AP Precalculus Unit 4 Notes

studied byStudied by 105 people
5.0(1)
get a hint
hint

Parametric functions

1 / 42

43 Terms

1

Parametric functions

Functions that are expressed in terms of a parameter, often denoted as t, and encapsulate two functions, x(t) and y(t), dependent on the parameter.

New cards
2

Graphing and Tabulating Parametric Functions

The process of creating a graph by plotting points from a table of values for parametric functions, aiding in the analysis and interpretation of their properties.

New cards
3

Parametric Function Overview

Parametric functions involve two dependent variables, x and y, that are influenced by a single independent variable, t.

New cards
4

Horizontal and Vertical Extrema

The furthest points in the horizontal and vertical directions of a parametric function, determined by the maximum and minimum values of x(t) and y(t), respectively.

New cards
5

Direction and Rate of Change

The analysis of the direction and rate of change in parametric planar motion functions by evaluating the functions x(t) and y(t) as the parameter t increases.

New cards
6

Average Rate of Change

The average rate of change in parametric functions provides insights into the horizontal and vertical changes in a particle's position over time.

New cards
7

Function and Its Inverse

The parametrization of a function f of x involves expressing both x and y as functions of a parameter t, while the parametrization of its inverse involves expressing x and y as functions of t.

New cards
8

Parametric Formulas for Conic Sections

Specific formulas for parametric representation of conic sections, such as parabolas, ellipses, and hyperbolas, using trigonometric functions.

New cards
9

Characteristics of Vectors

Vectors are defined as directed line segments with both direction and magnitude, and can be decomposed into components in the two-dimensional plane.

New cards
10

Representation in the Plane

Placing a vector in the plane involves defining its tail and head, representing the beginning and end of the line segment, respectively.

New cards
11

Vector Components

Vectors can be represented as (a,b) or ⟨a,b⟩, with a and b being the components calculated along each dimension.

New cards
12

Magnitude and Direction

The magnitude of a vector is determined by the Pythagorean theorem, while the direction aligns with the line segment from the origin to the point represented by the vector.

New cards
13

Scalar Multiplication

Scaling a vector by multiplying it with a scalar results in a vector parallel to the original vector.

New cards
14

Vector Addition

The sum of vectors involves combining their corresponding components.

New cards
15

Dot Product

The dot product of two vectors is the sum of the products of their corresponding components, and is related to magnitudes and angles.

New cards
16

Unit Vectors

Unit vectors have a magnitude of 1 and represent direction, often obtained by dividing a vector by its magnitude.

New cards
17

Angle Between Vectors

The angle between two vectors can be computed using the arccosine function.

New cards
18

Law of Cosines

The Law of Cosines relates the cosine of an angle between vectors to their magnitudes and dot product.

New cards
19

Law of Sines

The Law of Sines relates the sine of angles and the lengths of sides in vector-formed triangles.

New cards
20

Position Vectors

Vectors that describe a particle's position using a vector originating from the origin and ending at the particle's location.

New cards
21

Position Vector

A vector-valued function that represents the position of a particle in a plane.

New cards
22

Velocity Vector

A vector-valued function that represents the velocity of a particle in a plane.

New cards
23

Magnitude of Velocity

The speed of a particle at a specific time, calculated by taking the square root of the sum of squared x and y components of the velocity vector.

New cards
24

Matrix Dimensions

The number of rows and columns in a matrix, denoted as n×m.

New cards
25

Square Matrix

A matrix with an equal number of rows and columns, denoted as n×n.

New cards
26

Row Matrix

A matrix with only one row, denoted as 1×n.

New cards
27

Column Matrix

A matrix with only one column, denoted as n×1.

New cards
28

Matrix Elements

The individual values in a matrix, denoted as a_mn, representing the element in the m-th row and n-th column.

New cards
29

Matrix Equality

Two matrices are equal if they have the same order and each corresponding element is equal.

New cards
30

Matrix Multiplication

The multiplication of two matrices is possible if the number of columns in the first matrix equals the number of rows in the second matrix. The resulting matrix has dimensions m×n, where m is the number of rows of the first matrix and n is the number of columns of the second matrix.

New cards
31

Identity Matrix

A special square matrix with 1's on the main diagonal and 0's elsewhere.

New cards
32

Determinant

A scalar value that has applications in invertibility and vector areas. For a 2×2 matrix A=[ab​cd​], the determinant is calculated as det(A)=ad−bc.

New cards
33

Area of Parallelogram

The absolute value of the determinant represents the area of the parallelogram formed by vectors.

New cards
34

Inverse of a Matrix

A square matrix A has an inverse (A−1) if and only if the determinant of A is not equal to 0. The inverse of a 2×2 matrix [ab​cd​] is given by A−1=det(A)1​[d−b​−ca​].

New cards
35

Linear Transformation

A function that maps input vectors to output vectors while preserving vector addition and scalar multiplication operations.

New cards
36

Transformation Matrix

For a linear transformation L from 2R2 to 2R2, there exists a unique 2×2 matrix A such that L(v)=Av for all vectors in 2R2.

New cards
37

Composition of Linear Transformations

Linear transformations can be composed, where one transformation is applied followed by another. The composition of two linear transformations is itself a linear transformation.

New cards
38

Inverse of a Linear Transformation

The inverse of a linear transformation undoes the effect of the original one. It is represented by the inverse of the matrix for the original transformation.

New cards
39

Matrix Models

Matrices can be used as dynamic models to represent transitions between different states.

New cards
40

Transition Matrix

A square matrix, often 2×2, that encodes the rates of transition between different states.

New cards
41

Predictive Matrix Multiplication

The systematic multiplication of the transition matrix with the current state vector to predict future states.

New cards
42

Steady State

A dynamic equilibrium undisturbed by subsequent transitions, predicted through iterative multiplication of the transition matrix.

New cards
43

Retrospective Projection

Unraveling the past by projecting past states through multiplication with the current state vector using the inverse of the transition matrix.

New cards

Explore top notes

note Note
studied byStudied by 17 people
Updated ... ago
5.0 Stars(1)
note Note
studied byStudied by 203 people
Updated ... ago
5.0 Stars(1)
note Note
studied byStudied by 12 people
Updated ... ago
5.0 Stars(1)
note Note
studied byStudied by 13 people
Updated ... ago
5.0 Stars(1)
note Note
studied byStudied by 14 people
Updated ... ago
5.0 Stars(1)
note Note
studied byStudied by 2871 people
Updated ... ago
4.9 Stars(26)
note Note
studied byStudied by 7 people
Updated ... ago
4.0 Stars(1)
note Note
studied byStudied by 12 people
Updated ... ago
5.0 Stars(1)

Explore top flashcards

flashcards Flashcard41 terms
studied byStudied by 2 people
Updated ... ago
5.0 Stars(1)
flashcards Flashcard50 terms
studied byStudied by 18 people
Updated ... ago
5.0 Stars(1)
flashcards Flashcard30 terms
studied byStudied by 7 people
Updated ... ago
5.0 Stars(1)
flashcards Flashcard125 terms
studied byStudied by 8 people
Updated ... ago
5.0 Stars(1)
flashcards Flashcard84 terms
studied byStudied by 2 people
Updated ... ago
5.0 Stars(1)
flashcards Flashcard20 terms
studied byStudied by 4 people
Updated ... ago
5.0 Stars(1)
flashcards Flashcard160 terms
studied byStudied by 13 people
Updated ... ago
5.0 Stars(1)
flashcards Flashcard26 terms
studied byStudied by 63 people
Updated ... ago
5.0 Stars(4)