![]() I've drawn a picture, and now this picture does not have to necessarily be a good picture. Okay, so here I've done step one, or at least the first part of step point. So let me walk you through this example that we have here about the spherical balloon and try toe boil it down into doing these three steps. So you relate the variables, and then you use implicit differentiation to get a relationship between their rates of change. That's your relationship between the relevant variables and then the last step is to take the derivative. So maybe you just go look up on the Internet, you know, what's the volume of the sphere? Four thirds pi? A cubed. So can I write down with the volume of sphere is in terms of the radius. So maybe, you know, in this example it looks like we have volume and radius. So this normally just involves riding down in equation. The second step is to really key on the two quantities that you're trying to relate or, more specifically, the two quantities whose race of change with time you're trying to relate. So I'm sort of just taking the word problem and putting it in kind of schematic format in a picture and then just bare bones. So the quantities that air changing with time and I can also kind of state exactly what I know about those quantities and exactly what I'm looking for. And also in that picture I can label the relevant quantities. if I can portray it is a picture, then it's gonna help me just visualize what's going on. So instead of reading about what's happening in this picture, are sorry in this problem. To do that, especially for a lot of learning styles, is to draw a picture. So because this is a word problem, I would like to kind of extract the key Information is quickly as possible in one great way. Hopefully, the question is written well, so you can figure out what is asking. ![]() It may not be exactly clear what the question is asking. And one of the reasons why this section is a little bit difficult is because it involves word problems, and it involves problems where maybe the wording of things could be confusing. There's always variations to this, you know, sort of bullet point methodology of solving problems. So I tried to boil it down into three steps, and now this is not a catch All. ![]() So let's see the strategy for solving a related rates problem like the one we have here. Or, in other words, you want to relate their derivatives. You want to relate their rates of change. Problem is, you have two quantities that air changing with time. So this time, when we say rates of change of a quantity, we really mean the special case where the independent variable is time so two quantities that air changing with time, and the goal is just to relate those rates of change. So we're really zooming in on the case where we have two quantities that air changing with time. ![]() In a related rates problem, it's nice because the name of the problem actually tells you exactly what you're supposed to do to solve the problem. Yeah, it's like I said, this is called a related rates problem related rates. In this problem that I have written here, it's a word problem is an example of a related rates problem. And so we're gonna begin talking about types of problems called related rates problems. But we're ready to really dig into how derivatives can help us solve problems. We've basically gone over all the differentiation rules we want to know, probably more than we want to know how I would guess. We’ve labeled the length of each side of the square x.So we're ready to begin talking about applications of differentiation. ![]() Draw a picture of the physical situation. Are you wondering why that $\dfrac$ term.ġ. ![]()
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