Assignments, MAT227

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.):

1. 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?
2. Why limits? Isn't it enough to use 10 rectangles to approximate the area under a curve?
3. 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?
4. What is the difference between the three rectangular methods? What advantages and disadvantages can you see to each?
5. What's so special about rectangles? Could we have used other geometrical objects to approximate areas under a curve?
6. 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):

1. How is a definite integral related to area?
2. What is a Riemann sum? What's a partition?
3. What happens to the value of the integral when you change the order of the limits of integration? Why does this make sense?
4. How are the sample points determined for the three major rules (midpoint, left endpoint, right endpoint)?
5. Do the ``intermediate points'' have to be equally spaced?
6. 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?

1. Suppose you use one antiderivative for f and I use a completely different one: will we get the same result for the integral?
2. 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?

• 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.

Page 325: 4, 6, 11, 14, 18, 20, 24, 27, 36, 38 (due Monday, 10/11)

pp. 364--: 7, 11, 15, 19, 21, 27, 47, 57, 85 Additional questions:

1. How do you know, by inspecting a graph, that it's the graph of an invertible function?
2. Given the graph of an invertible function, how do you find the graph of the inverse function?
3. Carefully draw a diagram of a growing exponential and dying exponential pair; then draw in their inverses.