If all segments in the horizontal direction on a slope field have the same slope, then the differential equation does not contain an x-term. ![]() If all line segments in the vertical direction on a slope field have the same slope, then the differential equation does not contain a y-term. The students look at the slope field to see if all line segments on the slope field have the same slope: if they do, the differential equation will be of the form dy/dx = constant. ![]() The knowledge the students gain by making these observations helps when I ask them to match a differential equation to a slope field. This time the students notice that all of the points that have the same y-coordinate have the same slope because this differential equation contains a y-term but no x-term. This time the students should notice that all of the points on the graph that have the same x-coordinate have the same slope because our differential equation has an x-term but no y-term.Īfter we complete the slope field for dy/dx = x +1, we draw a slope field for another differential equation, such as dy/dx = 2y. I ask the students to name other points that have the same slope. We pick a starting point on our grid and draw a tiny line segment that passes through our point and has the slope that we found. Next I have the students draw a slope field for the differential equation dy/dx = x +1. When we move to another point, we notice that the slope will also be 2 and that the slope will be 2 no matter what point we consider since dy/dx = constant. We pick a starting point on our grid and draw a tiny line segment that passes through our point and that has a slope of 2. Then I hand them a sheet of grid paper and a ruler, and we start with a differential equation such as dy/dx = 2. When I teach my students to draw a slope field, I first review how to graph a line, given a point and a slope. Students can look at the slope field and visualize the family of antiderivatives and can also sketch the solution curve through a particular point. Another way to show the family of antiderivatives is to draw a slope field for dy/dx = 2x. I ask them to sketch several of these antiderivatives on the same graph grid so that they can see the family of antiderivatives. Student answers might include y = x 2, y = x 2 + 3, y = x 2 - 1, and so forth in other words, y = x 2 + C. When I introduce antiderivatives to my students, I ask them to name a function whose derivative is 2x. ![]() Slope fields also give us a great way to visualize a family of antiderivatives. When an explicit solution to a differential equation is not possible, the slope field provides a way to solve the equation graphically. When solving differential equations explicitly, students can use slope fields to verify that the explicit solutions match the graphical solutions. ![]() No matter how busy you are, this 5 Steps to a 5 guide will help you make the most of your last-minute study to build the skills you need in a minimal amount of time.Slope fields are an excellent way to visualize a family of solutions of differential equations. Written by expert AP teachers who know the exam inside and out, the questions closely resemble those you’ll face on exam day, and include detailed review explanations for both right and wrong answers.ĥ Steps to a 5: 500 AP Calculus AB/BC Questions to Know by Test Day, Fourth Edition is updated for the latest exam, featuring only the type of questions you’ll see on this year’s exam, plus a super-helpful 20 Question Diagnostic quiz to test your knowledge. You’ll find the smartest, most effective test prep in 5 Steps to a 5: 500 AP Calculus AB/BC Questions to Know by Test Day, Fourth Edition. The only study guide you’ll need for the AP Calculus AB/BC test-revised and updated, now with a 20-question Diagnostic QuizĬonfidence is key when taking any exam, and it will come easier if you spend your test prep time wisely-even if you’ve been so busy that you’ve put off preparing until the last weeks before the exam.
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