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# 1.01.Growth - [email protected] = a x b The calculation reveals that...

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Growth Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 1.01 Growth BASICS B.1) Growth of line functions f @ x D = a x + b A line function f @ x D is any function whose formula is f @ x D = a x + b where a and b are constants. Here's a plot of a line function: Clear @ f, x D ; a = 0.5; b = 4; f @ x _ D = a x + b; Plot @ f @ x D , 8 x, - 2, 8 < , PlotStyle -> 88 Red, Thickness @ 0.015 D<< , AxesLabel -> 8 "x" , "f @ x D " < , PlotLabel -> "A line function" D - 2 2 4 6 8 x 4 5 6 7 8 f @ x D A line function There's steady growth as x advances from left to right. Here's another: Clear @ f, x D ; a = - 0.3; b = 2; f @ x _ D = a x + b; Plot @ f @ x D , 8 x, - 2, 8 < , PlotStyle -> 88 Red, Thickness @ 0.015 D<< , AxesLabel -> 8 "x" , "f @ x D " < , PlotLabel -> "A line function" D - 2 2 4 6 8 x 0.5 1.0 1.5 2.0 2.5 f @ x D A line function Steady (negative) growth as x advances from left to right. Play with other choices of a and b until you get the feel of a line function. · B.1.a.i) Constant growth rate The most important feature of a line function f @ x D = a x + b is revealed by the following calculation. Clear @ f, a, b, x, h D ; f @ x _ D = a x + b; Expand @ f @ x + h D - f @ x DD a h What feature of line functions is revealed by this calculation? · Answer: The calculation reveals that when you take a line function f @ x D = a x + b , then you find that f @ x + h D - f @ x D = a h . This tells you that when x advances by h units, then f @ x D grows by a h units. Consequently a line function f @ x D = a x + b has constant growth rate of a units on the f @ x D -axis for each unit on the x -axis. · B.1.a.ii) The meaning of the growth rate. As you saw above, the growth rate of a line function f @ x D = a x + b measures out to a units on the f @ x D axis per unit on the x axis. What is the significance of a ? · Answer: Big positive a 's force big-time fast growth as the following true scale plot shows: Clear @ f, x D ; a = 8; b = 2; f @ x _ D = a x + b; bigpositive = Plot @ f @ x D , 8 x, - 1, 2 < , PlotStyle -> 88 Red, Thickness @ 0.015 D<< , PlotLabel -> "Big a > 0" , AspectRatio -> Automatic D 1.0 1.5 2.0 - 5 5 10 15 Big a > 0 Small positive a 's force slow growth as the following true scale plot shows: Clear @ f, x D ; a = 0.2; b = 2; f @ x _ D = a x + b; smallpositive = Plot @ f @ x D , 8 x, - 1, 5 < , PlotStyle -> 88 Red, Thickness @ 0.015 D<< , PlotLabel -> "Small a > 0" , AspectRatio -> Automatic D - 1 1 2 3 4 5 2.0 2.2 2.4 2.6 2.8 3.0 Small a > 0 a = 0 forces no growth at all as the following true scale plot shows:

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Clear @ f, x D ; a = 0; b = 2; f @ x _ D = a x + b; smallpositive = Plot @ f @ x D , 8 x, - 1, 5 < , PlotStyle -> 88 Red, Thickness @ 0.015 D<< , AxesLabel -> 8 "x" , "f @ x D " < , PlotLabel -> "a = 0" , AspectRatio -> Automatic D - 1 1 2 3 4 5 x 1 2 3 4 f @ x D a = 0 Small negative a 's force slow (negative) growth as the following true scale plot shows: Clear @ f, x D ; a = - 0.2; b = 2; f @ x _ D = a x + b;
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1.01.Growth - [email protected] = a x b The calculation reveals that...

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