$\frac{dy}{dx} = f(x, y)$

$y = F(x) + C$

Solve $\frac{dy}{dx} = 0$ or where $\frac{dy}{dx}$ is undefined.

$\int \frac{1}{1+x^2} dx = \arctan(x) + C$

$y = f(x) + C$, where C is an arbitrary constant.

Use initial condition $y(x_0) = y_0$ to solve for C in $y = f(x) + C$.

Slope at any point (x,y) is the difference between x and y.

$y = \int f(x) dx + C$

Look for horizontal or vertical tangents on the graph.

$\frac{dy}{dx} |_{(x_0,y_0)}$

Slope fields provide a graphical representation of the slope at various points, allowing us to approximate solution curves without explicitly solving the differential equation.

The constant of integration accounts for the fact that the derivative of a constant is zero, leading to a family of possible solutions differing by a constant value.

Initial conditions provide a specific point on the solution curve, allowing us to solve for the constant of integration and identify a unique solution.

The derivative is zero, indicating a potential maximum or minimum.

Positive slopes indicate increasing function, negative slopes indicate decreasing function.

Denser segments indicate faster changes in the function's value.

Slope field visually represents the derivative of a function at various points.

Observe how the slopes change; increasing slopes indicate concave up, decreasing slopes indicate concave down.

Equilibrium solutions are constant solutions where $\frac{dy}{dx} = 0$ for all x, represented by horizontal lines in the slope field.

It represents the vertical shift of the solution curve.

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.