Warm-Up Warm-Up In 1990, the population of Houston, In 1990, the population of Houston, TX was 1,637,859. In 1998, the TX was 1,637,859. In 1998, the population was 1,786,691. Assuming population was 1,786,691. Assuming the population increases by a the population increases by a certain percent each year, predict certain percent each year, predict the population of Houston in 2014. the population of Houston in 2014. WITHOUT WITHOUT your calculator, sketch the your calculator, sketch the graphs of the following: graphs of the following: 2 3 4 4 2 4 4 x x x x y y y y
Transcript
Warm-UpWarm-Up In 1990, the population of Houston, TX was In 1990, the population of Houston, TX was
1,637,859. In 1998, the population was 1,637,859. In 1998, the population was 1,786,691. Assuming the population 1,786,691. Assuming the population increases by a certain percent each year, increases by a certain percent each year, predict the population of Houston in 2014.predict the population of Houston in 2014.
WITHOUTWITHOUT your calculator, sketch the your calculator, sketch the graphs of the following:graphs of the following:
2
3
4 4 2
4 4
x x
x x
y y
y y
Homework:Homework:
Section 11-3Section 11-3: The Number : The Number ee
We will answer theseWe will answer these burningburning questions…questions…
Since when is Since when is ee a number? a number? Can I perform mathematical operations Can I perform mathematical operations
on an on an ee ? ? How on earth am I supposed to use the How on earth am I supposed to use the
A = final amountA = final amount P = principle or initial investmentP = principle or initial investment r = annual interest rater = annual interest rate N = number of compoundings per yearN = number of compoundings per year t = the number of yearst = the number of years
1nt
rA P
n
Figure this out with your group.Figure this out with your group.
Mariza has $1500 she wants to Mariza has $1500 she wants to invest for 5 years. She can get 6% invest for 5 years. She can get 6% interest and is offered 4 different interest and is offered 4 different plans: Compounded yearly, monthly, plans: Compounded yearly, monthly, weekly or daily. How much will each weekly or daily. How much will each plan yield at the end of 5 years? plan yield at the end of 5 years? Which plan should she take?Which plan should she take?
Continuous CompoundingContinuous Compounding
Where:Where: P is the initial amount or investment, P is the initial amount or investment,
A is the final amount, A is the final amount, r is the interest rate r is the interest rate and t is the time in years. and t is the time in years.
P rtA e
Redo Mariza’s problem with Redo Mariza’s problem with continuous compoundingcontinuous compounding
So, just what is So, just what is ee??
ee is called the “Natural Number”. is called the “Natural Number”. ee is an irrational number which was is an irrational number which was
found to occur regularly in nature, found to occur regularly in nature, much as much as pipi does.does.
It is found using the sum of an It is found using the sum of an infinite series (more on those later).infinite series (more on those later).
So, just what is So, just what is ee??
ee is called the “Natural Number”. is called the “Natural Number”. ee is an irrational number which was is an irrational number which was
found to occur regularly in nature, found to occur regularly in nature, much as much as pipi does. does.
It is found using the sum of an It is found using the sum of an infinite series (more on those later).infinite series (more on those later).
1 1 1 1 11 ...
1 12 12 3 12 3 4 12 3 ... 2.718
en
Let us calculate with Let us calculate with ee..
5 33
2 1
2
e e e
e e e
Let’s graph with Let’s graph with ee!!
More Uses of the Number More Uses of the Number ee Formulas in terms of Formulas in terms of ee::
Where:Where: N is the final amount; N is the final amount;
N N00 is the initial amount, is the initial amount, k is a constant k is a constant and t is time. and t is time.
0ktN N e
Does this formula sound familiar?
When is it growth and when is it decay?
Let’s do some Physics…Let’s do some Physics…
According to Newton, a beaker of liquid According to Newton, a beaker of liquid cools exponentially when removed from a cools exponentially when removed from a source of heat. Assume the initial source of heat. Assume the initial temperature temperature TT11 is 90º F and that is 90º F and that kk = = 0.275.0.275. a. Write a function to model the rate at which a. Write a function to model the rate at which
the liquid cools.the liquid cools. b. Find the temperature b. Find the temperature TT of the liquid after 4 of the liquid after 4
minutes (minutes (t t ).). c. Graph the function on your calculator and c. Graph the function on your calculator and
use the graph to verify your answer in part b.use the graph to verify your answer in part b.
Let’s Partner Up!Let’s Partner Up!
You will work with a partner to You will work with a partner to complete Practice 11-3complete Practice 11-3
Take turns. One partner solves and Take turns. One partner solves and the other praises/coaches.the other praises/coaches.
The Number The Number ee
Page 714Page 714#1 – 9 all, 11(a and b), 13 #1 – 9 all, 11(a and b), 13 and 17and 17
Quiz!Quiz! Friday 11-1 to 11-4 Friday 11-1 to 11-4
