What is a slope field?
A visual representation of solutions to differential equations, showing slopes at different points.
What is a critical point?
A point where the derivative of a function is zero or undefined.
What is a differential equation?
An equation that relates a function with its derivatives.
What is the constant of integration?
An arbitrary constant (+C) added during integration, representing a family of solutions.
What is a family of functions?
A set of solutions to a differential equation, each differing by the constant of integration.
What do horizontal line segments in slope field indicate?
Indicate a slope of zero, potential critical points.
What do vertical line segments in slope field indicate?
Indicate an undefined slope, potential critical points.
What is the significance of steepness of line segments in slope field?
Represents the magnitude of the slope.
What is an initial condition?
A specific value used to determine the constant of integration (+C) and find a particular solution.
What is a particular solution?
A single solution from the family of functions, determined by an initial condition.
What are the differences between general and particular solutions?
General: Includes '+C', represents family of functions. | Particular: '+C' is solved for using initial conditions, represents one specific function.
What are the differences between stable and unstable equilibrium solutions?
Stable: Nearby solution curves approach the equilibrium. | Unstable: Nearby solution curves move away from the equilibrium.
What are the differences between slope fields and solution curves?
Slope fields: Show the slope at various points. | Solution curves: Represent a particular solution to the differential equation.
What is the difference between a critical point and an inflection point?
Critical Point: $\frac{dy}{dx} = 0$ or undefined, indicates local max/min. | Inflection Point: Change in concavity, second derivative is zero or undefined.
What is the difference between differential equation and integral?
Differential Equation: Equation involving derivatives. | Integral: The reverse process of differentiation.
What is the difference between direction field and slope field?
Direction Field: Another name for slope field. | Slope Field: Visual representation of solutions to differential equations.
What is the difference between increasing and decreasing function?
Increasing Function: $\frac{dy}{dx} > 0$. | Decreasing Function: $\frac{dy}{dx} < 0$.
What is the difference between definite and indefinite integral?
Definite Integral: Evaluated over a specific interval, yields a numerical value. | Indefinite Integral: Represents a family of functions, includes '+C'.
What is the difference between derivative and antiderivative?
Derivative: Rate of change of a function. | Antiderivative: Function whose derivative is the original function.
What is the difference between local and global extrema?
Local Extrema: Max/min in a specific interval. | Global Extrema: Absolute max/min over the entire domain.
Differential equation for slope field
$\frac{dy}{dx} = f(x, y)$
General solution of a differential equation
$y = F(x) + C$
How to find critical points?
Solve $\frac{dy}{dx} = 0$ or where $\frac{dy}{dx}$ is undefined.
Formula for integrating $\frac{1}{1+x^2}$
$\int \frac{1}{1+x^2} dx = \arctan(x) + C$
How to represent a family of functions?
$y = f(x) + C$, where C is an arbitrary constant.
How to find a particular solution?
Use initial condition $y(x_0) = y_0$ to solve for C in $y = f(x) + C$.
What does $\frac{dy}{dx}=x-y$ represent in slope field?
Slope at any point (x,y) is the difference between x and y.
How is the general solution represented?
$y = \int f(x) dx + C$
How to find the critical points graphically?
Look for horizontal or vertical tangents on the graph.
How to represent slope at a point?
$\frac{dy}{dx} |_{(x_0,y_0)}$
