Summary
| Subject keyword(s) | Mathematics, Mathematics -- Algebra, Mathematics -- Applied mathematics, Social studies, Social studies -- Economics |
|---|---|
| Grade level | High School, Higher Education, Vocational/Professional Development Education |
| Intended audience | Educator, Learner |
| Resource type | Instructional Material, Reference Material, Tool |
| Resource format | text, text/html |
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Sample Problems from Chapter 11 Chapter 11: Compound Interest to e and i The important thing about doing most math is that wereally want more than the answer, we want to generalize the answer, to find theanswer for any problem like we're workin on. Ian, at age 11, came to me with the problem- what would his Dad pay in monthlyinstallments on a house worth $10,000 at a 10% annual interest, with a 30 yearmortage- he really wanted a number for an answer. This got us working on thesimpler problem of investing and finding the interest. I must say that I don't know everything, and make a lotof mistakes, in spite of what my students might believe. So I wasn't ableto give Ian a quick answer, but he and I worked very hard to solve his problem.We started this way. 1. Simple interest. Find the amount you have in the bank after 2years if you put in $1, at a 6% annual rate of interest. (It will always be anannual rate of interest with all the problems below.) The Interest = Principal * rate * time I(2 yr) = $1 * .06 * 2 = $.12 and theamount (A) you would have after 2 years A = P + I = $1 + $.12 = $1.12 Notice that the interest is not added each yearto get a new principal. 2. Compound interest and 3, and 4. leading to a very important infinitesequence. Here the interest is added after each compounding period. Find the amount you have in the bank after 1 year, with a principal of $1, at@6%, compounded semi-annually or 2 times peryear. The interest earned during the first (1/2) year = I(first 1/2 year) =$1*.06*(1/2) = .06/2 ($.03) The Amount you have in the bank after the first (1/2) year A = P + I =(1 + .06/2) = $1.03 which will bethe new principal. I(2nd 1/2 year) = (1 + .06/2)* .06/2 A(after 2 (1/2)years)= P + I = (1 + .06/2) +(1 + .06/2)*.06/2 Factoring out the term (1 + .06/2), we get A(after 2 (1/2)years) = (1+ .06/2)2= $1.124 (more than the simple interest after 2 years). What would the amount you have in the bank after 1 year, putting in $1, at 6%,compounded quarterly (4 times per year)? What would the amount you have in the bank after 1 year, putting in $1, at 6%,compounded monthly (12 times per year)? What would the amount you have in the bank after 1 year, putting in $1, at 6%,compounded daily (365 times per year)? What would the amount you have in the bank after 1 year, putting in $1, at 6%,compounded 10,000 times per year? What would the amount you have in the bank after 1 year, putting in $1, at 6%,compounded continuously, (an infinite number of times per year)? New things, not in my books! See the answer pageto find out what Kirsten and I did with graphing and exploring (1+ .07/x)x 5. Naming exas an infinite series. Using the binomial expansion from chapter 9, the first 5 terms of (A + B)nare: 6. Patterns with i 7. Leading to the most exciting single mathematical statement, that containsthe 5 most important numbers in mathematics! This is notin my books, but happens regularly.When a student works on logs, as in Ch13., I'll have them find ln(-1). Looking at the graph at the right, you can seethat ln x is the inverse function of ex(they are mirror images of each other in the line y = x), but also there is noln of a negative number! Now I'll have the student find ln(-1) in Deriveand lo and behold we get ln x = i*Pi, which is nota real number but an imaginary number... and that shouldn't show up on our graphat the right. That means that ei*Pi= -1 by definition of logs. Adding 1 to each side we get ei*Pi+ 1 = 0. Here are the 5 mostimportant numbers in mathematics, in one statement!! When I firstencountered this statement in college, I was so excited about it, that I did awatercolor painting of a circular rainbow with this statement in the center! In my books I get this same statement using exas an infinite series, then ei*xas an infinite series, eventually in terms of cos(x) and sin(x), then put Pi infor x. 8. Using iteration to do compound interest. Not done here. 9. One result of Ian playing with powers of powers. Ian played with his calculator and powers of powers, during physics class. Hecame up with the following expression.If x goes to infinity what does this go to? Clickhere to see how Ian figured this out. 10. Graphing powers of i Continue the graph. What do you notice? Where is the point for i? 11. Graphing powers of ( 1 + i ) Notice we get another spiral! (See chapter 6 for other spirals). If you doublethe angle, what happens to the length of the radius vector? 11a. Graph of iterating in,starting with n = i Before Mathematica Inever would have seen this graph! Notice, another spiral! (Actually there are 4spirals here if you look at the endpoints of the segments). And iiis a real number! Amazing stuff here. One of the finestmath sites on the internet, which has interactive applets, is IESin Japan. One applet inspired by the graph above is atI^I. You can drag the point at 0+1i around thecomplex plane to get some amazing things!! 12. Back to Ian's problem. Ways young people have solved thesekinds of problems To orderDon's materials To choose sample problems from other chapters Mathman home