Graphing Nonlinear Functions Continued

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Graphing Nonlinear Functions(Continued)

• GE 111 - Lecture 5

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Graphing non-linear data

• Plot the data on rectilinear paper. See if there
  is linear trend.
• Yes ? Find y mxb.
• No ? Plot data on semi-log paper. See if there is
  linear trend.
• Yes ? Find y bemx.
• 1. Take logarithm. Plot ln y vs x. Find ln
  yln bmx.
• 2. Plot y vs x on semi-log paper.
• - Read intercept(b) and slope(mRise/Run)
  directly.
• - Use the method of selected points.
• No ? Plot data on log-log paper. See if there is
  linear trend.
• Yes ? find y bxm.
• 1. Take logarithm. Plot log y vs log x. Find
  logylogbmlogx.
• 2. Plot y vs x on log-log paper.
• - Read intercept(b) and slope(mRise/Run)
  directly.
• - Use the method of selected points.

• Method of selected points
• Pick two points on the straight line P1(x1, y1),
  P2(x2, y2)
• Sustitute into eq.
• y1mx1b
• y2mx2b
• Solve for m and b
• Verify your answer

• Method of selected points
• Pick two points on the straight line P1(log x1,
  log y1), P2(log x2, log y2)
• Sustitute into eq.
• log y1log b m log x1
• log y2log b m log x2
• Solve for m and b
• Verify your answer

• Method of selected points
• Pick two points on the straight line P1(x1, ln
  y1), P2(x2, ln y2)
• Sustitute into eq.
• ln y1ln b mx1
• ln y2ln b mx2
• Solve for m and b
• Verify your answer

• Method of selected points
• Pick two points on the straight line P1(x1, y1),
  P2(x2, y2)
• Sustitute into eq.
• ln y1ln A Bx1
• ln y2ln A Bx2
• Solve for A and B
• Verify your answer

• Method of selected points
• Pick two points on the straight line P1(x1, y1),
  P2(x2, y2)
• Sustitute into eq.
• log y1log b m log x1
• log y2log b m log x2
• Solve for m and b
• Verify your answer

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Example Ball Drop
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Solution 1 linear graph
P2(0.7, 2.1)
P3(0.3, 1.3)
log x 0 log y log b y b
log y log b m log x
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Solution 1 Linear graph

• Take the logarithm
• ybxm ? log y log b m log x (form of basic
  line)
• Plot log y versus log x on rectilinear paper.
• Calculate the slope by reading rise and run
  directly off the axes. The scale is linear, so
  interpolation is easier to read and often more
  accurate.
• When you pick 2 points, sometimes you can save
  work by picking one point at the y intercept,
  measured at log x 0, which leads to log y log
  b, and can be read directly off of the axis scale.

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Solution 1 Linear graph

• From the example, the plot line (not data point)
  has a y-intercept (at log x 0) of
• log b 0.69. Therefore b 4.9
• Plug P2(0.7,2.1) into eq. and get
• m (2.1-0.69)/(0.7) 2.0
• Equation of the line is d 4.9 t2, corresponding
  to the familiar d ½ g t2 where we know g to be
  9.81m/s2
• g is acceleration due to gravity

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Solution 2 log-log scales
P2(4.35, 100)
P3(2, 19)
x1 yb
ybxm
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Solution 2 log-log scales

• Solution - pick 2 points and calculate
• Given the general form y bxm
• and taking the log of both sides
• log y log b m log x
• Using 2 points P1(x1, y1) and P2(x2, y2)
• log y1 log b m log x1
• log y2 log b m log x2

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Solution 2 log-log scales

• and solving for m,
• m log(y2)-log(y1) / log(x2)-log(x1)
• m log (y2 / y1 ) / log (x2 / x1 )
• From the original equation, solving for b
• b y/xm
• This allows points P1(x1, y1) and P2(x2, y2) to
  be read directly off of the log-log paper and
  substituted in to solve b and m

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Solution 2 log-log scales

• Using this method for the example, pickingy1
  10 and y2 100, then reading values x11.35 and
  x24.35 from the chart,P1(1.35,10) and
  P2(4.35,100)
• Solving for m m log(100/10) / log(4.35/1.35)
• m1.97 or m2.0 (sig figs)
• Solving for b by/xm, b10/1.352 5.48, b5.5
• Demonstrates that estimation/interpolation on a
  non-linear log scale can be difficult, resulting
  in some error. (b should be 4.9 based on
  original data).
• Note that using a tool such as Excel can improve
  the accuracy of curve fit and coefficient
  calculation

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Solution 2 log-log scales

• Solution Check (read values off of plot)
• b can be determined directly from the plot, as
  the y intercept, at x 1, or in the example, at
  time t1, we can see that b is slightly less than
  5.0, or b4.9 (confirming our calculations)

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Solution 2 log-log scales

• m can be determined as rise/run, but both rise
  and run must be read in log cycles.
• Measure rise (and later, run), and transfer
  distance to the start of one of the log cycles on
  the corresponding axis, and reading how many
  cycles are spanned
• Calculate mrise/run.
• In the example, if one full cycle of time (run)
  is used, the rise can be measured to be 2.0
  cycles (if you extend the plot line)
• Therefore m2.0

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Tips for using log paper

• Look at your data and decide of log cycles.
• Carefully mark numbers on both axes.
• Plot your data.
• With help of Excel, take a look at the straight
  line.
• If possible, calculate the intercept by reading
  off the graph.
• ybemx when x0, then yb.
• ybxm when x1, then yb.
• Pick two points on the grid lines.
• Give proper unit.
• To verify, pick one of x value from given data,
  and compare the computed y value to the given y
  value.

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Tips for determining units
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Example
1. Plot R vs A on rectilinear paper
2. Plot log R vs log A on rectilinear paper
and solve the equation.
3. Plot R vs A on log-log paper and solve
equation
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1. R vs A on rectilinear paper
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2. log R vs log A on rectilinear paper
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3. R vs A on log-log paper
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4. Excel Generated