Reflect on the strategies we discuss in class on Monday (start your notebook!); consider your own strategies; then look over the strategies the book offers up. Can you see how to do some of the problems? Which ones have you stumped?
Turn in the following, due Wednesday, 9/1: Page 258: 2, 3, 11, 13, 22. Also answer the following (write a short paragraph for each -- feel free to include figures, doodles, etc.):
- Ever hear of the "dirt" formula? d=rt -- distance equals rate time time. What formula does our author prefer to this one, and does it offer any advantages?
- Why limits? Isn't it enough to use 10 rectangles to approximate the area under a curve?
- All of the methods for computing areas in this section involve chopping the x-axis into equally sized chunks -- any reason they have to be equally sized?
- What is the difference between the three rectangular methods? What advantages and disadvantages can you see to each?
- What's so special about rectangles? Could we have used other geometrical objects to approximate areas under a curve?
- Explain sigma notation to a friend (a classmate is fine), using an example. How did it go?
Due Wednesday, 9/8: pp 270--: 1, 4, 8, 12, 14, 32, 44, 46, 48, 52, 70 Also answer the following (usual conditions apply -- see above):
- How is a definite integral related to area?
- What is a Riemann sum? What's a partition?
- What happens to the value of the integral when you change the order of the limits of integration? Why does this make sense?
- How are the sample points determined for the three major rules (midpoint, left endpoint, right endpoint)?
- Do the ``intermediate points'' have to be equally spaced?
- There's a lot going on in this section (check out the summary on p. 269). Pay special attention to all those properties of integrals: which ones are natural, and role off your tongue? Which ones seem strange?
Page 276: 2, 10, 22, 25, 26, 29, 32,36, 44, 49
- Suppose you use one antiderivative for f and I use a completely different one: will we get the same result for the integral?
- Like the "dummy" index in sigma notation, there is a "dummy" variable in integrals. Where do we find it, and how can we interpret it?
Page 282: 4, 14, 18, 22, 24, 20, 28, 30, 32, 39. Other questions to answer:
- Why are differentiation and integration not perfect inverse operations?
What analogy can you draw with the functions
and
?
- Observe the discussion of the chain rule in Example 4. The generalization is in the section summary, p. 281. Give an example of your own.
- There are functions -- very important functions -- which don't have elementary anti-derivatives, so we can use the integral to understand them. Give an example.