Mooculus 7: Applications of differentiation
11 videos • 6,235 views • by Jim Fowler Last week we learned about derivatives of trigonometric functions and inverse trigonometric functions. So at this point, we can differentiate a ton of functions—with the chain rule from Week 5, we're can differentiate complicated compositions, too. What do we do with this new knowledge? This week, we cover two applications: l'Hôpital's rule and related rates. It is a bit surprising that derivatives can help us to evaluate limits. This is l'Hôpital's rule, and initially, it is only helpful for limits where the numerator and denominator are heading towards zero, but it is possible to transform quite a few limit problems so that l'Hôpital's rule applies. As a bit of a warning: sometimes students get so very excited about l'Hôpital's rule that they want to use it everywhere, but please, do not fall in love with l'Hôpital. The second topic this week is related rates. If you have an equation relating two quantities, and both those quantities are changing, you can explore the relationships between those rates of change. The basic technique to solving such a problem goes as follows. 1) Draw a picture, labeling the relevant quantities. 2) Write down an equation, using your diagram as a starting point. 3) Differentiate the equation, remembering to regard quantities as depending on a variable t, for time. 4) Evaluate the result to find whatever you wanted to know about. In practice, it gets tricky, because these are word problems so there are many opportunities for confusion! A classic question asks you to consider how fast a shadow's length changes as you walk. Another classic is studying how fast a ladder slides down a building. You can see these things in the real world by thinking about bowls filling with water or balloons filling with air. Often you will have to use certain geometric facts, like the volume of a cone in order to make progress.