# Unit 7 Differential Equations

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ENDURING UNDERSTANDING
FUN-7 Solving differential equations allows us to determine functions and develop models.

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 Topic Name Essential Knowledge 7.1 Modeling Situation with Differential Equations 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.

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Differential Equations The basics and definitions.

Good Question 2: 2002 BC 5 A differential equation that cannot be solved by separating the variables is investigated anyway. Most of this question is AB material.

A Family of Functions  Further investigation of the general solution of the equation discussed above in Good Question 2. Most of this question is AB material.

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 7.2 Verifying Solutions for Differential Equations LEARNING OBJECTIVE FUN-7.B Verify solutions to differential equations. FFUN-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.

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An Exploration in Differential Equations  An exploration covering pretty much all of the ideas in differential equation based on 2018 AB 6.  The exploration is here and the solutions here..

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 7.3 Sketching Slope Fields 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.

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Slope Fields Graphical solutions: The solution is lurking in the slope field.

Good Question 2: 2002 BC 5 A differential equation that cannot be solved by separating the variables is investigated anyway. Most of this question is AB material.

A Family of Functions  Further investigation of the general solution of the equation discussed above in Good Question 2. Most of this question is AB material.

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 7.4 Reasoning Using Slope Fields LEARNING OBJECTIVE FUN-7.C Estimate solutions to differential equations. FUN-7.C.3 Solutions to differential equations are functions or families of functions.

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Slope Fields Graphical solutions: The solution is lurking in the slope field.

Good Question 2: 2002 BC 5 A differential equation that cannot be solved by separating the variables is investigated anyway. Most of this question is AB material.

A Family of Functions Further investigation of the general solution of the equation discussed above in Good Question 2. Most of this question is AB material.

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 7.5 Approximating Solutions Using Euler’s Method BC ONLY 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

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Euler’s Method (1-12-2015) Numerical solutions (BC only topic)

Euler’s Method for Making Money (2-25-2015) The connection between compound growth (compound interest) and Euler’s Method.

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 7.6 Finding General Solutions Using Separation of Variables 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.

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Accumulation and Differential Equations  (2-1-2013) Solving differential equations without the “+C

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 7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables 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 $F\left( x \right)={{y}_{0}}+\int_{a}^{x}{{f\left( t \right)dt}}$ is a particular solution to the differential equation $\frac{{dy}}{{dx}}=f\left( x \right)$ dy = fx ()dx , satisfying $F\left( a \right)={{y}_{0}}$ FUN-7.E.3 Solutions to differential equations may be subject to domain restrictions.

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Domain of a Differential Equation (4-7-2017) notes and examples on finding the domain of the solution of a differential equation

Accumulation and Differential Equations  (2-1-2013) Solving differential equations without the “+C

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 7.8 Exponential Models with Differential Equations LEARNING OBJECTIVES FUN-7.F Interpret the meaning of a differential equation and its variables in context. FUN-7.G Determine general and particular solutions for problems involving differential equations 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 $\frac{{dy}}{{dt}}=ky$ . FUN-7.G.1 The exponential growth and decay model, $\frac{{dy}}{{dt}}=ky$ , with initial condition $y={{y}_{0}}$ when t = 0, has solutions of the form $y={{y}_{0}}{{e}^{{kt}}}$ .

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 7.9 Logistic Models with Differential Equations BC ONLY 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 $\frac{{dy}}{{dt}}=ky\left( {a-y} \right)$. 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

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Logistics Growth – Real and Simulated

Type 6 Questions: Differential Equations  Summary and review notes.

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