copyright © 2011 pearson education, inc. modeling our world

Copyright © 2011 Pearson Education, Inc.

Modeling Our World

Copyright © 2011 Pearson Education, Inc. Slide 9-3

Unit 9C

Exponential Modeling


9-C

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 = Q0 (1+r)t

whereQ = value of the exponentially growing quantity at time tQ0 = initial value of the quantity (at t = 0)

r = fractional growth rate for the quantityt = 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

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,Q0), the initial value at t = 0.

For an exponentially growing quantity, the value of Q is 2Q0 (double the initial value) after one doubling time (Tdouble), 4Q0 after two doubling times (2Tdouble), 8Q0 after three doubling times (3Tdouble), and so on.

For an exponentially decaying quantity, the value of Q falls to Q0/2 (half the initial value) after one half-life (Thalf), Q0/4 after two half-lives (2Thalf), Q0/8 after three half-lives (3Thalf), and so on.

Graphing Exponential Functions


9-C

Exponential Growth

To graph exponential growth, first plot the points (0,Q0), (Tdouble,2Q0), (2Tdouble,4Q0), (3Tdouble,8Q0), and so on. Then fit a curve between these points, as shown tothe right.


9-C

Exponential Decay

To graph exponential decay, first plot the points (0,Q0), (Thalf,Q0/2), (2Thalf,Q0/4), (3Thalf,Q0/8), and so on. Then fit a curve between these points, as shown to the right.


9-C

If given the growth or decay rate r, use the form

If given the doubling time Tdouble, use the form

If given the half-life Thalf, use the form

Forms of the Exponential Function


9-C

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 2028.

If t = 0 represents 2008, Q0 = 2.1, r = 0.08, and t = 20 years.

Q = Q0 (1+r)t = 2.1 (1 + 0.08)20 = 2.1 (1.08)20 ≈ 9.8

China’s predicted coal consumption is about 9.8 billion tons.


9-C

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

copyright © 2011 pearson education, inc. modeling our world

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