2 Natural Base, eThe natural base, e, is used to estimate the ages of artifacts and to calculate interest that is compounded continuously.
3 Natural 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
4 Evaluate f(x)=ex to the nearest thousandth for each value of x below:
5 Exponential 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
6 Try This! (10 ) Simplify the expression. 1. e7 e4 e11 4. e –4x 3 1000 2.2e–36e512e2Use a calculator to evaluate3.24e84e56e32.117
8 Graph 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.
9 Graph 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.
10 Try This! Graph the function. State the domain and range. y 2e 0.5x = domain: all real numbers, range: y>0
11 Try This! Graph the function. State the domain and range. 1 2 f(x) = e -x+1domain: all real numbers, range: y>1
12 Continuous 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
13 An 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
14 Find the value:Of $500 after 4 years invested at an annual interest rate of 9% compounded continuously.$716.66
15 HomeworkGreen Book!!Page 142 # 1-25odd#27 and 28