Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1
Section 3.4 Continuous Growth and the Number e 2
You invest $1 in a bank account that pays: Interest Percent Frequency 100% 1 x / year Annually 50% 2 x / year Semi-Annually 25% 4 x / year Quarterly 8.33333333% 12 x / year Monthly 1.92307692% 52 x / year Weekly.2739726% 365 x / year Daily.01141552511% 8760 x / year Hourly Page 130 3
You invest $1 in a bank account that pays: Interest Percent Frequency 100% 1 x / year Annually 50% 2 x / year Semi-Annually 25% 4 x / year Quarterly 8.33333333% 12 x / year Monthly 1.92307692% 52 x / year Weekly.2739726% 365 x / year Daily.01141552511% 8760 x / year Hourly Page 130 Here is the model that calculates the balance: 4
Annual growth rate for an investment of $1 with interest rate of 100% that is paid n times / year: 1 1 n n Page N/A 5
Annual growth rate for an investment of $1 with interest rate of 100% that is paid n times / year: 1 1 1 n n Page N/A 6
Let's use our calculator. 1 1 n n Page 130 7
The number e = 2.71828182 is often used for the base, b, of the exponential function. Base e is called the natural base. Page 131 8
Graph the following on the calculator: y y y y 10 3 2 e x x x x Page 131 9
Window settings: Window Value Xmin 0 Xmax 5 Xscl 1 Ymin 0 Ymax 5 Yscl 1 Page 131 10
Results: Page 131 11
Exponential Functions with Base e Represent Continuous Growth Page 131 12
Exponential Functions with Base e Represent Continuous Growth Any positive base b can be written as a power of e: b e k Page 131 13
Exponential Functions with Base e Represent Continuous Growth Any positive base b can be written as a power of e: b e If b > 1, then k is positive; if 0 < b < 1, then k is negative. k Page 131 14
Exponential Functions with Base e Represent Continuous Growth The function Q = ab t can be rewritten in terms of e: Q ab t a( e k ) t ae kt Page 131 15
Exponential Functions with Base e Represent Continuous Growth The function Q = ab t can be rewritten in terms of e: Q ab t a( e k ) t ae kt The constant k is called the continuous growth rate. Page 131 16
In general: For the exponential function Q = ab t, the continuous growth rate, k, is given by solving e k = b. Then If a is positive, Q ae If k > 0, then Q is increasing. If k < 0, then Q is decreasing. kt Page 131 Blue Box 17
The value of the continuous growth rate, k, may be given as a decimal or a percent. If t is in years, for example, then the units of k are given per year; if t is in minutes, then k is given per minute. Q ae kt Page 132 18
Give the continuous growth rate of each of the following functions and graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Page 132 Example #1 19
Give the continuous growth rate of each of the following functions and graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Continuous growth rates? Page 132 Example #1 20
Give the continuous growth rate of each of the following functions and graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Continuous growth rates? The value of the continuous growth rate, k, may be given as a decimal or a percent. Q ae kt Page 132 Example #1 21
Give the continuous growth rate of each of the following functions and graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Continuous growth rates? The function P = 5e 0.2t has a continuous growth rate of 20%. Page 132 Example #1 22
Give the continuous growth rate of each of the following functions and graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Continuous growth rates? The function Q = 5e 0.3t has a continuous 30% growth rate. Page 132 Example #1 23
Give the continuous growth rate of each of the following functions and graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Continuous growth rates? The function R = 5e 0.2t has a continuous growth rate of 20%. The negative sign tells us that R is decreasing instead of increasing. Page 132 Example #1 24
Now let's graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Page 132 Example #1 25
Now let's graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Page 132 Example #1 Window Value Xmin -5 Xmax 5 Xscl 1 Ymin 0 Ymax 10 Yscl 1 26
Now let's graph each function: 0.2t 0.3t 0.2t P 5 e, Q 5 e, R 5e Page 132 Example #1 27
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. Page 132 Example #2 28
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the key phrase here? Page 132 Example #2 29
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the key phrase here? Page 132 Example #2 30
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the key phrase here? This implies that e is involved. Page 132 Example #2 31
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the general formula? Page 132 Example #2 32
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the general formula? Q ae kt Page 132 Example #2 33
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the specific formula? Page 132 Example #2 34
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the specific formula? P 7.3e 0.022t Page 132 Example #2 35
A population increases from 7.3 million at a continuous rate of 2.2% per year. Write a formula for the population, and estimate graphically when the population reaches 10 million. What's the specific formula? P 7.3e 0.022t Let's graph. Page 132 Example #2 36
P 7.3e 0.022t Window Value Xmin 0 Xmax 20 Xscl 2 Ymin 0 Ymax 20 Yscl 2 Page 132 Example #2 37
Estimate graphically when the population reaches 10 million. P 7.3e 0.022t Page 132 Example #2 38
Estimate graphically when the population reaches 10 million. P 7.3e 0.022t Add y 2 = 10 Page 132 Example #2 39
P 7.3e 0.022t x=14.305034, y=10 Page 132 Example #2 40
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? Page 132 Example #3 41
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? Page 132 Example #3 42
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? What is out exponential function here? Page 132 Example #3 43
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? What is out exponential function here? Q ae kt Page 132 Example #3 44
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? What is out exponential function here? A 100e 0.17t Page 132 Example #3 45
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? What is out exponential function here? A 100e 0.17t A = starting amount of caffeine, Decay rate = 17% / hour Page 132 Example #3 46
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? A 100e 0.17t How do we calculate? Page 132 Example #3 47
Caffeine leaves the body at a continuous rate of 17% per hour. How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? A 100e 0.17t How do we calculate? Key into main screen on calculator. Page 132 Example #3 48
How much caffeine is left in the body 8 hours after drinking a cup of coffee containing 100 mg of caffeine? A 100e 0.17*8 25.6660777 Answer: 25.6660777 mg of caffeine Page 132 Example #3 49
The Difference between Annual and Continuous Growth Rates Page 133 50
The Difference between Annual and Continuous Growth Rates If P = P 0 (1.07) t, with t in years, we say that P is growing at an annual rate of 7%. Page 133 51
The Difference between Annual and Continuous Growth Rates If P = P 0 (1.07) t, with t in years, we say that P is growing at an annual rate of 7%. If P = P 0 e 0.07t, with t in years, we say that P is growing at a continuous rate of 7% per year. Page 133 52
The Difference between Annual and Continuous Growth Rates If P = P 0 e 0.07t, with t in years, we say that P is growing at a continuous rate of 7% per year. Since e 0.07 = 1.072508181, we can rewrite P 0 e 0.07t = P 0 (1.072508181) t. Page 133 53
The Difference between Annual and Continuous Growth Rates In other words, a 7% continuous rate and a 7.2508181% annual rate generate the same increases in P. We say the two rates are equivalent. Page 133 54
The Difference between Annual and Continuous Growth Rates In other words, a 7% continuous rate and a 7.2508181% annual rate generate the same increases in P. We say the two rates are equivalent. The continuous growth rate is always smaller than the equivalent annual rate. Page 133 55
End of Section 3.4 56