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Q&A #251

Teachers' Lounge Discussion: New examples of growth in calculus

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From: RON <rselby@cinci.rr.com>
To: Teacher2Teacher Public Discussion
Date: 2003050916:08:22
Subject: Re: Re: NEW EXAMPLES OF GROWTH IN CALCULUS


SEMI-LOG GRAPH PAPER AND LOG-LOG GRAPH PAPER CAN BE USED FOR
"STRAIGHTENING OUT" A CURVE FOR IDENTIFICATION PURPOSES. THIS IS USED
IN A LABORATORY SETTING WHEN DATA IS AVAILABLE BUT THE EXACT EQUATION
PRODUCING THE DATA IS BEING SOUGHT.

FOR EXPONENTIAL EQUATIONS, SUCH AS Y=2^X, USING SEMI-LOG PAPER WILL
RESULT IN THE GRAPH OF A LINE. THE CALCULATIONS FOLLOW:

Y=2^X
LOG(Y)=LOG(2^X)
LOG(Y)=X*LOG(2)

NOW, SUBSTITUTING Y' FOR LOG(Y), AND SINCE LOG(2)=.301

Y'=.301X  WHICH IS A LINE WITH SLOPE=.301

NOTE THAT ONLY THE "Y" VARIABLE IS LOGARITHMIC. "X" IS AS BEFORE.

FOR POWER EQUATIONS, SUCH AS Y=X^2, USING LOG-LOG PAPER WILL RESULT IN
THE GRAPH OF A LINE. THE CALCULATIONS FOLLOW:

Y=X^2
LOG(Y)=LOG(X^2)
LOG(Y)=2*LOG(X)

NOW, SUBSTITUTING Y' FOR LOG(Y) AND X' FOR LOG(X)

Y'=2X'  WHICH IS A LINE WITH SLOPE=2.

NOTE THAT BOTH "Y" AND "X" ARE LOGARITHMIC.

AS FOR GRAPHING ZERO ON LOG GRAPH PAPER, THE ANSWER IS NO. YOU ARE
CORRECT THAT THE AXIS IS MARKED AS POWERS OF TEN, AND SINCE 10^X=0 HAS
NO REAL NUMBER SOLUTION, THERE IS NO ZERO ON THE AXIS.

NOTE THAT THE LOG FUNCTION, Y=LOG(X), IS ASYMPTOTIC TO X=0 AND THE
EXPONENTIAL FUNCTION, Y=10^X, IS ASYMPTOTIC TO Y=0.

RON

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