Comprehensive Guide to AP Calculus BC Unit 7

Differential Equation (DE)

An equation that relates a function y to its derivative(s).

First-Order Differential Equation

A differential equation that involves the first derivative (dy/dx).

General Solution

A family of functions containing an arbitrary constant C.

Particular Solution

A specific function obtained by applying an initial condition.

Initial Condition

A value that allows us to find the specific solution to a differential equation.

Verify Solutions

The process of checking if a proposed function satisfies a differential equation.

Left Hand Side (LHS)

The left component of a differential equation when set equal to the right side.

Right Hand Side (RHS)

The right component of a differential equation when set equal to the left side.

Slope Field

A graphical representation of a differential equation showing slopes at various points.

Direction Field

Another name for a slope field, illustrating the direction of solution curves.

Slope at Point (x,y)

The value of dy/dx at a specific coordinate (x,y) in a differential equation.

Flow of Solution Curves

The pattern followed by solution curves indicated by slope fields.

Constructing Slope Fields

The method of creating a slope field, which involves calculating slopes at points.

Zero Slope Isoclines

Lines where the slope of a function is zero, indicating horizontal tangent lines.

Euler's Method

A numerical method to approximate values of a function using its tangent line.

Local Linearity

The concept that a function behaves like its tangent line around a point.

Step Size (h)

The interval used in Euler's method for approximating function values.

Approximate Value of a Function

The value calculated using numerical methods like Euler's method.

Concavity

The curvature of a function indicating whether it is curving up or down.

Separation of Variables

An algebraic method for solving differential equations by separating variables.

SIPPY Method

A mnemonic for the steps in solving differential equations: Separate, Integrate, Plus C, Plug In, Y Equals.

Integrating Factor

A function used to simplify solving linear differential equations.

Exponential Growth Model

A model where the rate of change of y is proportional to y itself.

Carrying Capacity (M)

The maximum population size an environment can sustain for the given resources.

Logistic Differential Equation

A model for population growth that considers carrying capacity.

Inflection Point

The point at which the rate of growth is maximized, typically at half the carrying capacity.

Absolute Value in Integration

The necessity to include absolute value in the logarithmic integration of y.

Constant of Integration (C)

An arbitrary constant added to the solution of an indefinite integral.

Overapproximation

An estimate of a value that is higher than the actual value.

Underapproximation

An estimate of a value that is lower than the actual value.

Negative Solutions

Solutions to equations that are less than zero; important for correct domain understanding.

Concave Up

When the function curves upwards, indicating that the slope function is increasing.

Concave Down

When the function curves downwards, indicating that the slope function is decreasing.

Slope Calculation

The process of determining the slope at given points in slope fields or Euler's method.

System of Differential Equations

A set of multiple differential equations that are related and solved together.

Linear Differential Equation

A differential equation where the dependent variable and its derivatives appear linearly.

Homogeneous Equation

A differential equation where every term is a function of the dependent variable and its derivatives.

Particular Integral

A specific solution found by applying initial or boundary conditions to a general solution.

Autonomous Differential Equation

A differential equation in which the independent variable does not appear explicitly.

Stability of Equilibrium

The behavior of equilibrium solutions of a differential equation as they respond to perturbations.

Systems of Differential Equations

Multiple interrelated differential equations that can be solved together to find relationships of several functions.

Implicit Solutions

Solutions for which y cannot be explicitly isolated on one side of the equation.

Fixed Point

A point that is mapped to itself by a function, indicating stability in differential equations.

Bounded Solutions

Solutions that remain within prescribed bounds and do not diverge.

Velocity Field

A vector field that represents the velocity of a dynamic system at different points.

Parameter Independence

The property that a solution does not depend on specific parameters, allowing for generalization.

Perturbation Methods

Techniques used to find an approximate solution to a problem, which is modified slightly from a problem that is solvable.

Behavior Near Singularity

A study of how solutions act near points where they are not well-defined.

Numerical Stability

The property that a numerical method produces bounded solutions under small perturbations.