2Natural Base, eThe natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.
3Natural Exponential Function The exponential function with base e, f(x)=ex, is called the natural exponential function.And e is called the natural base.1You calculator has a FANCY ex button you can use!!! 2
4Evaluate f(x)=ex to the nearest thousandth for each value of x below:
5Exponential Properties Are the samefor base e!!Simplify the expression.a.e2e5=e2 + 5=e7b.12e43e3=e4 – 34=4e(5)c.e –3x2=52(e –3x)2=25e –6x=25e6x
6Try This! (10 ) Simplify the expression. 1. e7 e4 e11 4. e –4x 3 1000 2.2e–36e512e2Use a calculator to evaluate3.24e84e56e32.117
8Graph natural base functions Graph the function. State the domain and range.a.y=3e 0.25xSOLUTIONBecause a = 3 is positive and r = 0.25 is positive, the function is an exponential growth function. Plug in 0 and 1.x y1The domain is all real numbers, and the range is y > 0.
9Graph natural base functions Graph the function. State the domain and range.b.y=e –0.75(x – 2)+ 1SOLUTIONa = 1 is positive and r = –0.75 is negative, so the function is an exponential decay function. Translate the graph of y = right 2 units and up 1 unit.e –0.75xThe domain is all real numbers, and the range is y > 1.
10Try This! Graph the function. State the domain and range. y 2e 0.5x = domain: all real numbers, range: y>0
11Try This! Graph the function. State the domain and range. 1 2 f(x) = e -x+1domain: all real numbers, range: y>1
12Continuous Compounding Formula If P dollars are invested at an interest rate, r, that is compounding continuously, then the amount, A, of the investment at time t is given by:A = Pert
13An investment of $1000 earns an annual interest rate of 7.6% Compare the final amounts after 8 years for interest compounded quarterly and for interest compounded continuously.Compounded Quarterly Continuously
14Find the value:Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously.$716.66
15HomeworkGreen Book!!Page 142 # 1-25odd#27 and 28