Video: Identifying the Concavity and Monotonicity of a Function from Its Graph

Consider the given slope field graph representing a differential equation. If the solution of the differential equation contains point 𝑆, which point can also belong to the solution?

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Video Transcript

Consider the given slope field graph representing a differential equation. If the solution of the differential equation contains point 𝑆, which point can also belong to the solution?

Let’s trace solution curves through each of the points in the direction of the pattern of slopes mapped out through the line segments. Starting at the point in question, 𝑆, we can see that the solution curve could go through the point 𝐢. Although 𝐢 doesn’t lie exactly on the line segment, it is within the pattern of the slope field marked out by a solution through the point 𝑆. So point 𝐢 is a contender for the solution through 𝑆. Let’s also check the points 𝐴, 𝐡, 𝐷, and 𝐸. Following the pattern of the slope field through the point 𝐴, the solution does not coincide with the point 𝑆. So the point 𝐴 cannot belong to the same solution as 𝑆. Following the pattern through the point 𝐡, again this solution does not coincide with the point 𝑆. The same applies for point 𝐷 and also for point 𝐸, so that only point 𝐢 could belong to a solution containing the point 𝑆.

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