Science:Math Exam Resources/Courses/MATH102/December 2014/Question A 03

MATH102 December 2014

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[hide] Question A 03
Consider a differential equation $dy/dt=f(y)$. Shown in A-D is the phase line (state space) diagram ($f(y)$ versus $y$). Which of the following is the correct pairing of these sketches with the sketch of a solution $y(t)$ to the differential equation? Diagram A1, B4, C2, D3 A2, B4, C1, D3 A2, B4, C3, D1 A4, B2, C3, D1 A4, B2, C1, D3

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[show] Hint
If the derivative of a function is positive/negative in some interval or maximized/minimized at certain point, what doest it mean?

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[show] Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We will pair each graph of $y'$ with the graph of corresponding function $y$ by the sign and the magnitude of $y'$. First look at diagram (A). It shows the $y'$ is always positive in the domain of interest, which means function $y$ must be a increasing function. Hence we know the solution sketch corresponding to (A) might be (2) or (4). Next we notice $y'$ has a local minimum in the mediate $y$ value which means the increasing rate of $y$ achieves its lowest value when the value of $y$ is about in the middle, which indicates (2) is the correct choice. Similarly, the plot of $y'$ in (B) indicates $y$ is an increasing function and its increasing rate first increases before the middle point and then decreases again. The only correct pairing is (B)(4). Next, from $y'$ in (C) we know $y$ is a decreasing function. Its decreasing rate first increases as the magnitude (absolute value) of $y'$ reaches its maximum in the middle and then decreases. Only (1) matches it. Finally, from (D) we get $y$ is a decreasing function but its decreasing rate slows down in the middle, which matches (3). So the correct answer is (b).

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