AP Calculus AB, unit: Differential Equations. Conceptual understanding and key rules for this unit, explained in plain language alongside the formula. Front: a concept or short problem cue. Back: the explanation, rule, or answer.
37 cards · basic cards · AI-written, checked twice. Edit anything.
- What is a differential equation?
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An equation relating a function to its derivatives- What is a slope field?
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A graphical representation showing the slope of solutions to a differential equation at various points- What is separation of variables used for?
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Solving a differential equation by isolating each variable on opposite sides before integrating- What is a general solution to a differential equation?
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A family of solutions including an arbitrary constant- What is a particular solution to a differential equation?
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A specific solution satisfying a given initial condition- What is an initial condition, in a differential equation problem?
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A known value of the function at a specific point, used to find a particular solution- What is exponential growth modeled by?
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A differential equation where the rate of change is proportional to the current amount- What is the general solution to dy/dx = ky?
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y = C times e^(kt), where C is determined by the initial condition- What does a positive value of k represent in exponential growth models?
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Growth, where the quantity increases over time- What does a negative value of k represent in exponential models?
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Decay, where the quantity decreases over time- What is Euler's Method used for?
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Numerically approximating solutions to a differential equation using small linear steps- What is logistic growth?
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A growth model where the rate of change slows as the quantity approaches a carrying capacity- What does verifying a solution to a differential equation involve?
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Substituting the proposed function and its derivatives back into the equation to confirm it holds- What is an autonomous differential equation?
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A differential equation where dy/dx depends only on y, not on the independent variable x.- What is an equilibrium solution to a differential equation?
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A constant solution y = c where dy/dx = 0 for all x.
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