Precalculus 3.5 notes

3.5 Exponential and Logarithmic Models

5 most common:

1). Exponential growth model: y = ae^(bx), b>0.

2). Exponential decay model: y = ae^(-bx), b>0.

3). Gaussian model: y = ae^{(-((x-b)2)/c)}

4). Logistic growth model: y = a/(1+be^(-rx))

5). Logarithmic models:

a. y = a + b ln x; b. y = a + b log x

Graphs:

Examples:

Exponential Growth: Given P = 240360e^(0.012t)

t = 0 represents 2000, when will the population reach 275,000?

275000 = 240360 e^(0.012t)

1.144117158 = e^(0.012t)

ln 1.144117158 = 0.012 t

11.21944152 = t

Therefore, 2000 + 11 = 2011.

Example 2:

Country: Croatia, in 1997 had 5.0 million people and is estimated to have 4.8 in 2020, how many will there be in 2030?

Two points: (0, 5) and (23, 4.8)

Find the exponential growth model using y = ae^(bt)

5 = ae^(b*0)

5 = ae⁽⁰⁾

5 = a

4.8 = ae^{(b * 23)}

4.8 = 5e^(23b)

.96 = e^(23b)

ln .96 = 23b

-.0017748693 = b

Croatia’s equation is y = 5e^{(-.0017748693t)}

So in 2030, the population is 4.7 million

The “b” is the growth rate.

Since “b” is negative in Croatia, the population is decreasing.

Gaussian Models y = ae^-((x-b)2)/c

The model is commonly used in probability and statistics to represent populations that are NORMALLY DISTRIBUTED. For STANDARD normal distributions, the model takes the form:

y = (1/(standard deviation times the square root of 2 pi) )times (e^{((-x^2)/(2 standard deviation squared))})

The graph of a Gaussian model is called a bell-shaped curve.

Example:

The SAT scores in 1997 followed:

y = 0.0036e^{((-(2-511)^2)/(25088))} where 200 is less than or equal to x which is less than or equal to 800.

The maximum point is (511.11112, 0.0036)

A logistic growth curve: is a rapid growth followed by a declining rate of growth OR a sigmoidal curve.

y = a / ( 1 + be^(-rx))

Example: Spread of a virus (flu)

y = 5000/(1 + 4999e^(-.8r))

a. How many students are infected after 5 days?

54 people

b. If classes are cancelled when 40% or more are infected of a campus of 5000 students, how long will it take? 10.14 days.

Magnitudes of Earthquakes

R = log (I/ I_o)

Magnitude R of an earthquake of Intensity of I where I₀ usually equals 1.

Example: Find the magnitude of R of an earthquake of Intensity I, let I₀ = 1.

a). Taiwan in 1999, I = 39811000

R = log 39811000 = 7.600003087 = 7.6

Example 2: What is the Intensity if R = 8.6?

8.6 = log I

10^(8.6) = I

I = 398107170.6

Homework: #30 pg. 266; #1-29 odd, 39