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Unit 7: Differential Equations

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  1. 7.1 Modeling Situations with Differential Equations

    1. FUN-7.A—Interpret verbal statements of problems as differential equations involving a derivative expression

      • 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.

  2. 7.2 Verifying Solutions for Differential Equations

    1. FUN-7.B—Verify solutions to differential equations

      • 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.

  3. 7.3 Sketching Slope Fields

    1. FUN-7.C—Estimate solutions to differential equations

      • 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.

  4. 7.4 Reasoning Using Slope Fields

    1. FUN-7.C—Estimate solutions to differential equations—Topic 7.4

      • 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.

  5. 7.5 Approximating Solutions Using Euler’s Method

    1. FUN-7.C—Estimate solutions to differential equations—Topic 7.5

      • FUN-7.C Estimate solutions to differential equations. • FUN-7.C.4 Euler’s method provides a procedure for approximating a solution to a differential equation or a point on a solution curve. • Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

  6. 7.6 Finding General Solutions Using Separation of Variables

    1. FUN-7.D—Determine general solutions to differential equations

      • 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.

  7. 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

    1. FUN-7.E—Determine particular solutions to differential equations

      • 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.

  8. 7.8 Exponential Models with Differential Equations

    1. FUN-7.F—Interpret the meaning of a differential equation and its variables in context

      • 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.

    2. FUN-7.G—Determine general and particular solutions for problems involving differential equations in context

      • 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.

  9. 7.9 Logistic Models with Differential Equations

    1. FUN-7.H—Interpret the meaning of the logistic growth model in context

      • FUN-7.H Interpret the meaning of the logistic growth model in context. • FUN-7.H.1 The model for logistic growth that arises from the statement “The rate of change of a quantity is jointly proportional to the size of the quantity and the difference between the quantity and the carrying capacity” is dy/dt = ky(a − y). • FUN-7.H.2 The logistic differential equation and initial conditions can be interpreted without solving the differential equation. • FUN-7.H.3 The limiting value (carrying capacity) of a logistic differential equation as the independent variable approaches infinity can be determined using the logistic growth model and initial conditions. • FUN-7.H.4 The value of the dependent variable in a logistic differential equation at the point when it is changing fastest can be determined using the logistic growth model and initial conditions. • Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.