Chapter 9 Review

Alright so here's a quick review and some sample problems from chapter 9. The test is on Thursday June 2nd. There is a calculator and non calculator part so remember your green cards!

9.1 Exponential Growth

The exponential curve is different than a parabala or hyperbola. It is also an asymptote.

Some formulas associated with exponential growth are as follows:
Geometric Sequence-
Compound Interest-
Exponential Function- y=ab^x

9.2 Exponential Decay. This is very similar to exponential growth, but it depreciates. The growth factor is between 0 and 1, so there is a decrease.


9.3 Continuous Compounding. This is tricky, so watch out on the test for the wourd CONTINUOUSLY!!!! This is differnt than y=ab^x, and it is guaranteedthat some of will goof this up on the test. The formula for this is known as Pert. A = Pe^rt
P is the initial condition, e is on your calculator, r is the rate and t is time.

$7808 is invested at 11.9% and compounded continuously for 17 years.
7808e^1.0119x17= $59036.08

9.4 Fitting Exponential Models to Data
For this your calculator is essential. In order to get into a lists and spreadsheets, hit home, 1, 4, and input date into the spreadsheet. Once your data is in hit menu, 4, 1, A, in order for it to analyze the exponential regression.
9.5 Common Logarithims

Exponential form 1o^7=10000000 Log Form log10 10000000=7
The answer is always the power, so they way to think of it is 10 to the x power equals 10000000.

9.7 Logorithims to bases other than 10

It is the same concept, think about the exponent as the answer. 3^4=81. In log form, that looks like log3 81=4. Although the base isn't 10 don't let it throw you off, it is the exact same thinking and way yo put it into log form.

9.8 natural logarithms
n is the natural logarithm of m, written n=ln m, if and only if m=e^n.
e is also the inverse of ln.

9.9 Properties of Logarithims
x^0=1
x^m x x^n= x^m+n
x^m/x^n=x^m-n
(x^m)^n=x^mn

9.10 using logarithms to solve exponential equations
In this lesson, you have to solve b^x=a. There are a couple ways to do it. Of course you can use CAS but you can use logarithms to solve the equation.

ln2=ln9e^12r)
ln2=12r
r=ln2/12
.0578

9.5

Today in class, we learned about logarithms. We first went over how to find the "log" button on a calculator. You press CTRL 10^x. We then started on the 9.5 notes and sketched a graph of the exponential function y=10^x. From there we found the inverse by switching x and y. This is where the new function, logarithms, comes in. We found that the inverse of y=10^x is x=10^y, or the log formula. After graphing both we compared and contrasted the two equations.

y=10^x y=log x or x=10^y

Domain: all reals x>0

Range: y>0 all reals

x-int.: none (1,0)

y-int.: (0,1) none

Asymptote: x-axis y-axis

After, we did some examples without our calculators. The first problem we looked at was log 10^100. In order to figure it out, we set in up as 10^? = 100. From there it was easy to figure out that the answer was two. We then moved on to decimals. The next problem was log 10^0.1. In order to figure it out we changed it to 10^? = 1/10. We then figured out that our answer was -1.

Some more examples are:

log 10^ 1,000 = 3 log 10^ 10 = 1 log 10^ 1000000 = 6

log 10^ 5 = .669 log 10^ 30 = 1.477

It is important to notice that log 10^ (-100) does not work. This is because you can't raise 10 to a value and get a negative number. Another important thing to notice is that the base does not have to be 10. A problem can be log 9^ 81.

The next set of problems that we did were very similar to the previous examples. However, they did not included a base number. When logs do not have a base number, it can be assumed that 10 is the number. "Log x is called the common logarithm, so log 10^x is the same as log x."

Examples:

log 10000= 4 -This is because you assume the equation is written as log 10^ 10000.

log (o.oo1) = 3 log 10 = 1 log 7 = .8451 log 316 = 2.4997

We were then introduced to another set of equations. Log x = 4. It looked difficult, however, using the two methods we previously learned, it was pretty simple. First we put in 10 as the base because we learned that you can assume a log without a base is 10. Then we changed the equation around to read 10^4 = ?. This equation resembled the ones we had done previously. We then used our green cards and found that x = 10,000.

The last lesson of the day was the conceptual question., "Between what two integers is log 7598?" To solve this we asked ourselves 10^ ? = 7598. We then plugged in number for 10^ ? and found minimum and maximum values of 3 and 4. These two numbers worked because 10^ 3 = 1,000 and 10^ 4 = 10,000. We knew that 3 and 4 were the two integers because 7598 fit in the middle of them.

Examples:

log 172: log .4:

10^ ? = 172 10^ ? = .4

10^ 2 = 100 10^ 0 = 1

10^ 3 = 1,000 10^ -1 = .1

Answer is 2 and 3 because 172 fits in Answer is 0 and -1 because .4 fits in

between 10^2 and 10^3. between 10^0 and 10^-1.

If you still need help the 9.5 notes are located on moodle. Also the homework for tonight is 905 - 1, 3-11, 14-16, 18, 21-23.

