1. Exponential Applications Exponential Growth and Decay Models

2. 8/14/2013 Exponential Applications 2 Exponential Functions Increasing and Decreasing Exponential growth : f(x) = C a x for a > 1 and C > 0 Example : f(x) = C2 x Exponential decay : x y f(x) = C2 x (0, C) Domain = R Range = { x x > 0 } Exponential Growth Exponential Compounding

3. 8/14/2013 Exponential Applications 3 Exponential Functions Increasing and Decreasing Exponential growth : f(x) = C a x Exponential decay : g(x) = f(–x) = C a –x for a > 1... a reflection of f(x) h(x) = Cb x x y f(x) = C2 x g(x) = f(–x) (0, C) Domain = R Range = { x x > 0 } OR 1 a b =b =, say a = 2, for 0 < b < 1 and = C2 –x h(x) = C ( ) 1 2 x Exponential Decay

4. 8/14/2013 Exponential Applications 4 Exponential Functions Increasing and Decreasing Exponential growth : f(x) = C a x Exponential decay : g(x) = f(–x) = C a –x x y f(x) = C2 x g(x) = f(–x) (0, C) Domain = R Range = { x x > 0 } = C2 –x Questions: Intercepts ? Asymptotes ? Growth factor a ? Decay factor a ? One a > 1 0 < a < 1 Exponential Growth Exponential Decay Exponential Compounding

5. 8/14/2013 Exponential Applications 5 Exponential Functions Increasing and Decreasing Exponential growth : f(x) = C a x Exponential decay : g(x) = f(–x) = C a –x x y f(x) = C2 x g(x) = f(–x) (0, C) Domain = R Range = { x x > 0 } = C2 –x f = { (x, 2 x ) x R } g = { (x, 2 –x ) x R } As ordered pairs, with C = 1, and

6. 8/14/2013 Exponential Applications 6 Exponential Functions Increasing and Decreasing Exponential growth : f(x) = C a x Exponential decay : g(x) = f(–x) = C a –x x y f(x) = C2 x g(x) = f(–x) (0, C) Domain = R Range = { x x > 0 } = C2 –x In tabular form, with C = 1, –2 ¼ 4 –1 ½ 2 011011 12½12½ 24¼24¼ 3 8 –3 8 x 2 x 2 –x 16 4 16 16 16 –4

7. 8/14/2013 Exponential Applications 7 Think about it !

8. 8/14/2013 Exponential Applications 8 Solving Equations: Examples 1. World population 195019601970198019902000 20102020203020402050 15 14 13 12 11 10 9 8 7 6 5 4 3 2 t P(t) 1950 2.5098 1960 3.0000 1970 3.5859 1980 4.2862 1990 5.1233 2000 6.1239 2010 7.3199 2020 8.7495 2030 10.458 2040 12.500 2050 14.942 t P( t) P1P1 P2P2 P3P3 ( x 10 9 ) P(t) = 3(1.018) t–1960

9. 8/14/2013 Exponential Applications 9 Spare Parts Slide 2 2 | | | ± ½ ½ ½ ¼ ¼ ¼