Materials: Hw #9-6 answers handout; Do Now and answers overhead; Special note-taking template; Pair Work and answers overhead; hw #9-7

Transcription

1 Pre-AP Algebra 2 Unit 9 - Lesson 7 Compound Interest and the Number e Objectives: Students will be able to calculate compounded and continuously compounded interest. Students know that e is an irrational number arrived at by the limit x, (1 + 1/x) x use the natural logarithm in the same way as other log bases. Students will be able to model exponential situations and solve exponential models for both independent and dependent variables. Materials: Hw #9-6 answers handout; Do Now and answers overhead; Special note-taking template; Pair Work and answers overhead; hw #9-7 Time Activity 5 min Homework Check 15 min Do Now Students work on modeling exponential situations. Review answers on overhead. 40 min Direct Instruction Put the compound interest scaffolding transparency on the overhead. The interest rate is 8% and the principal is $100. How much money will there be after 1 year? Label the first period yearly, fill in 8% on the bar, and the yield is $108. To compound interest means to calculate interest on top of the accumulated interest. Show how this works with twice a year compounding. Label the second period semi-annually. Cut the bar into two pieces, labeling each with 4%. Under the first piece, write 100(1.04) = 104 and under the second piece 104(1.04) = Write this under the yield. Note that this method earns $0.16 more than just doing it once. Why does this happen? Because you get interest during the second half of the year on both the principal and the $4 of interest that was already earned. Continue on with quarterly, monthly, and daily compounding periods. Note that the yield is increasing each time, but not by much. It seems to flatten out the next part will show why. Move on to the back page and have students complete the table using their calculators. Note that we are approaching a specific number: , which is an irrational number like, and is given the name e. We can work with e just like any other number. Concept: You have already met the Common Log, which is the other name for log 10. It s notation is just log. There is another special log called the Natural Log, which is the other name for log e. That s right, the irrational number e that you learned in the last class (e ) is the base of this logarithm! Its notation is ln. The reason it is backward, and not written nl, is because it comes from the French: logarithme naturel. In summary: log x is pronounced log of x and means log 10 x. ln x is pronounced L.N. of x and means log e x. You use ln exactly the same way as any other logarithm. If you get confused, you can always rewrite ln x as log e x. For example, ln e 4 = log e e 4 = min Pair Work Students work on the pair work handout, copied to the back of the Do Now. Review answers on overhead if there is time. Homework #9-7: The Number e and ln

2 Pre-AP Algebra Do Now Name: Warm-Up: Exponential Models You put $2000 into a savings account that grows by 8% each year. a) Write an exponential function that models this situation. b) How much money will you have after 10 years? c) How long will it take for you to have $10,000 in the account?

3 Pre-AP Algebra Pair Work Name: Working with e 1. You deposit $975 in an account that pays 5.5% annual interest compounded continuously. What is the balance after 6 years? 2. The number e is a number like any other, and we work with it in the same way. Keep that in mind, and the next few problems should make sense! Simplify: a. e 3 e 5 b. 3e 3x 2 c. 3 27e 6x Evaluate (using a calculator): a. e 2 3 b. e 3.2 c. 0.02e Determine whether the function is an example of exponential growth or decay. a. f (x) 5e 3x c. f (x) 1 4 e x b. f (x) 1 8 e5x d. f (x) 5 3 e7x 4. The air pressure P (measured in lb/in 2, pronounced pounds per square inch, or p.s.i. for short) at sea level is about 14.7 lb/in 2. As the altitude h (in feet above sea level) increases, the air pressure decreases. This relationship can be modeled by P(h) = 14.7e h. Mount Everest rises to a height of 29,028 feet above sea level. What is the air pressure at the peak? 5. The area of a wound decreases exponentially with time. The area A of a wound after t days can be modeled by A = A 0 e -0.05t, where A 0 is the initial area of the wound. For a wound that starts off at 4 square centimeters, about how much of the wound is left after 2 weeks?

