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Q&A #251

Teachers' Lounge Discussion: New examples of growth in calculus

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From: RON

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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