CC 8 Logs, Exponentials, Logistic Growth and Inverses.

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CC 8 Logs, Exponentials, Logistic Growth and Inverses

Transcript of CC 8 Logs, Exponentials, Logistic Growth and Inverses.

CC 8 Logs, Exponentials, Logistic Growth and Inverses

What question comes to mind?

How tall will Chloe be as an adult?

Date Height (inches)

7/16/2010 21.25

9/13/2010 23.75

1/17/2011 27.13

4/18/2011 28.25

7/18/2011 30.5

1/16/2012 34

7/17/2012 36.25

7/22/2013 39.25

7/30/2014 42.5

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I. Logistic Growth

A. Definition

Logistic Growth models real-life quantities whose growth levels off because the rate of growth changes from an increasing growth rate to a decreasing growth rate.

On your desk sketch a graph that changes

from increasing to decreasing

B. Visual

Why does the graph

continue to increase if the growth

rate is decreasing?

C. Process

Analyze the function

C. Process

Goal Problems

Recall & Reproduction

If, at the end of two years a savings account has a balance of $1172.60, and the interest rate is compounded monthly at 3.2%, then what is the original amount deposited two years ago?

Routine

Based on recent census data, a logistic model for the population of Dallas, t years after 1900 is as follows.

According to this model, when was the population 1 million?

Non-Routine

Goal Problems

Recall & Reproduction Routine

1985

Non-Routine

Goal Problems

R& R What are the key characteristics of log, exponential and logistic

growth functions? When is compound interest used vs. continuous compound

interest? How do you know when the model is growth or decay?

Routine Why is the y value of the point of maximum growth ? Why do we start out with t = 0?

Non-routine In what context could a domain of all real numbers make sense?

Active Practice

R& R: Analyzing graphs of logs, exponentials and logistic

growth Compound Interest Exponential Growth

Routine: Application of logs, exponentials and logistic growth

Non-Routine Update Plan: How tall will Chloe be as an adult

Task

Exponential & Logarithmic Fcns

LT 3G: I understand the inverse relationship between exponential and logarithmic functions and can use this relationship to explain the definition of a logarithm. I can use the definition of a logarithm to find the value of a variety of given logarithms, including natural logarithms. I can use the inverse relationship between exponential and logarithmic functions to solve algebraic equations or to sketch logarithmic and exponential functions. I can explain when technology is a tool or a hindrance in computing logarithms (e.g. ln1, lne2)

A. Definition

Logarithm – a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

B. Visual

Log636=2

62=36

What is the general form for the transformation above?

y = bxx = logby

Rewrite the following equation to a more familiar form.

Based on the definition, label the corresponding parts

between the two forms.

.

A. Definition

Logarithm – a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

y = bxx = logby

C. Process

3216log6 236log6

Given

1216

36log6

Justify why the following equation is a true statement:

Write the general form of this relationship as a logarithm and an exponential.

Rules of Logarithms Rules of Exponents

x = logby

logb(mn) = logbm + logbn

logb(m÷n) = logbm – logbn

logb(mn) = nlogb(m)

Change-of-base formula

C. Process

Notation: Common Logarithm: log x =

log10xNatural Logarithm: ln w = logew

Rules of Logarithms Rules of Exponents

x = logby y = bx

logb(mn) = logbm + logbn

(am)(an) = am+n

logb(m÷n) = logbm – logbn

(am)÷(an) = am-n

logb(mn) = nlogb(m) (am)n = amn

Change-of-base formula

Leading into the Weekend

Reminder: Quick Check Monday (2/23) Topic: Unit 3 CC 7 – Matrices

Agenda

Active Practice CC 7 - Matrices CC 8 - Logs/Exp

Concept Check

Concept Check (Individually)On your own, write down the answers to these

questions:

Explain what happens when you multiply a matrix with its inverse?

Compare and contrast “logistic” vs “logarithmic”?

How are the log rules related to the exponent rules?

Prove that: logb(jk) = logbj + logbk

WAIT! Don’t share thoughts… we will do that after everyone has had time to process on their own.

Concept Check (as a Group)

As your group shares:

In your notes: Write down questions and answers that your group shared that helped you. Tally for yourself, how many times someone writes down one of your questions/answers.

Explain what happens when you multiply a matrix with its inverse?

Compare and contrast “logistic” vs “logarithmic”?

How are the log rules related to the exponent rules?

Prove that: logb(jk) = logbj + logbk

Leading into the Weekend

Reminder: Quick Check Monday (2/23) Topic: Unit 3 CC 7 – Matrices

Make a Plan to prepare for the QC

List either: Questions regarding material that need to

be answer this weekend Problems types that need to be reviewed

Goal: Finding insights to add to your notes

Goal ProblemsRecall & Reproduction:

1. Simplify:

a. log4(16)

b. 6log(100)*e0.

c. log(1/2)32

d. (log100)÷(log10)

e. What is the value of n that satisfies logn64 = 2

Routine:Solve: ln(x + 5) = ln(x – 1) – ln(x + 1)

Non-Routine:The half-life of radioactive radium (226Ra) is 1556 years. What percent of a present amount of radioactive radium will remain after 100 years?