Exponential Functions x01234 y x01234 y

Exponential Growth and Decay

Applications Exponential Growth: – Compound Interest – Population Growth Exponential Decay: – Time Dependent Heat Transfer – Attenuation of Light traveling through a medium – Time dependence of Voltage/Current in electronic circuits – Radioactive processes/ Half Life – Probability dependence of small particles (Quantum Physics)

Modeling Exponential Growth and Decay

Example 1: Population Growth (Humans) The population of Jacksonville was 3,810 in 2007 and is growing at an annual rate of 3.5%. At this growth rate, what should its population be in 2020?  What is the growth rate, expressed as a decimal?  What is the growth factor?  What is the initial population?  What is the value of time (t) we need to use?  What is the population at this time t?

Example 2: Compound Interest You invested $1000 in a bank account that pays 5% annual interest. How much will be in the account after 6 years?  What is the growth rate, expressed as a decimal?  What is the growth factor?  What is the initial amount?  What is the value of time (t) we need to use?  What is the amount at this time t? How many years will it take for the account to have at least $1500 in it? (Use a graphing calculator and the intersect function)

How do we Transform Exponential Functions? Recall: For a Parent Function: f(x) We transform as: g(x) = af(x-h) + k Applied to the exponential function: f(x) = b x We have: g(x) = ab (x-h) + k Using your graphing calculator: Graph:  f 1 (x) = 2 x  f 2 (x) = 2 (x-4)  f 3 (x) = 3·2 (x+4) + 3

How can we Model Data using Exponential Functions? Example: Cooling Coffee Coffee is often brewed at around 200 ◦ F and is considered cool enough to drink at 185 ◦ F. Considering the given data, how much time should we wait before drinking coffee? Assumption: Coffee cools down eventually to 68 ◦ F. Time (min) Temp ( ◦ F)

Coffee Cooling cont’d 1)Enter data into lists: time, temp 2)Plot the data to see if an exponential model makes sense. (If you do exponential linear regression on this plot it will be incorrect because calculator assumes a zero asymptote; Try it.) 3)Make a third list: tempshift = temp-68 (to adjust asymptote), Plot tempshift vs. time. 4)Do an exponential linear regression on this graph: f(x) = (.956) x 5)Shift up by 68 to adjust asymptote and obtain our cooling function: T(t) = 134.5(.956) t )Plot the cooling function and y = 185 to find the time it takes for the coffee to be cool enough to drink.

Compound Interest Revisited

Compound Interest cont’d Percentage wise, we get: We calculate it like this:We get: 1100% applied once 250% applied twice 425% applied four times 1010% applied ten times 1001% applied 100 times % applied 1000 times

Compound Interest cont’d Percentage wise, we get: We calculate it like this:We get: 1100% applied once$ % applied twice$ % applied four times $ % applied ten times $ % applied 100 times $ % applied 1000 times $2.717

Continuously Compounded Interest

“Hwk 33” Pages , 22, 31, 32, 41, 36

Logarithmic Functions: The Inverse of the Exponential Function

Logarithmic and Exponential Forms of an Equation

More Examples

Logarithm Bases

Logarithmic Scales When a function describing a quantity varies over a very large range, we often describe it by referring to its logarithm. Example: Earthquake Magnitude: the Richter Scale Uses the logarithms (base 10) of wave amplitudes to compare strengths of earthquakes. An increment by 1 on the scale corresponds to a tenfold increase in “intensity” (amplitude size). (This actually corresponds to an increase in released energy of about 31 times). An Earthquake measuring 4.0 on the Richter scale is: Ten times more intense than one measuring 3.0 and One hundred times more intense than one measuring 2.0.

Richter Scale Example

Another Richter Scale Example In 1995, Mexico received an 8.0 earthquake and in 2001, Washington state received one of magnitude 6.8. How many times more intense was the quake that hit Mexico?

The pH Scale FoodApple Juice ButtermilkCreamKetchupShrimp Sauce Strained Peas 3.2× × × × × ×10 -6

The pH Scale (cont’d)

HWK 34

Graphing Logarithmic and Exponential Functions

How can we Graph a Logarithmic Function using its Inverse?

How do we Transform Logarithmic Functions?

Properties of Logarithms

Examples

More Examples

Change of Base Formula

Examples

Solving Exponential Equations

Another Example

Solving Logarithmic Equations

Natural Logarithms

Some Examples

Example: Stable Orbit about Earth(pg480) By the way: Are they right about v=7.7 km/s giving a stable orbit at a height of 300 km above the Earth? (Earth Radius = 6.4×10 6 m and Earth Mass = 6.0×10 24 kg). Also Given: c = 2.8 km/s

BAD SLIDE.Compound Interest Revisited