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Graphing Nonlinear Functions Continued

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Graphing Non-Linear Functions II. GE 111 Engineering Problem Solving. 1. Graphing Nonlinear. Functions (Continued) GE 111 - Lecture 5. Graphing Non-Linear Functions II ... – PowerPoint PPT presentation

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Title: Graphing Nonlinear Functions Continued


1
Graphing Nonlinear Functions(Continued)
  • GE 111 - Lecture 5

2
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

3
Example Ball Drop
4
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
5
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.

6
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

7
Solution 2 log-log scales
P2(4.35, 100)
P3(2, 19)
x1 yb
ybxm
8
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

9
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

10
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

11
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)

12
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

13
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.

14
Tips for determining units
15
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
16
1. R vs A on rectilinear paper
17
2. log R vs log A on rectilinear paper
18
3. R vs A on log-log paper
19
4. Excel Generated
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