1. Pg 482 15) D: (–∞, ∞) R: (–∞, 0) Asy: y = 0 19) R: (–1, ∞) Asy: y = –1
23)
R: (–∞, 1)
Asy: y = 1 3) C 4) A 5) B 7)
D: (–∞, ∞)
R: (–∞, 0)
Asy: y = 0
11) \
24) B
25) D
27)
The graph should be moved 3 units to the right and not 3 units to the left
28)
y= 1219 (1.12)t
30)
y= 450(1.06)t
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7.3 – The Natural Base

2. Pg 489 3) Exp Decay 4) Exp Growth 5) 6) 7) D: (–∞, ∞) R: (0, ∞)
Asy: y = 0
11)
R: (–∞, 0)
Asy: y = 1
15) B
19)
D: (–∞, ∞)
R: (–1, ∞)
Asy: y = 1
23)
R: (–2, ∞)
Asy: y = –2
26)
The decay factor should be 0.98 and not 0.02
27) D
31a)
y = 200(0.75)t; $84.38
31c)
2.5 years
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7.3 – The Natural Base

3. Natural Bases and Application
Section 7.3
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7.3 – The Natural Base

4. Euler’s Number
Leonhard Euler (pronounced OILER NOT Yuler) worked with this unusual number [e] which is approximately … coming from and named it after himself  
Graphing exponentials like y = ex using e to be around The e button is located [2nd] [ln] button or [2nd] [÷] button if raised to the first power.
Solving all exponential equations involving e just like any other exponential equations.
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7.3 – The Natural Base

5. The Natural Base
Suppose that $1 is invested at 100% interest (r = 1) compounded n times for one year
As n gets very large, interest is continuously compounded.
2.6
2.8 3 2
2.2
2.4 1 4 12
365
8760
525,600
31,536,000
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7.3 – The Natural Base

6. The Natural Base
Suppose that $1 is invested at 100% interest (r = 1) compounded n times for one year
As n gets very large, interest is continuously compounded.
Compound N
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
(1+(1/2))2 = 2.25
Quarterly 4
2.4414
Monthly 12
2.6130
Daily
365 ….
Compound n
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
(1+(1/2))2 = 2.25
Quarterly 4
2.4414
Monthly 12
2.6130
Daily
365 ….
Hourly
8760 …
Compound n
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
(1+(1/2))2 = 2.25
Quarterly 4
2.4414
Monthly 12
2.6130
Compound n
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
(1+(1/2))2 = 2.25
Quarterly 4
2.4414
Monthly 12
2.6130
Daily
365 ….
Hourly
8760 …
Minute
525,600 …
Second
31,536,000 ….
Compound n
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
(1+(1/2))2 = 2.25
Quarterly 4
2.4414
Monthly 12
2.6130
Daily
365 ….
Hourly
8760 …
Minute
525,600 …
Compound n
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
(1+(1/2))2 = 2.25
Quarterly 4
2.4414
Compound n
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
(1+(1/2))2 = 2.25
Compound n
Total (1+(1/n))n
Annually 1
(1+(1/1))1 = 2
Semiannually 2
Quarterly 4
Monthly 12
Daily
365
Hourly
8760
Minute
525,600
Second
31,536,000
e = …
2.6
2.8 3 2
2.2
2.4 1 4 12
365
8760
525,600
31,536,000
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7.3 – The Natural Base

7. What is an easy way to remember this?
The Natural Base
What is an easy way to remember this? 2. 7
459045
23…
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7.3 – The Natural Base

8. Exponent Rules Product: Quotient: Power: n + m n – m n* m
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7.3 – The Natural Base

9. Example 1 Simplify e7 • e9 (No Decimals) n + m 7 + 9
7.3 – The Natural Base

10. Example 2 Simplify (No Decimals) n – m 10 9/22/2018 10:20 AM
5.3A – The Natural Base
7.3 – The Natural Base 10

11. Example 3 Simplify (3e5x)2 n • m 9/22/2018 10:20 AM
7.3 – The Natural Base

12. Your Turn
Simplify
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7.3 – The Natural Base

13. Example 4 Simplify e2x • e5x • e3 9/22/2018 10:20 AM
7.3 – The Natural Base

14. Example 5
Simplify
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7.3 – The Natural Base

15. Your Turn
Simplify
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7.3 – The Natural Base

16. Example 6
Determine whether f(x) = e–2x is exponential growth or decay.
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7.3 – The Natural Base

17. Your Turn
Determine whether f(x) = –2ex is exponential growth or decay.
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7.3 – The Natural Base

18. Graphing the Natural Base
A function of the form, y = aerx is called the Natural Base Exponential Function
If a > 0 and r > 0, the function is an exponential growth function
If a > 0 and r < 0, the function is an exponential decay function
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7.3 – The Natural Base

19. Example 7 Graph y = ex and state the range and asymptote 1 1 x y x y1 … x y 1
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7.3 – The Natural Base

20. Example 8 Graph y = e–x and state the range and asymptote 1 1 x y x y1 x y 1
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7.3 – The Natural Base

21. Example 9 Graph y = –ex + 3 and state the range and asymptote 2 1 1 x2 1 … x y 1
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7.3 – The Natural Base

22. Your Turn Graph y = 2ex + 1 and state the range and asymptote 3 1 1 x3 1
6.4365 x y 1
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7.3 – The Natural Base

23. Assignment Worksheet 23 9/22/2018 10:20 AM 9/22/2018 10:20 AM
5.3A – The Natural Base
7.3 – The Natural Base 23