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Pre-Calculus

Pre-Calculus. Chapter 3 Exponential and Logarithmic Functions. 3.5 Exponential and Logarithmic Models. Objectives: Recognize the five most common types of models involving exponential or logarithmic functions.

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Pre-Calculus

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  1. Pre-Calculus Chapter 3 Exponential and Logarithmic Functions

  2. 3.5 Exponential and Logarithmic Models Objectives: • Recognize the five most common types of models involving exponential or logarithmic functions. • Use exponential growth and decay functions , Gaussian functions, logistic growth functions, and logarithmic functions to model and solve real-life problems.

  3. Exponential Growth Model • Model: y = aebx, b > 0

  4. Exponential Decay Model • Model: y = ae–bx, b > 0

  5. Gaussian Model • Model:

  6. Logistic Growth Model • Model:

  7. Logarithmic Models • Models: y = a + blnx y = a + b log10x

  8. Example – Exponential Growth • Estimates of the world population (in millions) from 1995 through 2004 are shown in the table. Use your graphing calculator to create a scatter plot of the data.

  9. Example – Exponential Growth • An exponential growth model that approximates this data is given by where P is the population (in millions) and t = 5 represents 1995. According to the model, when will the world population reach 6.8 billion?

  10. Example – Create the Model • In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 100 flies, and after 4 days there are 300 flies. How many flies will there be after 5 days?

  11. Carbon Dating Model • In living organic material, the ratio of the content of radioactive carbon isotopes (carbon 14) to the content of nonradioactive carbon isotopes (carbon 12) is about 1 to 1012. • When organic material dies, its carbon 12 content remains fixed, whereas its radioactive carbon 14 begins to decay with a half-life of 5730 years. • To estimate the age of dead organic material, scientists use the formula which denotes the ratio of C14 to C12 present at any time t (in years).

  12. Example – Carbon Dating • The ratio of carbon 14 to carbon 12 in a newly discovered fossil is • Estimate the age of the fossil.

  13. Example – Gaussian Model • In 2002, the SAT math scores for college-bound seniors roughly followed the normal distribution where x is the SAT score for math. Graph this function and estimate the average SAT score.

  14. Example – Logistic Growth • On a college campus of 5000 students, one student returns from vacation with a contagious flu virus. The spread of the virus is modeled by where y is the total number infected after t days. The college will cancel classes when 40% or more of the students are infected. • How many students are infected after 5 days? • After how many days will the college cancel classes?

  15. Logarithmic Model – Richter Scale • On the Richter scale, the magnitude R of an earthquake of intensity Iis given by where Io = 1 is the minimum intensity used for comparison. • Intensity is a measure of the wave energy of an earthquake.

  16. Example – Logarithmic Model • In 2001, the coast of Peru experienced an earthquake that measured 8.4 on the Richter scale. In 2003, Colima, Mexico experienced an earthquake that measured 7.6 on the Richter scale. Find the intensity of each earthquake and compare the two intensities.

  17. Homework 3.5 • Worksheet 3.5 # 1 – 6 matching, 23 – 26, 28, 29 – 43 odd, 47, 48

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