MATHEMATICS · Multi-part · AP · Calculus AB · Differential Equations · Differential Equations
Problem
For a certain differential equation dxdy=g(x,y), the slopes of the direction segments at integer lattice points (x,y) for 0≤x≤3 and 0≤y≤3 are recorded in the table.
y = 0 y = 1 y = 2 y = 3
x = 0 0 1 2 3
x = 1 −1 0 1 2
x = 2 −2 −1 0 1
x = 3 −3 −2 −1 0
A particular solution curve passes through the point (1,1).
(a) Based on the numerical pattern in the table, conjecture an explicit formula for g(x,y) in terms of x and y. Explain how the data support your conjecture.
(b) Use your conjectured formula to predict the slope at the point (1.5,2.5). Verify that this prediction is consistent with the trend in the table.
(c) Determine the y-coordinate of the point where the solution curve through (1,1) first attains a horizontal tangent as x increases. Provide a clear justification that references the slope information.
(d) Using slope values near the point found in part (c), discuss the concavity of the solution curve for 1≤x≤2. Support your conclusion with appropriate reasoning based on changes in slope.