4 Pre-AP Algebra 2 Homework #9-7 Name: Homework #9-7: The Number e 1. You deposit $2500 into a savings account that pays 6% ANNUAL interest. Use the formulas in your notes to calculate the account balance after one year, if the interest is compounded a. annually (i.e. once a year) b. semiannually (i.e. twice a year) c. quarterly d. monthly e. daily f. continuously What do you notice about these results? 2. Simplify: 1 a. 4 e2 3 b. 6e3x 3 c. 64e15x 2 4e 3. Solve for x: a. e 2x1 e 8 b. e3 e 5x 1 e 2 c. e4 x e x2 e Expand each logarithm using the correct properties. Convert all roots and exponents into coefficients. a. ln e5 4 b. ln 5 2 x 5. Condense each logarithm into a single expression. Remember to convert coefficients first, and use the order of operations. a. ln3 1 2 ln x b. log x 3 2 log 2 x 5 c. 2ln x lnx 5 ln(x 1)

5 6. The number of watts w provided by a space satellite s power supply after a period of d days is given by the function w(d) = 50e d. a. Is this a growth or decay function? How can you tell? Does this make sense in terms of the problem? b. How much power will be available after 30 days? c. How much power will be available after 1 year? 7. A model for the number of people N in a high school community who have heard a certain rumor is N = P(1 e -0.15d ), where P is the total population of the school and d is the number of days that have elapsed since the rumor began. Assuming that a DHS has 2,000 students, and I start a mean rumor today, how many students will have heard my rumor after 3 days?

6 8. Currently, there are 1000 people that live on Mystery Island. Each month, a boat brings in 150 more. a. Write a function P(t), where P is the population and t is the number of months, that models this situation. b. How many people will be on the island in 2 years? c. How long will it take for there to be 10,000 people on the island? 9. The air pressure P (measured in lb/in 2, pronounced pounds per square inch, or p.s.i. for short) at sea level is about 14.7 lb/in 2. As the altitude h (in feet above sea level) increases, the air pressure decreases. This relationship can be modeled by P(h) = 14.7e h. d. Mount Everest rises to a height of 29,028 feet above sea level. What is the air pressure at the peak? e. How high would a mountain have to be for the air pressure to be half the pressure at sea level? 10. You put $2500 into a savings account that has a 5.4% annual interest rate, compounded continuously. f. Write a function A(t) that models this situation, where A is the amount in dollars and t is the time in years. g. How much money will you have after 4½ years? h. You want to buy a car that costs $8500. How long will you have to wait before you can afford it?

7 HW 9-6 Tally Sheet 1) 2) 3) 4) 5) 6) 7) 8) 9) Chart Part 2: Exponential Models 1) a. b. Graph c.i) ii) 2) a) b) c) 3) a) a. b. c. b) a. b. c. c) a. b. c. d. e. f.

8 HW 9-6 Answer Sheet 1) ) ) 400 4) 1.8 9) 5) $ ) $ ) 17.9% decrease 8) 8.3% increase Hour # Bacteria Part 2: Exponential Models 1) a. 18, 55, 166, 504, 1526, 4619 b. c. x = ) a) initial value = $500, growth, 3.5% b) initial value = 200 mg, decay, 35% c) initial value = km 3, decay, 24% 3) a) t a. C ( t) 35(.77) b. 12.3mg c hours b) t a. A ( t) 100(1.75) b months c. 1.9 billion c) t a. P ( t) 42(1.12) b. 104 c yrs d. 9.7 yr e yr f. 34 yr

9 Lesson Name: Compounded Interest and the Number e Date: Student: Concepts Examples Compounded Interest Yearly Interest: Principal Amount: Period year Yield P = n = r = t = A(t) = Compounded Interest Formula: Continuously Compounded Interest Formula:

10 Concepts Examples The number e Use your calculator to fill in the table. Round decimals off after 7 places. n 1 1 n n

11 Concepts Examples The Natural Log What is ln? Use the properties of logarithms to condense the following expressions 1) 5ln x ln4 2) ln(x 2) ln(x 2) 3) x ln5 5ln x 2 3 ln27 Use the properties of logarithms to expand the following expressions What is 4 ln e? Summary of the Logarithm Properties: Multiplication : log b AC log b A log b C A Division : log b C log A log C b b Power : log b A C C log b A 4) ln(7x 2 ) 5) ln(7x) 2 6) ln 4e5 3 7) A local bank, in response to a rival s successful promotion, is offering a Super-Duper Fantastic Savings account with a continuously compounded annual interest rate of 4.5%. This offer can be modeled by a function where P 0 is the initial investment in dollars, r is the annual interest rate, t is the time in years, and v(t) is the total value of the account. Find the function that models the growth. Find the number of years it will take you to triple your initial investment.