Copyright © 2011 Pearson Education, Inc. Modeling Our World

Copyright © 2011 Pearson Education, Inc. Slide 9-3 Unit 9C Exponential Modeling

9-C Copyright © 2011 Pearson Education, Inc. Slide 9-4 Exponential Functions An exponential function grows (or decays) by the same relative amount per unit time. For any quantity Q growing exponentially with a fractional growth rate r, Q = Q 0 (1+r) t where Q = value of the exponentially growing quantity at time t Q 0 = initial value of the quantity (at t = 0) r = fractional growth rate for the quantity t = time Negative values of r correspond to exponential decay. Note that the units of time used for t and r must be the same.

9-C Copyright © 2011 Pearson Education, Inc. Slide 9-5 To graph an exponential function, use points corresponding to several doubling times (or half-lives, in the case of decay). Start at the point (0,Q 0 ), the initial value at t = 0. For an exponentially growing quantity, the value of Q is 2Q 0 (double the initial value) after one doubling time (T double ), 4Q 0 after two doubling times (2T double ), 8Q 0 after three doubling times (3T double ), and so on. For an exponentially decaying quantity, the value of Q falls to Q 0 /2 (half the initial value) after one half-life (T half ), Q 0 /4 after two half-lives (2T half ), Q 0 /8 after three half-lives (3T half ), and so on. Graphing Exponential Functions

9-C Copyright © 2011 Pearson Education, Inc. Slide 9-6 Exponential Growth To graph exponential growth, first plot the points (0,Q 0 ), (T double,2Q 0 ), (2T double,4Q 0 ), (3T double,8Q 0 ), and so on. Then fit a curve between these points, as shown to the right.

9-C Copyright © 2011 Pearson Education, Inc. Slide 9-7 Exponential Decay To graph exponential decay, first plot the points (0,Q 0 ), (T half,Q 0 /2), (2T half,Q 0 /4), (3T half,Q 0 /8), and so on. Then fit a curve between these points, as shown to the right.

9-C Copyright © 2011 Pearson Education, Inc. Slide 9-8 If given the growth or decay rate r, use the form If given the doubling time T double, use the form If given the half-life T half, use the form Forms of the Exponential Function

9-C Copyright © 2011 Pearson Education, Inc. Slide 9-9 China’s Coal Consumption China’s rapid economic development has lead to an exponentially growing demand for energy, and China generates more than two-thirds of its energy by burning coal. During the period 1998 to 2008, China’s coal consumption increased at an average rate of 8% per year, and the 2008 consumption was about 2.1 billion tons of coal. Use these data to predict China’s coal consumption in If t = 0 represents 2008, Q 0 = 2.1, r = 0.08, and t = 20 years. Q = Q 0 (1+r) t = 2.1 ( ) 20 = 2.1 (1.08) 20 ≈ 9.8 China’s predicted coal consumption is about 9.8 billion tons.

9-C Copyright © 2011 Pearson Education, Inc. Slide 9-10 Exponential growth functions have rates of change that increase. Exponential decay functions have rates of change that decrease. Linear functions have straight line graphs and constant rates of change. Exponential functions have graphs that rise or fall steeply and have variable rates of change. Changing Rates of Change