DERIVE for Windows version 5.05 DfW file saved on 30 Sep 2002
f(x):=ê^(-x)·(x^4 - x^2)
hCross:=APPROX(- 1032258064516129/400000000000000)
vCross:=APPROX(- 19166666666666667/10000000000000000)
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\viewkind4\uc1\pard\qc\cf1\b\f0\fs24 The Relationship Between
\par The Graph of a Function & The Graph
\par of Its Derivative
\par \pard\b0
\par \cf0\f1 NOTE: \cf2 While working with this example we recommend that you open a 2D\f2 -p\f1 lot window. Under the \ul O\ulnone ptions menu turn on the options to "Approximate before Plotting" and "Change Plot Colors." Then return to this window and choose the option Window>Tile Windows Vertically. That way you will be able to work through the exploration and see the graphical display side by side.
\par \cf0
\par \b Displaying the Graph of a Function and Its Derivative\b0
\par
\par In the following expression we use Derive's function notation. The assignment operator (:=)is used for assigning values or expressions to a variable or function. Note: the equals sign (=) is used in equations and logical expressions. Do not confuse the use of these two different symbols.\cf1\f3
\par }
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\viewkind4\uc1\pard\cf1\f0\fs24 The function EXP(-x) is the exponential function which can also be written as \cf0\i\f1 e\up12\fs16 -x\up0\fs24 . \i0 This is the more common notation, but we used the function name to stress the fact that we are talking about the exponential function. \cf2 Draw the graph of this function in the 2D\f2 -p\f1 lot window. \f2 Set \f1 the Plot Range on the \i y\i0 -axis to -3 \ul <\ulnone \i y\i0 \ul <\ulnone 5. After doing this \f2 insert annotations in the plot window by \f1 pressing F12. Label each of the places where the curve is increasing (or rising) and each of the places where it is decreasing (or falling). There should be a total of four labels\f2 .\cf1\f1
\par
\par We let Derive differentiate this function by highlighting expression #1 and choosing the \f2 Find Derivative \f1 button labeled with a rounded "d" on the \f2 Algebra Window's \f1 toolbar. This will yield an expression similar to #2.\f3
\par }
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{\colortbl ;\red255\green0\blue0;\red0\green0\blue0;}
\viewkind4\uc1\pard\cf1\f0\fs24 Simplify\f1 this expression and graph the result. \f0 Create annotations \f1 to label all the places where the derivative is negative (graph lies below the \i x\i0 -axis) and where it is positive (graph lies above the \i x-\i0 axis).\cf2 As you see, this makes for a rather crowded graph. The labels seem to be in the same regions. \cf1 Verify this by completing the following table with the appropriate intervals. You will need to use the cross in the 2-D plot window to find the ends of the intervals. Read the coordinates of the \f0 c\f1 ross in the lower left-hand corner of the status bar beneath the window.\f0 \f1 Use the right-click, Edit option to add your observations to the table.\f2
\par }
€8=À©Userð¿¤[["Sign of Derivative","Behavior of Graph of F(x)"],["------------------","------------------------"],[" Positive"," "],[" Negative"," "],[" Zero"," "]]€µç†ÿJ{\rtf1\ansi\ansicpg1252\deff0\deflang1033{\fonttbl{\f0\fmodern\fprq1 Times New Roman;}{\f1\fmodern\fprq1\fcharset0 Times New Roman;}{\f2\fmodern\fprq1\fcharset2 DfW5 Printer;}}
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\viewkind4\uc1\pard\cf1\ul\f0\fs24 Exercise\ulnone
\par \cf2 1) Write a one or two sentence summary of your observations as a result of labeling the graph and filling in the table in expression 3.
\par \cf1
\par We could extend our graphing and labeling to the graph of the second derivative; however, this would tend to make our plot window even more congested. It would also be very difficult to read. We will try something else. \cf2\f1 With the 2D-plot window active, delete all plots \f0 by pressing \f1 C\f0 trl\f1 +\f0 D. Then\f1 ,\f0 under the \ul E\ulnone dit menu choose the option to "Delete All Annotations". \cf1
\par
\par We will take advantage of Derive's "if" statement. This statement operates exactly like the "if" statement of a spreadsheet except that it can handle symbolic as well as numerical calculations. Its form is:
\par if( "Logical test", "action if true","action if false or null")
\par We use as our test the sign of the derivative of \i f\i0 (\i x\i0 ) and choose our action if true to be evaluation of \i f\i0 (\i x\i0 ) and null as our action if false.\f2
\par }
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\viewkind4\uc1\pard\cf1\f0\fs24 Graph this expression.\cf2 You will note that the intervals where the graph appears are the same as the first line that you entered in expression #3. \cf1 Now graph expression #5 and note the intervals that are filled in by this graph and compare to your entry in expression #3.\cf2\f1
\par }
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\viewkind4\uc1\pard\cf1\f0\fs24 What we have done is display the information we displayed before\f1 ,\f0 but in a more compact form. \f1 D\f0 ifferent plot colors are use\f1 d\f0 instead of labeling. \f1 Both \f0 plots are generated based on the sign of the derivative. \f1 However, o\f0 ne color graphs where the graph is increasing (rising). The other graphs where the graph is decreasing (falling).
\par
\par \ul Exercise\ulnone
\par \cf2 2) What is happening on the graph where the derivative is zero?\cf1
\par
\par \b Including Inforamation on the Second Derivative\b0
\par
\par To evaluate the second derivative of \i f\i0 (\i x\i0 ) we author the expression, dif(f(x),x,2). The result is shown in expression #6.\f2
\par }
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{\colortbl ;\red255\green0\blue0;\red0\green0\blue0;}
\viewkind4\uc1\pard\cf1\f0\fs24 Simplify\f1 this expression and draw the graph of the result in the 2-D plot window with the colored graph of \i f\i0 (\i x\i0 ). Label the regions where the grap\f0 h\f1 of \i f\i0 (\i x\i0 ) is concave upward (holds water) and where it is concave downward (spills water). Also label where the graph of the second derivative is positive and negative. Do the regions on each graph coincide? Do they agree with those in the table of expression #3?
