We will cover the following topics in this class:
We will cover some more ideas from chapter 2 - higher derivatives, and more examples of the chain rule. Then we'll go on to applications of derivatives, from chapter 3. We will be covering 3.1 and 3.5 of the text book.
In chapter 3, we see applications of derivatives, so this means we will be spending time on working out how to convert a problem given in words into a mathematics problem.
If you pull down at speed 30 m/s, for the above situation with the pulley and the weight, at what speed does the weight rise?
Answer
The proceedure for this kind of question is very similar to that for max/min problems, which is outlined on page 185 of the text book.
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In this example, we're interested in two speeds. Speed of the wieght, speed of the end being pulled. How can you express speeds as derivatives? A speed is a rate of change of a distance, so you need to find some appropriate distances, and lable the diagram accordingly:
In the diagram, dX/dt is the speed of the weight, and dY/dt is the speed you pull at. In the end, we want to find how these two speeds are related; that means first finding out how X and Y are related.
How are X, Y, and L related? This is shown in the following diagram:
we have
Differentiate this, and we get:
But the length of the rope is fixed, so L is not changing, so the rate of change of L is zero, so dL/dt = 0, so:
We are told that dY/dt = 30, so this means
So this means:
| the weight moves up at 6m/s. |
(Note, the minus sign just means that the weight is moving in the opposite direction from thhe direction you pull in.)
The following questions from the text book should be done for practice, but are not to be handed in for marking.
The following questions should be handed in Wednesday 4 March:
See the hints page for some hints. Let me know if you need any more hints!
Please send corrections to spelling mistakes, and any other comments, to me.
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