8.3 The Number e.

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Presentation transcript:

8.3 The Number e

Objectives/Assignment Use the number e as the base of exponential functions Use the natural base e Assignment: 17-75 odd

The Natural base e Much of the history of mathematics is marked by the discovery of special types of numbers like counting numbers, zero, negative numbers, Л, and imaginary numbers.

Natural Base e Like and ‘i’, ‘e’ denotes a number. Called The Euler Number after Leonard Euler (1707-1783) The number e is irrational – its’ decimal representation does not terminate or follow a repeating pattern. As n gets larger (n→∞), (1+1/n)n gets closer and closer to 2.71828….. Which is the value of e.

e3 · e4 = e7 (3e-4x)2 9e(-4x)2 9e-8x 9 e8x 10e3 = 5e2 2e3-2 = 2e Examples e3 · e4 = e7 (3e-4x)2 9e(-4x)2 9e-8x 9 e8x 10e3 = 5e2 2e3-2 = 2e

More Examples! (2e-5x)-2= 2-2e10x= e10x 4 24e8 = 8e5 3e3

Using a calculator Evaluate e2 using a graphing calculator 7.389 Evaluate e2 using a graphing calculator Locate the ex button you need to use the second button

Evaluate e-.06 with a calculator

f(x) = aerx is a natural base exponential function Graphing f(x) = aerx is a natural base exponential function If a>0 & r>0 it is a growth function If a>0 & r<0 it is a decay function

Graphing examples Graph y=ex Remember the rules for graphing exponential functions! The graph goes thru (0,a) and (1,e) (1,2.7) (0,1)

Graphing cont. Graph y=e-x (1,.368) (0,1)

Graphing Example Graph y=2e0.75x State the Domain & Range Because a=2 is positive and r=0.75, the function is exponential growth. Plot (0,2)&(1,4.23) and draw the curve. (1,4.23) (0,2)

A = Pert Using e in real life. In 8.1 we learned the formula for compounding interest n times a year. In that equation, as n approaches infinity, the compound interest formula approaches the formula for continuously compounded interest: A = Pert

Example of continuously compounded interest You deposit $1000.00 into an account that pays 8% annual interest compounded continuously. What is the balance after 1 year? P = 1000, r = .08, and t = 1 A=Pert = 1000e.08*1 ≈ $1083.29