Presentation on theme: "18 Days. Five Days What equations did you come up with to model your data in the penny lab?"— Presentation transcript:
What equations did you come up with to model your data in the penny lab?
Identify the following as either exponential growth, decay, or undefined.
Lower Dauphin HS had a student population of 1150 kids in the school year ending in 2010. Find the function that models the population if the rate of growth continues at a rate of 2.3% per year.
Find the equation of the exponential function that passes through (4,8) and (6,32).
pg 434 (# 1 - 19 odd)
pg 434 (32, 42 - 45 all)
Practice with Exponential Growth and Decay
Exponential functions are those that have An example of an exponential function is
We can shift exponential function using the same patterns from before.
The half life of a radioactive substance in the time it takes for half of the substance to decay.
Caffeine has a half life of 5.7 hours in the human body. If you drink a Coca-Cola at noon, how much caffeine will be in your body when you go to bed? What do we need to know? What time do you go to bed? 32mg of caffeine per 12oz can.
The exponential function with base e are very useful in describing continuous growth and decay.
What is the difference between simple interest and compound interest? What is continuously compounded interest?
An initial investment of $35000 is continuously compounded at 8.5% interest. How much is the investment worth after 5 years? After 15 years?
pg 442 (# 2-8 even, 16, 17, 24-26, 36, 37)
Exponential Growth and Decay WS
Simple and Compound Interest WS
8.1 - 8.2 Review
We know that in exponential functions the exponent is a variable. When we wish to solve for that variable we have two approaches we can take. One approach is to use a logarithm. We will learn about these in a later lesson. The second is to make use of a property called the Equality Property for Exponential Functions.
Basically, this states that if the bases are the same, then we can simply set the exponents equal. This property is quite useful when we are trying to solve equations involving exponential functions. Let’s try a few examples to see how it works.
(Since the bases are the same we simply set the exponents equal.) Here is another example for you to try: Example 1a:
How can we solve an equation when the bases are not the same?? Does anyone have an idea how we might approach this?
(our bases are now the same so simply set the exponents equal) Let’s try another one of these.
Solving Exponential Equations WS
The logarithm to the base b exponential is defined as:
3 2 =9 x a+b =9
The common logarithm is a log that uses base 10. It can be written in either of the two forms:
Evaluate the following logs:
Page 450 (#6-25 all, 41-48 all, 53-61 all)
8-3 WS (# 1 - 49 odd)
We can use the properties of logs to re-write a logarithmic expression as a single log so that we can evaluate, or solve, the logarithm. Ex:
Guidelines: ◦ When solving an exponential equation, you must isolate the exponential (or write as one exponential on each side) before you take the log of both sides. ◦ When solving a logarithmic equation, you must isolate the logarithm (or write as a single log on each side of the equation) before you raise both sides using a base b.
Our calculators can only handle logs that are either base 10, or base e. So, we need a way to re-write any log so that it can be entered into the calculator. The Change of Base Formula: