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7.1 Modeling Situations with Differential Equations

FUN-7.A Interpret verbal statements of problems as differential equations involving a derivative expression.
FUN-7.A.1 Differential equations relate a function of an independent variable and the function’s derivatives.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

FUN-7.B Verify solutions to differential equations.
FUN-7.B.1 Derivatives can be used to verify that a function is a solution to a given differential equation.
FUN-7.B.2 There may be infinitely many solutions to a differential equation.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

FUN-7.C Estimate solutions to differential equations.
FUN-7.C.1 A slope field is a graphical representation of a differential equation on a finite set of points in the plane.
FUN-7.C.2 Slope fields provide information about the behavior of solutions to first-order differential equations.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

FUN-7.C Estimate solutions to differential equations.
FUN-7.C.3 Solutions to differential equations are functions or families of functions.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

FUN-7.D Determine general solutions to differential equations.
FUN-7.D.1 Some differential equations can be solved by separation of variables.
FUN-7.D.2 Antidifferentiation can be used to find general solutions to differential equations.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

FUN-7.E Determine particular solutions to differential equations.
FUN-7.E.1 A general solution may describe infinitely many solutions to a differential equation. There is only one particular solution passing through a given point.
FUN-7.E.2 The function F defined by F(x) = y₀ + ∫ₐˣ f(t)dt is a particular solution to the differential equation dy/dx = f(x), satisfying F(a) = y₀.
FUN-7.E.3 Solutions to differential equations may be subject to domain restrictions.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

FUN-7.F Interpret the meaning of a differential equation and its variables in context.
FUN-7.F.1 Specific applications of finding general and particular solutions to differential equations include motion along a line and exponential growth and decay.
FUN-7.F.2 The model for exponential growth and decay that arises from the statement “The rate of change of a quantity is proportional to the size of the quantity” is dy/dt = ky.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

FUN-7.G Determine general and particular solutions for problems involving differential equations in context.
FUN-7.G.1 The exponential growth and decay model, dy/dt = ky, with initial condition y = y₀ when t = 0, has solutions of the form y = y₀eᵏᵗ.
Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.