Unit 7: Differential Equations
- Syllabus
- 2020
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In this section
Topic 7.1
7.1 Modeling Situations with Differential Equations
Objectives in this topic
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.
Topic 7.2
7.2 Verifying Solutions for Differential Equations
Objectives in this topic
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.
Topic 7.3
7.3 Sketching Slope Fields
Objectives in this topic
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.
Topic 7.4
7.4 Reasoning Using Slope Fields
Objectives in this topic
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.
Topic 7.5
7.5 Approximating Solutions Using Euler’s Method
Objectives in this topic
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.
Topic 7.6
7.6 Finding General Solutions Using Separation of Variables
Objectives in this topic
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.
Topic 7.7
7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
Objectives in this topic
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.
Topic 7.8
7.8 Exponential Models with Differential Equations
Objectives in this topic
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.
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.
Topic 7.9
7.9 Logistic Models with Differential Equations
Objectives in this topic
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.