3D Geometry and Functions Quiz

Points: Defined by coordinates (x, y, z) in three-dimensional space.
Lines: Represented by parametric equations or segment endpoints.
Planes: Defined by a point and a normal vector or through the equation Ax + By + Cz + D = 0.
Distance:
- Between two points ( P_1(x_1, y_1, z_1) ) and ( P_2(x_2, y_2, z_2) ):
- Distance = ( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} )
Volume:
- Cubes: ( V = a^3 )
- Spheres: ( V = \frac{4}{3}\pi r^3 )
- Cylinders: ( V = \pi r^2 h )
Surface Area:
- Cubes: ( SA = 6a^2 )
- Spheres: ( SA = 4\pi r^2 )
- Cylinders: ( SA = 2\pi r(h + r) )

Function: A relation that assigns each input exactly one output; described as ( f(x) ).
Domain and Range: Domain is the set of all possible inputs; range is the set of all possible outputs.
Limit: The value that a function approaches as the input approaches a given value.
- Notation: ( \lim_{x \to c} f(x) = L )
- One-sided limits: ( \lim_{x \to c^-} f(x) ) and ( \lim_{x \to c^+} f(x) )
Continuity: A function is continuous at a point if ( \lim_{x \to c} f(x) = f(c) ).

Matrix Addition: Sum of two matrices of the same dimension is obtained by adding their corresponding elements.
Matrix Multiplication:
- Result of multiplying an ( m \times n ) matrix by an ( n \times p ) matrix is an ( m \times p ) matrix.
- Element at position (i, j) is computed as ( \sum_{k=1}^{n} a_{ik} b_{kj} ).
Determinant: A scalar value that can be computed from the elements of a square matrix, indicating if the matrix is invertible.
Inverse: The matrix ( A^{-1} ) such that ( AA^{-1} = I ), where I is the identity matrix.
Transposition: Flipping a matrix over its diagonal, switching the row and column indices (( A^T )).

Random Variable: A variable whose values depend on the outcomes of a random phenomenon.
Probability Distribution: Describes how the probabilities are distributed over the values of the random variable.
- Discrete Distribution: Assigns probabilities to discrete values (e.g., binomial, Poisson).
- Continuous Distribution: Assigns probabilities over intervals (e.g., normal, exponential).
Expected Value: The long-term average value of random variables, calculated as ( E(X) = \sum x_i P(x_i) ) for discrete and ( E(X) = \int x f(x) dx ) for continuous variables.
Variance: Measures the spread of a distribution, calculated as ( Var(X) = E[(X - E[X])^2] ).

Descriptive Statistics: Summarizes and describes the characteristics of a data set.
- Measures of Central Tendency: Mean, median, mode.
- Measures of Dispersion: Range, variance, standard deviation.
Inferential Statistics: Makes predictions or inferences about a population based on a sample.
- Hypothesis Testing: Procedure to determine if a hypothesis about a population parameter is true.
- Confidence Intervals: A range of values used to estimate a population parameter with a specified level of confidence.
Correlation and Regression:
- Correlation: Measures the strength and direction of the linear relationship between two variables (e.g., Pearson correlation coefficient).
- Regression Analysis: Predicts the value of a dependent variable based on one or more independent variables.

Points are identified by three coordinates (x, y, z) within three-dimensional space.
Lines can be expressed through parametric equations or defined by segment endpoints.
Planes can be described using a point paired with a normal vector or via the standard equation (Ax + By + Cz + D = 0).
The distance between two points (P_1(x_1, y_1, z_1)) and (P_2(x_2, y_2, z_2)) is given by the formula ( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} ).
Volume calculations include:
- Cubes: (V = a^3)
- Spheres: (V = \frac{4}{3}\pi r^3)
- Cylinders: (V = \pi r^2 h)
Surface area formulas include:
- Cubes: (SA = 6a^2)
- Spheres: (SA = 4\pi r^2)
- Cylinders: (SA = 2\pi r(h + r))

A function (f(x)) links each input to exactly one output.
The domain represents all permissible inputs, while the range encompasses all possible outputs.
Limits signify the value a function approaches as the input nears a specific value, denoted as ( \lim_{x \to c} f(x) = L ).
One-sided limits can be distinguished as ( \lim_{x \to c^-} f(x) ) (from left) and ( \lim_{x \to c^+} f(x) ) (from right).
Continuity at a point occurs when ( \lim_{x \to c} f(x) = f(c) ).

Matrix addition combines two matrices of the same size by summing corresponding elements.
Matrix multiplication involves multiplying an (m \times n) matrix by an (n \times p) matrix to yield an (m \times p) result.
Each element in the resulting matrix at position (i, j) is calculated using ( \sum_{k=1}^{n} a_{ik} b_{kj} ).
The determinant of a square matrix provides a scalar indicative of its invertibility.
The inverse matrix (A^{-1}) satisfies the equation (AA^{-1} = I), with (I) being the identity matrix.
Transposition of a matrix, denoted (A^T), involves swapping rows with columns.

A random variable represents outcomes dependent on a random process.
Probability distributions outline how probabilities are allocated over the values of random variables.
Discrete distributions provide probabilities for distinct values (examples include binomial and Poisson distributions).
Continuous distributions assign probabilities across intervals (examples include normal and exponential distributions).
The expected value, or long-term average, is computed as (E(X) = \sum x_i P(x_i)) for discretized values and (E(X) = \int x f(x) dx) for continuous scenarios.
Variance quantifies the spread of a distribution and is calculated as (Var(X) = E[(X - E[X])^2]).

Descriptive statistics offer summaries of data characteristics.
Measures of central tendency include mean, median, and mode.
Measures of dispersion consist of range, variance, and standard deviation.
Inferential statistics make population inferences from sample data.
Hypothesis testing evaluates the validity of assumptions regarding population parameters.
Confidence intervals estimate population parameters within specified bounds of confidence.
Correlation indicates the strength and direction of linear relationships between variables via metrics like the Pearson correlation coefficient.
Regression analysis forecasts dependent variable values based on independent variable inputs.