AM6.1c To Define and Solve with Exponents and Euler’s Number 11-29-17 & 11-30-17 AM6.1c To Define and Solve with Exponents and Euler’s Number

Opener: READ ALL THE WAY FIRST! Make sure you have a calculator. To find ex on your calculator, look for the ln or LN button. It is the second function (TI-30: 2nd ln, you will see e^). It has an “automatic” exponent. That is, once you type it, it knows that the next number you put in is the exponent for e. Turn to page 188, work Class Exercise Problems 1-5, to four decimal places.

Answers: 1. 7.3891 2. 24.5325 3. 0.0183 4. 4.1133 2.7183 (*** value for e ***) We will come back to e later.

Active Learning Assignment?

Ex. ln x = 1.7 Ex. e x = 24 Try. ln x = 5.8 Try. e x = 100 LESSON (I) Solve (1 dec. pl.): Ex. ln x = 1.7 Ex. e x = 24 Try. ln x = 5.8 Try. e x = 100

LESSON (II) (Don’t copy this page, just watch): If I invest $100.00 at 8% for a year at simple interest, I will receive $108.00 at the end of the year. But… If I invest $100.00 at 8% for a year compounded quarterly, you get a quarter of the interest, every quarter. 1st quarter: $100 * .02 = $2.00 Net: $102.00 2nd quarter: $102.00 * .02 = 2.04 Net: $104.04 3rd quarter: $104.04 * .02 = $2.08 Net: $106.12 4th quarter: $106.12 * .02 = 2.12 Net: $108.24 The difference? 24 ¢! However, if we expand that original amount to $1,000,000,000, that changes the game!

Formula for interest compounded “n” times a year: * A = final amount. (Also A t ) P = principle. (Also A 0 ) r = rate in decimal form. n = number of compoundings a year. t = time in years. Remember growth? A t = A 0 (1 + r)t 1. If I have $1000, compounded quarterly, at 6%, how much will I have in 5 years? 1000 0.06 4 5 A = final amount P = r = n = t = A = 1000 * (1 + .06/4) ^ (4 * 5) I will have $1346.86 in 5 years.

2. TRY: If I have $57,890. 00, compounded monthly, at 6 2. TRY: If I have $57,890.00, compounded monthly, at 6.2%, how much will I have in 10 years? 57890 .062 12 10 A = final amount P = r = n = t = A = 57890 * (1 + .062/12) ^ (12 * 10) I will have $107,441.71 in 10 years.

ln or LN is the natural logarithm and e is it’s inverse ln or LN is the natural logarithm and e is it’s inverse. They reflect natural growth and decay. We saw that the value for e is 2.7183 and it’s called Euler’s Number (pronounced like oiler’s). It is a transcendental number like pi, it is irrational, and it goes on forever without an exact repeating pattern. The formula for e is: (n is the number of times that you evaluate it. This finds the nth term.) Which comes from:

* Formula for compounding continuously: 3. If I have $1000, compounded continuously, at 6%, how much will I have in 5 years? A = final amount P = 1000 r = .06 t = 5 A = 1000 * e ^ (.06 * 5) I will have $1349.86 in 5 years. Last time it was $1346.86. What’s the difference between the two? $3 How can that possibly matter?

That’s the beauty of math. NCIS-LA: Sam and Callen are having a conversation. Sam is going undercover as an analyst in the world of high finance. (This is an actual transcript of their conversation!) Good luck! Don’t need it. Why not? That’s the beauty of math.

(Notice the .5 is for the half-life rate! Can you see a shortcut?) Formula for radioactive decay: * (Notice the .5 is for the half-life rate! Can you see a shortcut?) 4. If I have 100 g of a radioactive element whose half life is 230 years, how much will I have in 785 years? (1 dec. pl.) A 785 = final amount A 0 = 100 t = 785 h = 230 A 785 = 100 * .5 ^ (785/230) I will have 9.4 grams in 785 years. How does Carbon 14 dating work?

Active Learning Assignment: Now, it’s funny! Active Learning Assignment: P. 196: 43abc & 44abc & Handout