Even before you learn techniques for solving differential equations, you may be able to analyze equations qualitatively. As an example, look at the nonlinear equation dy (y – 4)°(y + 5) dt You are going to analyze the solutions, y, of this equation without actually finding them. You will be asked to sketch three solutions of the differential equation on the graph below based on qualitative informat differential equation. In what follows, picture the t-axis running horizontally and the y axis running vertically. There is no scale on the t axis but imagine it is large enough to display the behavior of the solutions as t approaches + a) For what values of y is the graph of y as a function of t increasing? Use interval notation for your answer. (-5,4)U(4,INF) b) For what values of y is the graph of y concave up? (-INF,-5)U(4,INF) For what values of y is it concave down? (-INF.-4)U(-2,4) (Help with interval notation.) What information do you need to answer question about concavity? Remember that y is an implicit function of t. (How to enter answer] Parts c),d), e) of this question ask you to modify the blue, red, and green curves in the plot below to make them represent graphs of particular solutions of the differential equation, To modify the blue curve, click the "blue curve" button below the plot to expose blue points and tangents. Solid blue points lie on the curve. With your mouse click and hold each solid blue point, and move it i position. If the solution curve crosses an edge of the viewing region then the corresponting solid point should be very near the edge, left or right, top or bottom. Improve the shape of the curve between the so moving the open points that lie on the dashed tangents. Experiment to see how the shape changes. Modify the red or green curve in a similar way, after clicking the corresponding button to expose its points and tangents. I recommend moving solid points into good positions first, then move the open points shape between the solid points. dy = (y – 4)°(y + 5). 4.
Even before you learn techniques for solving differential equations, you may be able to analyze equations qualitatively. As an example, look at the nonlinear equation dy (y – 4)°(y + 5) dt You are going to analyze the solutions, y, of this equation without actually finding them. You will be asked to sketch three solutions of the differential equation on the graph below based on qualitative informat differential equation. In what follows, picture the t-axis running horizontally and the y axis running vertically. There is no scale on the t axis but imagine it is large enough to display the behavior of the solutions as t approaches + a) For what values of y is the graph of y as a function of t increasing? Use interval notation for your answer. (-5,4)U(4,INF) b) For what values of y is the graph of y concave up? (-INF,-5)U(4,INF) For what values of y is it concave down? (-INF.-4)U(-2,4) (Help with interval notation.) What information do you need to answer question about concavity? Remember that y is an implicit function of t. (How to enter answer] Parts c),d), e) of this question ask you to modify the blue, red, and green curves in the plot below to make them represent graphs of particular solutions of the differential equation, To modify the blue curve, click the "blue curve" button below the plot to expose blue points and tangents. Solid blue points lie on the curve. With your mouse click and hold each solid blue point, and move it i position. If the solution curve crosses an edge of the viewing region then the corresponting solid point should be very near the edge, left or right, top or bottom. Improve the shape of the curve between the so moving the open points that lie on the dashed tangents. Experiment to see how the shape changes. Modify the red or green curve in a similar way, after clicking the corresponding button to expose its points and tangents. I recommend moving solid points into good positions first, then move the open points shape between the solid points. dy = (y – 4)°(y + 5). 4.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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