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ENDURING UNDERSTANDING
FUN7 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 FUN7.A Interpret verbal statements of problems as differential equations involving a derivative expression. 
FUN7.A.1 Differential equations relate a function of an independent variable and the function’s derivatives. 
Blog Posts
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 FUN7.B Verify solutions to differential equations. 
FFUN7.B.1 Derivatives can be used to verify that a function is a solution to a given differential equation.

FUN7.B.2 There may be infinitely many general solutions to a differential equation. 
Blog Posts
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 FUN7.C Estimate solutions to differential equations. 
FUN7.C.1 A slope field is a graphical representation of a differential equation on a finite set of points in the plane. 
FUN7.C.2 Slope fields provide information about the behavior of solutions to firstorder differential equations. 
Blog Posts
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 FUN7.C Estimate solutions to differential equations. 
FUN7.C.3 Solutions to differential equations are functions or families of functions. 
Blog Posts
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 FUN7.C Estimate solutions to differential equations. 
FUN7.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 
Blog Posts
Euler’s Method (1122015) Numerical solutions (BC only topic)
Euler’s Method for Making Money (2252015) 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 FUN7.D Determine general solutions to differential equations. 
FUN7.D.1 Some differential equations can be solved by separation of variables. 
FUN7.D.2 Antidifferentiation can be used to find general solutions to differential equations. 
Blog Posts
Accumulation and Differential Equations (212013) Solving differential equations without the “+C”
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7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables
LEARNING OBJECTIVE FUN7.E Determine particular solutions to differential equations. 
FUN7.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. 
FUN7.E.2 The function F defined is a particular solution to the differential equation dy = fx ()dx , satisfying  
FUN7.E.3 Solutions to differential equations may be subject to domain restrictions. 
Blog Posts
Domain of a Differential Equation (472017) notes and examples on finding the domain of the solution of a differential equation
Accumulation and Differential Equations (212013) Solving differential equations without the “+C”
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7.8 Exponential Models with Differential Equations
LEARNING OBJECTIVES FUN7.F Interpret the meaning of a differential equation and its variables in context. FUN7.G Determine general and particular solutions for problems involving differential equations in context. 
FUN7.F.1 Specific applications of finding general and particular solutions to differential equations include motion along a line and exponential growth and decay. 
FUN7.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 .  
FUN7.G.1 The exponential growth and decay model, , with initial condition when t = 0, has solutions of the form . 
Blog Posts
none
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7.9 Logistic Models with Differential Equations BC ONLY
LEARNING OBJECTIVE FUN7.H Interpret the meaning of the logistic growth model in context. bc only 
FUN7.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 . BC ONLY 
FUN7.H.2 The logistic differential equation and initial conditions can be interpreted without solving the differential equation. BC ONLY  
FUN7.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  
FUN7.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 
Blog Posts
Logistics Growth – Real and Simulated
Type 6 Questions: Differential Equations Summary and review notes.
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