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University of Notre Dame

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Problem 35

The temperature at a point $(x, y)$ is $T(x, y)$, measured in degrees Celsius. A bug crawls so that its position after $t$ seconds is given by $x = \sqrt{1 + t}$, $y = 2 + \frac{1}{3}t$, where $x$ and $y$ are measured in centimeters. The temperature function satisfies $T_x(2, 3) = 4$ and $T_y(2, 3) = 3$. How fast is the temperature rising on the bug's path after 3 seconds?

$$2^{\circ} \mathrm{C} / \mathrm{s}$$

## Discussion

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## Video Transcript

Hi there. In this problem, we have a temperature function T which depends on X and y. So it's right down what we know we have X. The X coordinate of the bugs path is given by squared of one plus t, and the Y coordinate is given by two plus 1/3 time's teeth. Okay, were also given partial derivatives. Perfect t respect to X at our 0.23 is four impartial with respect of why at the same point is three. How fast is the temperature rising on the path after three seconds. So think about what's going on. We care about the temperature function, Um, and what we want? Well, we have t depending on two variables X and Y and e x and y both depend on little which is time, right? So what we want in the end, we want the big T by deed little t hopefully as quickly as his temperature changing with respect to time And Jane Rule tells us, uh, in order to do that, you will have to evaluate Uh huh. Partial of tea. Respect to x and then time t x t t. That's following this tree down here, plus partial of TV respect. Why times D Y E T. It's important, of course, that time is three now, since they only gave us 1.2 comma three were pretty much expecting that when we plugged three and for tea. That's what we'll get for X and Y, but it's worth worth checking. But if you look up here and one plus three is four square to four is indeed two and over for why two plus 1/3 of three is three. So we have exactly the point that we need for this value of time. Okay, so this will be quick. Marshal of Tea by X is given. That's But they told us that before Partial of X with respect to t well, we'll have to look a TTE our definition of X right here. So it's one plus t to the 1/2. So the derivative of that is 1/2 one Krusty to the native 1/2 we should do the chain rule. But in this case, the inner derivative of a derivative of one plus tea is just one, uh, that plus partial t by why again is given. That's three and D Y T T. This is a little easier. We can just glance at why here and the derivative of that with respect to tease just 1/3. Okay, so we have four times 1/2 is to We can think of that negative 1/2 power as a square root on bottom and one plus t. Well, at this point, T is three. That ends up being four down there three times on third is one. So we get to over the score to four to over two is one close one equals two. That's your final answer. Let's just get the unit's right. We're measuring temperature, which in this case is measured in degrees Celsius and it's a rate. So let's look at our unit of time, which is seconds, two degrees Celsius for a second. And that's your final answer. Hopefully, that help