Students will work in groups of two or three to match the three types of. Bc2, imsa slope matching s 03 bc 2 slope fields name. At each sample point x, y, there is a small line segment. Match the gradient field of each function to the contour plot of that function. Which of the following could be a specific solution to that differential equation.
Slope fields show up on both the ap calculus ab and bc tests. Slope fields have been a topic on the ap calculus bc exam since 1998 and on the ap calculus ab exam since 2004. Here is the slope field for the differential equation. This set of 108 cards 27 sets of four cards matching cards contains 27 differential equations. Given a slope field, sketch a solution curve through a given point. Match each slope field with the equation that the slope field could represent. Sketch at least 5 solution curves for each differential equation. We also acknowledge previous national science foundation support under grant numbers 1246120, 1525057, and 14739. Slope fields nancy stephenson clements high school sugar land, texas draw a slope field for each of the following differential equations. Then use your tangent line equation to estimate the value of f 1. I will say that my favorite of what is below is the packet put together by nancy stephenson. Draw a slope field for each of the following differential equations. Sketch the slope field for each differential equation. Ap calculus students need to be able to relate differential equations, slope fields, descriptions about the slopefielddifferential equation, and an example of a particular solution.
Example 8 matching slope fields with differential equations use slope analysis to match each of the following differential equations with one of the slope fields a through d. Work on the relationship between differential equations and the slope field that represents them. At the right is a slope field from a certain differential equation. Integrate the 3 equations that it is possible to integrate. For details on both methods, see the next subsection. You will be asked to match slope fields with their differential equations, or to match differential equations with their slope fields. Matching slope fields to differential equations ex4. Graphing calculators and programs like winplot will draw slope fields. Slope fields an introduction intro activity give all students a small square cut from an overhead transparency with a dot on it. Solve the differential equation dy ky dt by using separation of variables and assuming k is a constant. Then give the class a problem such as or and talk about what this means slope of tangent line at the point.
Slope field matching lesson plan for 11th 12th grade. Slope field matching cards by math through discovery llc tpt. For each problem, find a differential equation that could be represented with the given slope field. Match the slope elds with their di erential equations. A slope field is a visual representation of a differential equation of the form dydx fx, y. Slope fields we will consider differential equations of the form dy dt ft,y. Match each slope field with the equation of the general solution it could represent. Our stroll through the slope fields above gave some examples of things you can look for. Match each slope field with the correct differential equation. Each group will receive one set of cards and an answer sheet to record their answers. Students will work in groups of two or three to match the three types of cards. Draw a slope field at a specified number of points by hand. Matching connect each of the six slope fields shown below to their differential equations.
Slope fields, solution curves, and eulers method 3 example 1 recall that the logistic equation is the di erential equation dp dt kp 1 p p max where k and p max are constants. Slope field card match nancy stephenson clements high. Draw a slope field for each of the following differential. For this slope field worksheet, students match the slope field with the matching differential equation. Slope fields from the ab outline, students should be able to give a geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations. Sketch a particular solution on a given slope field. Match the differential equations with their corresponding. Slope fields and differential equations students should be able to. Ap calculus ab syllabus course overview this is a college level calculus course designed to meet the advanced placement curricular requirements for calculus ab equivalent to a onesemester college course. Slope field card match nancy stephenson clements high school sugar land, texas students will work in groups of two or three. Match the slope field with their differential equations.
Sketch a slope field for a given differential equation. To solve a differential equation means to find the unique equation that satisfies the given conditions among. Plot the graph of a function within an arbitrary viewing window, 2. Slope fields also called vector fields or direction fields are a tool to graphically obtain the solutions to a first order differential equation. Ap calculus slope fields this is a collection of files and websites accumulated from a cry out for help on the ap list serve for introducing slope fields in ap calculus. Slope fields and differential equations sss handoutfinish. This equation can be used to modeled the growth of a population in an environment with a nite carrying capacity p max. What i want to do in this video is do an exercise that takes us the other way, start with a slope field and figure out which differential equation is the slope field describing the solutions for. The slope, yx, of the solutions yx, is determined once we know the values for x and y, e. To determine y as a function of x when we are given its derivative and the value of the function at a given point is called an initial value problem.
Matching slope fields to differential equations part 1 youtube. Given the following slope field with equilibrium solutions, meaning the slopes are zero and a the existence of a horizontal asymptote on the solution graph, at y 0 and y 1, find the matching differential equation. Ap calculus ab worksheet 98 slope fields draw a slope field for each of the following differential equations. Given a differential equation in x and y, we can draw a segment with dydx as slope at any point x,y. An article on how a teacher introduces slope fields in her classroom. Match the slope fields with their differential equations. Slope fields make use of this by imposing a grid of points evenly spaced across the cartesian plane. A 1 dy yy dx b 1 dy yy dx c 1 1 dy dx y y d dy 1 eyy 1 dx e 1 dy y dx y 20. While at first this topic might seem daunting, the questions on the test are actually quite straightforward. If youre behind a web filter, please make sure that the domains. Find the zeros of functions solve equations numerically, 3. B let f be the function that satisfies the given differential equation. Match each differential equation with the corresponding solution curve. Using the tinspire calculator in ap calculus version 3.
Given the following slope field with equilibrium solutions, that means slopes of zero and a horizontal asymptote on the solution graph, at y 0 and y 1, find the matching differential equation. Slope fields nancy stephenson clements high school sugar. The slope field from a certain differential equation is shown above. Write an equation for the tangent line to the curve yfx through the point 1, 1. Draw conclusions about the solution curves by looking at the slope field. Match each slope field with the equation that the slope filed could represent. Each vector field shown is the gradient of a function fx, y. Match a slope field to a solution of a differential equation. The solution curve passing through the point o, l is also shown.
Slope fields on the ap exams the availability of technology to draw slope fields is relatively new. Stirling 201112 page 3 of 7 match the slope fields with their differential equations. This slope field matching lesson plan is suitable for 11th 12th grade. Solving this differential equation means getting an equation of the form y dy x y dx 2 dy x dx 1 dy. Slope fields differential equations conclusions sf 1 sf 2 sf 3.
At each point the value of the derivative is calculated and a short segment with that slope is drawn. These segments graphed together form the slope field. Some textbooks do not mention slope fields, so this is a topic that may need supplementing. The calculator drawn slope field for the differential equation xy is shown in the figure below. The slope field for a certain differential equation is shown above. Slope field card match nancy stephenson clements high school sugar land, texas students will work in groups of two or three to match the three types of cards. Sketch a solution that passes through a given point on a slope field. Matching differential equations with their slope fields. Matching slope fields to differential equations part 1.
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