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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.
How to find critical points from a slope field?
1. Identify horizontal line segments (slope = 0). 2. Identify vertical line segments (slope undefined). 3. These locations are potential critical points.
How to sketch a solution curve on a slope field given an initial condition?
1. Locate the initial point. 2. Follow the direction of the line segments, sketching a curve that is tangent to them.
How to solve a separable differential equation?
1. Separate variables. 2. Integrate both sides. 3. Add '+C'. 4. Solve for y if possible.
How to determine the behavior of a solution as x approaches infinity from a slope field?
1. Examine the slope field as x gets large. 2. Observe if the solution curves approach a horizontal asymptote or grow without bound.
How to find a particular solution given a slope field and initial condition?
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.
How to determine stability of equilibrium solution from slope field?
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.
How to solve $\frac{dy}{dx}=\frac{1}{1+x^2}$?
1. Integrate both sides. 2. $\int \frac{dy}{dx} dx = \int \frac{1}{1+x^2} dx$. 3. $y = \arctan(x) + C$.
How to identify regions where the solution is increasing?
1. Look for areas with positive slopes. 2. These areas indicate the solution is increasing.
How to sketch a solution curve?
1. Start at the initial point. 2. Follow the direction of the slope field lines.
How to find equilibrium solutions?
1. Set $\frac{dy}{dx} = 0$. 2. Solve for y.
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?
$\frac{dy}{dx} |_{(x_0,y_0)}$