A visual representation of solutions to differential equations, showing slopes at different points.

A point where the derivative of a function is zero or undefined.

An equation that relates a function with its derivatives.

An arbitrary constant (+C) added during integration, representing a family of solutions.

A set of solutions to a differential equation, each differing by the constant of integration.

Indicate a slope of zero, potential critical points.

Indicate an undefined slope, potential critical points.

Represents the magnitude of the slope.

A specific value used to determine the constant of integration (+C) and find a particular solution.

A single solution from the family of functions, determined by an initial condition.

General: Includes '+C', represents family of functions. | Particular: '+C' is solved for using initial conditions, represents one specific function.

Stable: Nearby solution curves approach the equilibrium. | Unstable: Nearby solution curves move away from the equilibrium.

Slope fields: Show the slope at various points. | Solution curves: Represent a particular solution to the differential equation.

Critical Point: $\frac{dy}{dx} = 0$ or undefined, indicates local max/min. | Inflection Point: Change in concavity, second derivative is zero or undefined.

Differential Equation: Equation involving derivatives. | Integral: The reverse process of differentiation.

Direction Field: Another name for slope field. | Slope Field: Visual representation of solutions to differential equations.

Increasing Function: $\frac{dy}{dx} > 0$. | Decreasing Function: $\frac{dy}{dx} < 0$.

Definite Integral: Evaluated over a specific interval, yields a numerical value. | Indefinite Integral: Represents a family of functions, includes '+C'.

Derivative: Rate of change of a function. | Antiderivative: Function whose derivative is the original function.

Local Extrema: Max/min in a specific interval. | Global Extrema: Absolute max/min over the entire domain.

1. Identify horizontal line segments (slope = 0). 2. Identify vertical line segments (slope undefined). 3. These locations are potential critical points.

1. Locate the initial point. 2. Follow the direction of the line segments, sketching a curve that is tangent to them.

1. Separate variables. 2. Integrate both sides. 3. Add '+C'. 4. Solve for y if possible.

1. Examine the slope field as x gets large. 2. Observe if the solution curves approach a horizontal asymptote or grow without bound.

1. Sketch the solution curve through the initial condition on the slope field. 2. Solve the differential equation analytically and use the initial condition to find C.

1. Identify equilibrium solution. 2. Observe nearby solution curves. 3. If curves approach the equilibrium, it's stable. If they move away, it's unstable.

1. Integrate both sides. 2. $\int \frac{dy}{dx} dx = \int \frac{1}{1+x^2} dx$. 3. $y = \arctan(x) + C$.

1. Look for areas with positive slopes. 2. These areas indicate the solution is increasing.

1. Start at the initial point. 2. Follow the direction of the slope field lines.

1. Set $\frac{dy}{dx} = 0$. 2. Solve for y.