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# When you turn on a hot-water faucet, the temperature $T$ of the water depends on how long the water has been running.$$\begin{array}{l}{\text { (a) Sketch a possible graph of } T \text { as a function of the time } t} \\ {\text { that has elapsed since the faucet was turned on. }} \\ {\text { (b) Describe how the rate of change of } T \text { with respect to } t} \\ {\text { varies as } t \text { increases. }} \\ {\text { (c) Sketch a graph of the derivative of } T .}\end{array}$$

## a.)$\\$b.)$$\text { Description of the graph: It is flat line } y=0 \text { with a couple of blips }$$ $\\$c.)$$\text { Adequate viewing rectangle: }[0,10] \times[0,35]$$

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so looking at the graph, we want to determine what it looks like when we have a faucet that increases in temperature. We turn onto hot water. Um, and what happens? Well, we know that it starts off is cold. Um, it always starts off as cold, and then it quickly becomes warmer and warmer until it's the hottest temperature possible. Um, eso a decent representation of this would be this graph right here again, these mad. These numbers don't matter as much. You could do five or 10 whatever. But what we see is that we start off with a cold value. Um, the cold temperature if this y axis is temperature, and then as time goes on along the X axis, we see it's getting very hot quickly, and then it starts to stabilize. It breaches its maximum temperature because when we turn on a faucet, there's not a there's no way it could just keep heating up forever. There's there is a maximum point. So what that tells us, um, is Then if we take the derivative, we see what it looks like. Um, and this derivative is showing us if we look from zero to infinity that we have high temperature. Ah, very high rate of change. Rather, this derivative graph shows a high rate of change, a positive rate of change. So a quick increase in temperature. And then as time goes on, it's slower and slower. Um, as it increases in temperature, um, to the point where it becomes a minuscule amount of change.

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