle and whose axis is perpendicular to the base
Length of Regular Polygon Side
What is a circular sector?
A portion bounded by two radii and an arc
What are features of similar triangles?
The angles are the same and the sides are proportional in length
Area outside parabolic curve
Ratio of cross section area to base area for right circular cone
Area inside parabolic curve
Central angle of circular segment
Half Angle Identities
Regular polygon angle equations
What is a circular segment?
An region bounded by a circular arc and a chord
What is the sum of all angles in all triangles?
The sum of the angles is 180 degrees
Volume of right circular cone
What is cotangent?
adjacent of opposite
What is a regular polygon?
A polygon with equal sides and angles
General Identities
What is a polyhedron?
a 3-d object whose faces are all flat and whose edges are straight
Central angle of circular sector
What is a prismoid?
A polyhedron where all vertices are within two parallel planes
Area of regular polygon
What are adjacent angles?
angles with a common vertex and share one side
What is cosine?
adjacent over hypotenuse
Double Angle Identities
Area of circular segment
Prismoid volume
What are vertical angles?
Two equal angles with a common vertex and with sides mad by two intersecting lines.
What is tangent?
opposite over adjacent
What is a right triangle?
A trianhle with one angle of 90 degrees
Area of ellipse
Surface Area of Right Circular Cone
Section 3
(50 cards)
Cross Product Identities
Vector Addition
What is paraboloids?
The surface that is obtained by rotating a parabola about its axis
Division in polar form
Kth root of complex number
Classical Adjoint Equation
Matrix Transpose
In the form AX=B, which is the coefficient matrix?
Matrix A
Cramer's Rule Formula
If matrix A is zero and the system of equations in nonhomogeneous, what kind of solution exists?
If the determinant of all substitutional matrices are 0, there would be an infinite number of solutions. Otherwise, no solution exists.
What is the classical adjoint or adjugate?
The transpose of the cofactor matrix
If the system of linear equations in nonlinear and matrix A is nonzero, what kind of solution is there?
There is a unique answer to the set of equations
In the form AX=B, which is the variable matrix?
Matrix X
Logarithmic Base Change
What is the cross product of two parallel unit vectors?
the product vector is 0
Polar form of complex number
Vector Cross Product
Surface Area of right circular cylinder
What is a tensor?
It has magnitude in a direction but the direction is not unique. The vector is defined in 9 components
Identity Matrix
Vector Subtraction
Matrix Multiplication
What is a complex number?
A number contacting an real number and an imaginary number
If the system of linear equations is homogenous and matrix A is zero, what kind of solution is there?
There is an infinite set of solutions
What is the sign of the cofactor?
Positive when i+j is even Negative when i+j is odd
Dot product identities
Volume of right circular cylinder
Matrix Addition
What is the minor of entry a_ij?
The determinant of a submatrix resulting from the elimination of the single row i and the single column j
Multiplication in polar form
Determinant of 3x3 matrix
Determinant of 2x2 matrix
What is the dot product of two orthogonal unit vectors?
the product is 1
de Moivre's Formula
What is a complex conjugate?
The complex conjugate of a+jb is a-jb
Vector Dot Product
In the form AX=B, which is the constant matrix?
Matrix B
What is the dot product of two parallel unit vectors?
the product is 0
What is Cramer's Rule?
Calculate determinants of the original coefficient matrix A and of the n matrices resulting from the systematic replacement of a column in matrix A by the constant matrix B.
What is the vector cross product?
A vector that is orthogonal to the plane of the two vectors.
Polar-Rectangular coordinate conversion
What is the resultant vector?
The vector resulting from a mathematical operation of 2 or more vectors
Euler's Formula
What is the vector dot product?
A scalar that is proportional to the length of the projection of the first vector onto the second vector
If the system of linear equations is homogenous and matrix A is non-zero, what kind of solution exists?
Only the trivial solution
What are cofactors?
determinants of submatrices associated with particular entries in the original square matrix
What is Gauss Jordan Elimination?
1. The coefficient matrix is augmented by the constant matrix. 2. the elementary row operations are used to reduce the coefficient matrix into canonical form. All of the operations performed on the coefficient matrix are performed on the constant matrix. 3. The variable values that satisfy the simultaneous equations will be the entries in the constant matrix when the coefficient matrix is in canonical form
Volume of paraboloid
Logarithmic Identities
How can the unit vector representation of the cross product be calculated?
By determining the determinant of the 3x3 matrix formed by the two vectors and i,j,k
Section 4
(50 cards)
What is an arithmetic progression?
A standard sequence that divergence whose terms increase by a common difference
What is a series?
the sum of the terms of a sequence
Curvature Equation in rectangular coordinates
What is a progression or sequence?
an ordered set of numbers
What is the radius of curvature?
The radius of a curve whose center lies on the concave side of the curve and whose tangent coincides with the tangent of the curve at that point
What is a Taylor's Series?
A series composed of a derivate of the original function and a polynomial.
What is a Taylor Series used for?
To approximate the value of f(b) based on the value of F(a). The function needs to be continuous in the interval [a,b]
What is the interval of convergence for a power series?
An infinite power series can only be convergent between -1<x<1
What is the divergence of a vector field?
It is defined by the dot product of the del operator and the vector. It can be interpreted as the accumulation of flux in a small region
Distributive Properties of Series
Polar Moment of Inertia Equation
How does the derivative test determine a local minimum?
When the first derivative is zero and the second derivative is positive
What is integration?
the inverse of differentiation
Radius of Curvature Equation
Gradient Equation of scalar function
What is the gradient vector field of a scalar function?
It gives the maximum rate of change of a scalar function
What is the curl of a vector field?
The cross product of the del operator and the vector. The curl vector determines how fast the flux is rotating and in what direction the flux is going
Divergence Equation for vector field
When is a series said to converge?
When there is a sum, S, of all the terms. Hence, finite series are always convergant
First Moment of Area Equation
What is the derivative of a function?
The rate of change of a function
What are extreme points?
local maximums and minimums
What is the cross product of two normal unit vectors?
the product vector is orthogonal to both input vectors
What are convergence features of a geometric progression?
It convergence for a common ratio between -1 and 1. Otherwise it diverges
What is the sharpness of a curve?
The rate of change of inclination
What is a convergent sequence?
When the terms of the sequence approach infinity
Integral form of centroid of area
What is L'Hopital's rule?
It is used when both the numerator and the denominator are determinate ( I.e. both zero or both infinite)
Moment of Inertia Equation
Curvature Equation
What is a divergent sequence?
WHen the terms of the sequence approach infinity
Equation for sum of arithmetic progression
How does the derivative test determine a point of inflection?
When the second derivative is zero
What are critical points?
extreme points and inflections points
What is a partial derivative?
A derivative of a function with multiple independent variables with respect to only one of the independent variables
What are inflection points?
point of contraflexure
Area under the curve
L'Hopital's Rule Equation
Power Series Equation
What is an infinite series?
A series that has an infinite number of terms
What is the general term of a sequence?
The last term of the number set
What is an antiderivative?
An indefinite intergal
Equation for sum of geometric progression
How does the derivative test determine a local maximum?
when the first derivate is zero and the second derivative is negative
What is a power series?
An exponentiating series which is exponentiated by the value of i and is composed of the value of x.
Derivative Equation
What is an geometric progression?
A sequence where the quotient between adjacent term is constant at a common ratio
What is a scalar function?
A mathematical expression that returns a single value
Parallel Axis Theorem Equation
What is a limit?
the value that a function approaches when an independent variable approaches a target value
Section 5
(46 cards)