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

Consider a differential equation . Shown in AD is the phase line (state space) diagram ( versus ). Which of the following is the correct pairing of these sketches with the sketch of a solution to the differential equation?

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
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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Solution 

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