Exam 4 study questions and advice  Contemporary Mathematics Fall '09

Study questions:
Chapter 9:

1. What is the definition of the Fibonacci numbers?  (You'll need to give both a "starting place" and the rule for "getting to the next Fibonacci number.")
2. What are some situations or places in nature where Fibonacci numbers seem to arise?  (Including: Can you give the details on the "bunnies" situation?)
3. What is the golden ratio?  (This includes both its defining property, i.e. the proportionality it satisfies, as well as having a rough idea of its numeric value.)
4. Give some relationships between the Fibonacci numbers and the golden ratio:
1. something about ratios of consecutive Fibonacci numbers
2. something about integer powers of the golden ratio, i.e. ², ⁵, ⁷, etc.
3. estimating Fibnacci numbers from powers of the golden ratio
4. Binet's formula (how does this pertain to the golden ratio?)
5. What is a gnomon, precisely?  (the definition involves relationships between a geometric shape, an additional geometric shape, and the combination of the two.  Be specific about which role each shape plays.)
1. Be sure you can do the routine exercises like: "Given some diagram, how long would such-and-such a side need to be for one of the figures to be a gnomon to the other." (see the homework problems for examples).
   
6. What do gnomons have to do with the golden ratio?

1. What is a one-to-one correspondence between two sets?
   
2. How is counting an ordinary set of things (e.g. the students in our class) like making a one-to-one correspondence between the things (e.g. students) and a set of numbers? 
   
1. Give a diagrammatic example (see the cardinality handout's first figure for an example).
3. Two sets have the same cardinality if (what?)
1. Now if the sets are finite, there is a simpler way to say they have the same cardinality - a way that small children can easily understand - what is this "simpler way"?
2. However, as we've seen, if the two sets are infinite, things become trickier - can you give an example?  (Hint: see the last quiz or the Hotel Infinity story)
4. Say, what's our formal definition of a finite set, anyway? An infinite set?
   
5. As covered in class, Cantor made some startling discoveries about the cardinalities of infinite sets; perhaps you've already mentioned one of them above.
1. In particular, which of the following are countably infinite, and which one(s) are "bigger" than countably infinite?
1. The natural numbers 1,2,3,...
   
2. The integers:  ..., -3, -2, -1, 0, 1, 2, 3, ...
3. The positive fractions
4. The real numbers (every point on a number line)
2. Can you outline the reasoning behind the answers above (some are on the cardinality handout, all were covered in class.  The last two are hard, and Cantor is famous for figuring them out)

1. A percentage (e.g. 17%) means what, precisely?
2. When we talk about "17% of the price of a car", what precisely do we mean?  For example, perhaps someone claims that 17% of the price of a car is dealer profit.  Translate this into a statement that doesn't involve percentages.
3. Of course, be sure you can do the routine homework problems on percentages, for example:
   
1. a certain price goes down 30%, what's the new price?
2. The starting salary for some job goes up from (one given number) to (another given number), what's the percentage increase?
3. The number of unemployed people in some state was (some number) in 2007, it rose 5% in 2008, and 20% in 2009, what is it now?
4. ...and others like that; see the homework for specific examples
4. Simple interest problems (rare as they are in real life, they do occur) - make sure you can do them
5. If, in a compound interest problem, someone is paying 9% interest (per year) but it is compounded monthly, how do you figure the "interest rate per compounding period" and the "number of periods"?
6. How do you tell the difference between a basic compound interest problem and a "deferred annuity" (savings plan) or an installment loan?
7. Say, what's the reason why 6% interest for a year, compounded monthly, pays more than 6% simple interest for a year?
8. Describe how to compute the Effective Annual Yield or EAY or Annual Percentage Rate or APR for a certain interest rate and compounding frequency (as on the worksheet), e.g. "Give the APR for an interest rate of 9% compounded monthly."
   
9. In checking a "deferred annuity" (savings plan) problem, what's a quick way to check the final balance (hint: it has something to do with how much you actually paid into the plan).
10. Likewise, in checking an "installment loan" problem, what's a quick way to check if the amount of the monthly payment is correct?  How is this different than the previous answer?
11. For long-term loans (e.g. 30-year mortgages), describe how the amount owed changes over the early months/years of the loan (as seen on the worksheet).
    

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