Warm-Up In 1990, the population of Houston, TX was 1,637,859. In 1998, the population was 1,786,691. Assuming the population increases by a certain percent each year, predict the population of Houston in In 1990, the population of Houston, TX was 1,637,859. In 1998, the population was 1,786,691. Assuming the population increases by a certain percent each year, predict the population of Houston in WITHOUT your calculator, sketch the graphs of the following: WITHOUT your calculator, sketch the graphs of the following:

Homework:

Section 11-3: The Number e We will answer these burning questions… Since when is e a number? Since when is e a number? Can I perform mathematical operations on an e ? Can I perform mathematical operations on an e ? How on earth am I supposed to use the number e in real life? How on earth am I supposed to use the number e in real life?

Remember Compound Interest from Section 11-2? A = final amount A = final amount P = principle or initial investment P = principle or initial investment r = annual interest rate r = annual interest rate N = number of compoundings per year N = number of compoundings per year t = the number of years t = the number of years

Figure this out with your group. Mariza has $1500 she wants to invest for 5 years. She can get 6% interest and is offered 4 different plans: Compounded yearly, monthly, weekly or daily. How much will each plan yield at the end of 5 years? Which plan should she take? Mariza has $1500 she wants to invest for 5 years. She can get 6% interest and is offered 4 different plans: Compounded yearly, monthly, weekly or daily. How much will each plan yield at the end of 5 years? Which plan should she take?

Continuous Compounding Where: Where: P is the initial amount or investment, A is the final amount, r is the interest rate and t is the time in years. P is the initial amount or investment, A is the final amount, r is the interest rate and t is the time in years.

Redo Mariza’s problem with continuous compounding

So, just what is e? e is called the “Natural Number”. e is called the “Natural Number”. e is an irrational number which was found to occur regularly in nature, much as pi does. e is an irrational number which was found to occur regularly in nature, much as pi does. It is found using the sum of an infinite series (more on those later). It is found using the sum of an infinite series (more on those later).

So, just what is e? e is called the “Natural Number”. e is called the “Natural Number”. e is an irrational number which was found to occur regularly in nature, much as pi does. e is an irrational number which was found to occur regularly in nature, much as pi does. It is found using the sum of an infinite series (more on those later). It is found using the sum of an infinite series (more on those later).

Let us calculate with e.

Let’s graph with e!

More Uses of the Number e Formulas in terms of e: Formulas in terms of e: Where: Where: N is the final amount; N 0 is the initial amount, k is a constant and t is time. N is the final amount; N 0 is the initial amount, k is a constant and t is time. Does this formula sound familiar? When is it growth and when is it decay?

Let’s do some Physics… According to Newton, a beaker of liquid cools exponentially when removed from a source of heat. Assume the initial temperature T 1 is 90º F and that k = According to Newton, a beaker of liquid cools exponentially when removed from a source of heat. Assume the initial temperature T 1 is 90º F and that k = a. Write a function to model the rate at which the liquid cools. a. Write a function to model the rate at which the liquid cools. b. Find the temperature T of the liquid after 4 minutes (t ). b. Find the temperature T of the liquid after 4 minutes (t ). c. Graph the function on your calculator and use the graph to verify your answer in part b. c. Graph the function on your calculator and use the graph to verify your answer in part b.

Let’s Partner Up! You will work with a partner to complete Practice 11-3 You will work with a partner to complete Practice 11-3 Take turns. One partner solves and the other praises/coaches. Take turns. One partner solves and the other praises/coaches.

The Number e Page 714 Page 714 #1 – 9 all, 11(a and b), 13 and 17 #1 – 9 all, 11(a and b), 13 and 17 Quiz! Friday 11-1 to 11-4 Quiz! Friday 11-1 to 11-4