\par
\par \cf2 Actually, the first question in the previous paragraph is a little hard to answer since it is not easy to spot where the grap\f0 h\f1 turns from concave up to concave down. The answer to the question is "yes." Let's confirm this. \cf1 Once again \f0 delete all plots\f1 \f0 and \f1 annotations.\cf2
\par
\par What we are going to do is repeat the colored graphing exercise that we used for the first derivative except that we will use the second derivative. \cf1 Graph each of expressions #7 and #8.\f2
\par }
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\viewkind4\uc1\pard\cf1\ul\f0\fs24 Exercise\ulnone
\par \cf2 3) Click on the "Trace Plots" option in the 2D\f1 -p\f0 lot window and locate the points where the graph changes concavity. Use this information to make a table similar to the one in expression #3 for concavity and the sign of the second derivative.\cf1
\par
\par \b And Now, The Whole Story\b0
\par
\par If we know whether the graph of the function is increasing or decreasing and we know the concavity of the graph we are able to make a fairly good representation of the graph by plotting a very few points. In particular, if we know the points where the first derivative changes sign (a peak or a valley) and the points where the second derivative changes sign (an inflection point), then we can give a reasonably accurate picture of the graph of the function.
\par
\par This brings us to our final visualization of the function. We combine the first and second derivative tests using a logical and (\f2\'8f\f0 ) operator. Recal\f1 l\f0 that the "and" is true only when both of the tests are true. Since the first derivative test had two possibilities (we do not include zero at this point) and the second derivative test had two possibilities, we will need to graph our function in four parts. Each will be graphed in a different color.
\par
\par \cf2 Plot each of the following expressions.\cf1\f3
\par }
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\viewkind4\uc1\pard\cf1\f0\fs24 Using the trace function, fill in the following table. As before right click on the table and choose the \ul E\ulnone dit option to fill in your observations.\f1
\par }
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\viewkind4\uc1\pard\cf1\f0\fs24 Now fill in the next table with the action of the graph of \i f\i0 (\i x\i0 ) with respect to increasing or decreasing and the concavity.\cf2\f1
\par }
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\viewkind4\uc1\pard\cf1\f0\fs24 Finally, what about the points where the derivatives are zero? The following exercises address these points.
\par
\par \ul Exercise\ulnone
\par \cf2 4) List the points where the first derivative of \i f\i0 (\i x\i0 ) is zero. In the \ul O\ulnone ptions menu set the Display of Points to be Connected, "No" and Size, "Medium". Plot these points. What do they corresspond to on the graph of \i f\i0 (\i x\i0 )?
\par 5) List the points where the second derivative of \i f\i0 (\i x\i0 ) is zero. In the \ul O\ulnone ptions menu set the Display of Points to be Connected, "No" and Size, "Medium". Plot these points. What do they corresspond to on the graph of \i f\i0 (\i x\i0 )?
\par 6) Is it always the case that there will be a peak or a valley on the graph of \i f\i0 (\i x\i0 ) when the first derivative is zero? HINT: Consider the graph of \i\f1\fs20 x\up10\fs13 3\up0\fs20 .\i0\f0\fs24
\par 7) Is it always the case that there will be a point of inflection on the graph of \i f\i0 (\i x\i0 ) when the second derivative is zero? HINT: Consider the graph of \i\f1\fs20 x\up10\fs13 4 \up0\fs20 .\i0\f0\fs24
\par
\par \cf1\b Summary\b0
\par
\par In this exploration we used a function that you probably do not have a formula to compute the derivative. This was intentional because we wanted you to concentrate on what was happening on the graph. We also wanted to do something that is different from what you may find in your text. The objective was to get you to relate the behavior of the graph of the function to the signs of its first two derivatives.
\par
\par Each of the cases in expressions #9 through #12 yields a characteristic shape for the graph. Sometimes this shape is elongated. Sometimes it is truncated. However, the fact remains that if you know the signs of the first two derivatives on a given interval, you can draw a fairly accurate picture of the graph of the function. This gives you insight into the process that is described by the function.
\par
\par \ul Exercise
\par \cf2\ulnone 8)\cf1 \cf2 Duplicate the exploration of the shape of the graph of some of the following more familiar functions:
\par a. \i f\i0 (\i x\i0 ) = \i\f1\fs20 x\up10\fs13 3\up0\fs20 \i0\f0\fs24
\par b. \i f\i0 (\i x\i0 ) = \i\f1\fs20 x\up10\fs13 4 \i0\up0\f0\fs24
\par c. \i f\i0 (\i x\i0 ) = \i\f1\fs20 x\up10\fs13 3\up0\fs20 - 3x\up10\fs13 2\up0\fs20 + 4x + 1\i0\f0\fs24
\par d. \i f\i0 (\i x\i0 ) = \i\f1\fs20 x\up10\fs13 4\up0\fs20 - x\up10\fs13 2\up0\fs20 \i0\f0\fs24
\par e. \i f\i0 (\i x\i0 ) = sin( \i\f1\fs20 x\up10\fs13 2 \i0\up0\f0\fs24 )
\par f. \i f\i0 (\i x\i0 ) = \i\f1\fs20 3x\up10\fs13 1/3\up0\fs20
\par
\par \cf1\i0 Carl Leinbach: leinbach@gettysburg.edu\f2\fs24
\par }
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