Chap 9 Section 3: Continuous Compounding

Hello class. At the beginning of the day, Mr. Cope gave a 5 minute review on sections 9.1 and 9.2. He simply taught exponential growth (9.1) and exponential decay (9.2). He taught the equation y=ab^x. A being the number you begin with, b being the growth factor (growth b>1, decay 0<b<1) and the x being the number of years being used. This was just a quick review of what was taught on Monday and Tuesday.

Next Mr. Cope handed back the tests and homework quizzes we took last week. Unfortunately, Teacher Logic no longer shows the class average so there will be no comparison to period 1 in this blog post. Anyway, we were able to compare our answers to Mr. Copes key and then we moved on to today's lesson.

Today we learned about continuous compounding. We began with a chart showing a bank pays 100% interest for 1 year. So we went through annually, quarterly, monthly, daily, and hourly using the equation P(1+r/n)^nt. The final value (hourly) was very close to e (2.71828) or Euters number which is used to represent exponential growth. So the formal equation for Continuously Compounded Interest is a=Pe^rt. P stands for the original number, e stands for Euters number, r stands for either the growth of decay (growth=if number is positive, decay if number is negative), and t stands for the number of years. We then did an example 1 which was "Suppose $1800 is invested at an annual interest rate of 7% compounded continuously and the money is what happens in 5 years. So the equation would be 1800e^.07(5). You then would solve it on the calculator and your answer should be 2554.32.

Moodle notes: http://gbs-moodle.glenbrook225.org/moodle/mod/resource/view.php?id=13797

Woah!!!!!!

So I saw this pic whilst browsing the internet and I was like, BAM! MR. COPE!

Multiplying Polynomials

Hello class unfortunately I forgot to this blog the night i was assigned to but here is what we learned in section 2 of chapter 11.

We first started out by classifying polynomials by the number or terms

• Monomial- polynomial with one term (ex. 10 or 14x)
• Binomial- polynomial with two terms (ex. x+2 or 4x^4-7x)
• Trinomial- polynomial with three terms (ex. x+2+3x^2)

After getting a little off task with the usual 10-15 minute discussion we began examples on the back page of the notes.

(-y+6)(y^4+2y^2-3) first you must foil it by hand which we learned in some older chapters. when you foil that you get -y^6-2y^4+3y^2+6y^4+12y^2-18... then after you reduce that all down you end with -y^6+4y^4+15y^2-18.

But if you would rather not go through all that fun you can simply type it in to your calculator and get the answer fully reduced and without straining yourself in the slightest.

Hello class, yesterday we learned about factoring. There are 3 special cases of factoring

• Greatest common factor
• Difference of Perfect Squares
• Perfect Square Trinomials

Heres an example of a problem in which we need to solve using the Greatest Common Factor

5wz+25w^2(z)-35w^3(z)

First we need to find the GCF. All of the numbers are multiples of 5 so 5 would be a GCF so it would come outside the parentheses.

5(__________)

We can also see that every one of them has at least one w in it so that would also come outside the parentheses.

5w(_________)

We can also see that there is a z in every number so that too would come out of the parentheses and now you divide everything inside the parentheses by 5wz and your product should look like this

5wz(1+5w-7w^2)

Now heres an example problem with Difference of Powers Squared

x^2-1oo

the form for solving this kind of problem is a^2-b^2=(a+b)(a-b)

so we need to think. what times what equals x^2? well that would just be x

(x+__)(x-__)

Now what times what equals 100? well that would just be 10.

(x+10)(x-10)

and do not worry about the 10x and -10x because they cancel each other out.

And here is an example of a problem with perfect Square Trinomials. They are called that because the first and last numbers are perfect squares.

9x^2+6x+1

Now what times what equals 9x^2? 3x so,

(3x)(3x)

Now what times what equals 1? 1 so,

(3x+1)(3x+1)

and that is ok because when you distribute it 3x plus 3x equals 6x

Now we all know that something times itself is just that something squared so the final answer would look like this,

(3x+1)^2

I hope this blog is useful in understanding this concept. Thanks.

-Tim Kirby

10.4 (Day 2)

We learned about more Unit Cricle and Radians on 4/8/11

Radian is..
the angle created by a radius along the circumference of a circle.

Conversions

To convert from radians to degrees: multiply by 180/ㅠ

To convert from degrees to radians: multiply by ㅠ/180

For example,

11ㅠ/ 7

It's radian, so you multiply 180/ㅠ = 282.9 degrees

If you convert degrees to radians, then you leave ㅠ(pie) in answer

I can't put any images on here, because it doesn't work for me.

But, if you want to see completed Unit circle, go to

http://gbs-moodle.glenbrook225.org/moodle/file.php/1304/Chapter_10_Notes/10.4_Day_2_Notes_Key_2010.pdf

Based on 30-60-80 traingle and 45-45-90 tirangle,

you can find out the cosine and the sine of angle of unit circle.

For example,

30' has cosine of square root 3 over 2

You can get this by knowing 30-60-90 triangle

cosine is adj/hyp.

SO! it is squre root 3 over 2.

sine of 30' is 1/2.

sine is opp/hyp

So! it is 1/2

It's same way with 45-45-90 triangle.

ALSO

In Unit circle,

1 & 2 quadrants is sine POSITIVE.

2& 3 quadrants is cosine NEGATIVE.

you have to becareful of this

-Michelle S.