Differential Equations · ⁨สมการเชิงอนุพันธ์⁩

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective 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.

Learning Objective 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 general solutions to a differential equation.

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective 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.

Learning Objective 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.

Learning Objective 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. BC ONLY

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective 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.

Learning Objective 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 $F(x) = y_0 + \int_a^x f(t)\,dt$ is a particular solution to the differential equation $\dfrac{dy}{dx} = f(x)$, satisfying $F(a) = y_0$.
• 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.

Learning Objective 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 $\dfrac{dy}{dt} = ky$.

Learning Objective 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, $\dfrac{dy}{dt} = ky$, with initial condition $y = y_0$ when $t = 0$, has solutions of the form $y = y_0 e^{kt}$.

Enduring Understanding (FUN-7): Solving differential equations allows us to determine functions and develop models.

Learning Objective FUN-7.H: Interpret the meaning of the logistic growth model in context. BC ONLY

• 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 $\dfrac{dy}{dt} = ky(a - y)$. BC ONLY
• FUN-7.H.2 The logistic differential equation and initial conditions can be interpreted without solving the differential equation. BC ONLY
• 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. BC ONLY
• 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. BC ONLY