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KS
Numerade Educator

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Problem 67 Medium Difficulty

The president announces that the national deficit is increasing, but at a decreasing rate. Interpret this statement in terms of a function and its first and second derivatives.

Answer

If $D(t)$ is the size of the national deficit as a function of time $t,$ then at the time of the speech $D^{\prime}(t)>0$ (since the deficit is
increasing $),$ and $D^{\prime \prime}(t)<0$ (since the rate of increase of the deficit is decreasing).

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Video Transcript

the president announces that the deficit is increasing. But at a decreasing rate, we want to evaluate this function as well as the functions 1st and 2nd derivative. So to start, well value the function this is this is a deficit with respect to time. So we know it's increasing but we know that the rate that's increasing is decreasing. So as time passes at the the deepness of this local decrease now with respect to the first derivative, this regards the rate of change of this of the function. So we can see here that we start with a very steep slope and it decreases and decreases decreases more. So I'm going to drive here. We start with steepest steep slope and the function is a high distributive. So we'll start high and then it's decreasing over time. Now one thing we don't know is whether or not the rate that the change of deficit, the rate that the deficit changes decreasing, we don't know if that's linear or nonlinear. So although the deficits increasing, you know, the change indefinite decreasing and this could be something like this, In which case the 2nd derivative would be constant. So the rate of change will continue to decrease, but it will continue to decrease at the same rate. Or it could be something like this where the slope will decrease but it was stuck, but if longer it goes on, the faster it will start decreasing. In which case the second derivative would also be decreasing. The second derivative is the slope of this. Well this would be zero, but these would be negative so the second derivative would be decreasing. That is all. I hope it was helpful. Thank you.