2008 ap ab calculus multiple choice questions

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. Slope fields are an excellent way to visualize a family of solutions of differential equations.

2008 ap ab calculus multiple choice questions