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 9/22/2018 10:20 AM 7.3 – The Natural Base

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 9/22/2018 10:20 AM 7.3 – The Natural Base

Natural Bases and Application Section 7.3 9/22/2018 10:20 AM 7.3 – The Natural Base

Euler’s Number Leonhard Euler (pronounced OILER NOT Yuler) worked with this unusual number [e] which is approximately 2.71828182845904523… coming from and named it after himself.   Graphing exponentials like y = ex using e to be around 2.7. 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. 9/22/2018 10:20 AM 7.3 – The Natural Base

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 9/22/2018 10:20 AM 7.3 – The Natural Base

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 2.71457…. 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 2.71457…. Hourly 8760 2.718127… 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 2.71457…. Hourly 8760 2.718127… Minute 525,600 2.7182792… Second 31,536,000 2.7182825…. 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 2.71457…. Hourly 8760 2.718127… Minute 525,600 2.7182792… 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 = 2.718281828… 2.6 2.8 3 2 2.2 2.4 1 4 12 365 8760 525,600 31,536,000 9/22/2018 10:20 AM 7.3 – The Natural Base

What is an easy way to remember this? The Natural Base What is an easy way to remember this? 2. 7 18281828 459045 23… 9/22/2018 10:20 AM 7.3 – The Natural Base

Exponent Rules Product: Quotient: Power: n + m n – m n* m 9/22/2018 10:20 AM 7.3 – The Natural Base

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

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

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

Your Turn Simplify 9/22/2018 10:20 AM 7.3 – The Natural Base

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

Example 5 Simplify 9/22/2018 10:20 AM 7.3 – The Natural Base

Your Turn Simplify 9/22/2018 10:20 AM 7.3 – The Natural Base

Example 6 Determine whether f(x) = e–2x is exponential growth or decay. 9/22/2018 10:20 AM 7.3 – The Natural Base

Your Turn Determine whether f(x) = –2ex is exponential growth or decay. 9/22/2018 10:20 AM 7.3 – The Natural Base

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 9/22/2018 10:20 AM 7.3 – The Natural Base

Example 7 Graph y = ex and state the range and asymptote 1 1 x y x y 1 2.71828… x y 1 9/22/2018 10:20 AM 7.3 – The Natural Base

Example 8 Graph y = e–x and state the range and asymptote 1 1 x y x y 1 x y 1 9/22/2018 10:20 AM 7.3 – The Natural Base

Example 9 Graph y = –ex + 3 and state the range and asymptote 2 1 1 x 2 1 0.28172… x y 1 9/22/2018 10:20 AM 7.3 – The Natural Base

Your Turn Graph y = 2ex + 1 and state the range and asymptote 3 1 1 x 3 1 6.4365 x y 1 9/22/2018 10:20 AM 7.3 – The Natural Base

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