Slope Fields in AP Calculus

GA2PMT mini-conference -- October 18, 2002

 

Chris Harrow

The Westminster Schools

chrish@westminster.net

1424 West Paces Ferry Road, NW

404.609.6147

Atlanta, GA 30327

 

 

What is a slope field?

 

In the simplest terms, a slope field is the graphical representation of all possible solutions to a differential equation.  In this way, a slope field is very much like a graphical antiderivative.  It is very important to remember that functions (relations) are the solutions to both differential equations and antiderivatives.  Since a slope field is only a graphical representation of a differential equation, it should never be considered a proof of anything specific about the function in question or its derivative (i.e. point, slope, concavity, or equation(s)). 

 

Why are they troublesome?

 

A difference that throws many students is that differential equations don’t often look like antiderivative problems.  Rather than asking a student to solve , they are asked to find the function, , satisfying .  Seen another way, however, this is the core of the antiderivative concept - playing Jeopardy!  I’m giving you the derivative and challenging you to determine the form of the original function.  A differential equation gives you a derivative (same as an antiderivative), so all you have to do is identify the original….  Simple enough?

Since the only differential equation solution technique required by the AP is separable variables, the task is even easier; either you recognize that 2x is the derivative of anything in the form of  or you separate the variables and end up with the equivalent antiderivative:

 

 


Sketch multiple solutions to

[image]

Sketch a slope field for

[image]

 

Creating your own

 

Slope fields become more troublesome when their algebraic form is the result of implicit differentiation.  One easy example is:  .  Assuming you were only given the final result  and did not recall or want to do separation of variables, what could you do?  Graph the slopes to get an idea about the look of the solutions.

 

In general, I encourage my students to follow four basic guidelines:  1)  Where is the slope zero, 2)  Where is the slope undefined, 3)  Where is the slope , and 4)  Where is the slope positive and/or negative?

Slope field criteria

 

1)

 

 

 

2)

 

 

 

3)

 

 

 

4)

Sketch a slope field for

[image]

 

Notice that the function solutions to  are no longer simple vertical translations of a parent function; that only happens when  can be expressed as a function of x only.  Looking at your last picture, one might suspect circles to be the solutions:  .  Draw two different solutions on the slope field - start one from (1,0) and another from (-2,2).  Thinking like Euler’s method, allow each tangent line to “re-aim” your movement after each small step.  You should see two circles.  Now verify your suspicions algebraically:

 

So the C in the algebra solution is simply the  from our suspicion about the shape.  Notice that I did not claim the solution merely from the shapes suggested by the slope field, I used algebra to justify my answer while allowing the suspicion to guide my solution.

BC note:  In my opinion, explorations of slope fields and Euler’s method are perfectly suited for a unit of study.  Slope fields grant a rough picture which Euler’s attempts to more carefully quantify.

 

What about initial conditions?

 

An initial condition simply identifies an ordered pair through which a particular solution to the differential equation passes.  While a slope field visually represents all solutions to a differential equation, identifying an initial condition asks for the particular function (a specific “C”) passing through that point.

Looking back at the slope field for the differential equation, , identify again the solution curve passing through (1,0).  If a question asked for the equation of the curve through (1,0) satisfying the differential equation, looking at the slope field and saying  from the picture would not be sufficient.  Graphs are suggestions and guides – not proofs.  Rather, one would have to algebraically solve the differential equation and substitute:

You get the answer you suspected, but you used an algebraic proof guided by your informed intuitive knowledge of the slope field.

**Another potential use of slope fields in a multiple choice situation would be the presentation of a slope field without its defining differential equation and asking which of the given functions could satisfy the unstated differential equation.  Graphing the given functions would reveal whether they “followed” the pattern displayed in the slope field.

 

Are solutions unique?

 

Almost always!  From a slope field point of view, knowing where to start tells me where to go.  This does not mean that I will always be able to solve a differential equation algebraically, but I can always have a good idea about what the solution looks like!  It is important to note that since the solutions are generally unique (“always” is probably safe as far as the AP is concerned), then there can only be one solution curve passing through any given ordered pair in a slope field.  If two solution curves to a single differential equation intersect on its slope field, you probably have done something wrong.  Don’t confuse tight approaches to an asymptote with intersections.

 

Remember – these are SLOPE fields

 

Since one is graphing slopes, the differential equation will always be first order.  These are not “curvature” or “concavity” fields, although you can “see” the concavity in the slope field images.  Again, seeing concavity is insufficient proof of the existence of that concavity.  Algebraic means are required for proof.  As multiple choice questions do not require work, students may be able to push their visual intuition further on these questions than on free response queries.

 

Released AP Calculus BC questions on slope fields

 

As of this year, slope fields are only a BC topic, although they will be added to the AB curriculum within two years.  Also, since they are a relatively recent addition to BC (due to the 1990’s calculus reform movement), there aren’t that many questions.  The only multiple choice example is 1998[BC24] and the free response questions are 1998[BC4], 2000[BC6], and 2002[BC5]. 

I’m also including the problem set from slope fields section of the Harvard Calculus text to give additional examples.

References/problems included:

 

  • Hughes-Hallett, D., A. M. Gleason, et al. (1998). Calculus:  Single and Multivariable. New York, Wiley & Sons, 498-501.
  • AP Calculus BC released multiple choice question on slope fields:  1998 #24
  • AP Calculus BC released free response question on slope fields:  1998 #4, 2000 #6, 2002 #5

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1998 Exam scoring guidelines:  http://apcentral.collegeboard.com/repository/sg_calcbc_98_9670.pdf

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2000 Exam scoring guidelines:  http://apcentral.collegeboard.com/repository/sg_calculus_bc_00.pdf

Sample solutions to 2000 #6 at http://apcentral.collegeboard.com/repository/sample_calculus_bc_00_q6.pdf

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2002 Exam scoring guidelines:  http://apcentral.collegeboard.com/repository/sg_calculus_bc_02_11387.pdf

Sample solutions to 2002 #5 at http://apcentral.collegeboard.com/repository/sample_calculus_bc_q5_17718.pdf