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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 19721, 632]*) (*NotebookOutlinePosition[ 20582, 663]*) (* CellTagsIndexPosition[ 20538, 659]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{Cell["\<\ Exericse 03 Set, Rule Assignments, and SetDelayed\ \>", "Title", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Copyright ", StyleBox["\[Copyright]", FontSlant->"Plain"], " 1993 by Bill Titus, Carleton College, \nDepartment of Physics and \ Astronomy, Northfield, MN 55057-4025\nSeptember 6, 1993" }], "Subsubtitle", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Topics and Skills", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. Using symbols for calculations.\n2. Immediate assignment with ", StyleBox["Set", FontFamily->"Helvetica"], ".\n3. Multiple inputs in a single cell.\n4. Semicolon to suppress \ output.\n5. Help using ", StyleBox["?", FontFamily->"Courier"], "\n6. Immediate assignment with replacement rules.\n7. ", StyleBox["Table[]", FontFamily->"Courier"], " and ", StyleBox["TableForm[]", FontFamily->"Courier"], ".\n8. Lists - a brief introduction.\n9. ", StyleBox["Clear[]", FontFamily->"Courier"], ".\n10. MMA's symbol for the exponential constant e.\n11. Assignment \ with ", StyleBox["SetDelayed", FontFamily->"Helvetica"], "." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Problem", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ A ball is thrown vertically upward at 20.0 m/s from the top of the Olin \ Hall of Science. What's the ball's velocity and position for times 0.5, \ 1.0, 1.5, 2.0, 2.5, and 3.0 seconds after launch?\ \>", "Special1", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ The basic free fall equations, assuming the x-axis points vertically upward \ and the origin of the coordinate system is at the launch point, are\ \>", "Special1", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox[" v[t] = v[0] - gt\n x[t] = v[0]t - gt", FontFamily->"Courier"], StyleBox["2", FontFamily->"Courier", FontVariations->{"CompatibilityType"->"Superscript"}], StyleBox["/2", FontFamily->"Courier"] }], "Special1", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "The initial information is v[0] = 20.0 m/s and g = 9.8 m/s", StyleBox["2", FontVariations->{"CompatibilityType"->"Superscript"}], ". " }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Solution using Symbols", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell[TextData[{ "Step 1 - You're going to use symbols to help out in doing your multiple \ calculations. You'll first enter the general equations that you'll use. \ Type in the following commands, pressing ", StyleBox["", FontFamily->"Helvetica"], " each time you're done with a single line. Be sure there are blank \ spaces between the different symbols. Describe what happens." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ g = 9.8 vo = 20.0 v = vo - g t x = vo t - (1/2) g t^2\ \>", "Special3", CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Comment 1", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. Symbols are used to stand for quantites, including numbers, equations, \ and even collections of other symbols. Use lower case letters for your \ symbols. Why?\n2. The equal sign (also called ", StyleBox["Set", FontFamily->"Helvetica"], ") is an immediate assignment. How can you tell it is from what you just \ entered?\n3. What would happen if you didn't have a space between, say, g \ and t? Try it out by editing your previous input. What do you find?\n4. \ Note that all the inputs and outputs are in separate cells." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ "Step 2 - Now you're going to enter all the expressions together in a \ single cell. To do so, retype the symbols in Step 1, but now type a return \ after each line except the last one where you should press ", StyleBox["", FontFamily->"Helvetica"], ". Describe what happens." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Comment 2", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. Note that all the inputs are in a single cell but the outputs are in \ separate cells.\n2. To what does the ", StyleBox["In[]", FontFamily->"Courier"], " index refer in a cell containing multiple inputs?\n3. What are the \ advantages of having all the equations in a single cell? \n4. What if you \ didn't want to have the results for g and vo echoed to the screen. In \ MMA, a semicolon after a line suppresses its output." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ "Step 3 - Edit the previous input cell and put a semicolon after the lines \ corresonding to g and vo. Then send the calculations to the kernel by \ clicking the cursor anyplace in the cell and pressing ", StyleBox["", FontFamily->"Helvetica"], ". Describe what happens." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Comment 3", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ 1. What are the advantages and disadvantages of using semicolons to suppress \ output? 2. MMA doesn't care about extra spaces in equations, and you can use extra \ spaces, including indents, to make your equations more readiable.\ \>", "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ "Step 4 - Go to your previous set of equations and put in spaces so that \ the equal signs are all vertically aligned. Then ", StyleBox["", FontFamily->"Helvetica"], " the results." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Step 5 - You're now going to find v and x for t = 0.5. To do so, \ open a cell below and type in t = 0.5, followed by v and x, all on \ separate lines. Then press ", StyleBox["", FontFamily->"Helvetica"], ". What do you find? " }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Step 6 - Now, in the cell you just created, change t to 1.0 and press \ ", StyleBox["", FontFamily->"Helvetica"], ". What do you now find? Speculate on what MMA is doing." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Step 7 - Verify your guess by first entering ", StyleBox[" t", FontFamily->"Courier"], ", ", StyleBox["v", FontFamily->"Courier"], ", and ", StyleBox["x", FontFamily->"Courier"], " in a single cell, but on separate lines. Then, in separate cells, enter \ ", StyleBox["?t", FontFamily->"Courier"], ", ", StyleBox["?v", FontFamily->"Courier"], " , and ", StyleBox["?x", FontFamily->"Courier"], " . What do you find?" }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Comment 7", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. The symbol ", StyleBox["?", FontFamily->"Courier"], " is your friend. It gives help and is an extremely useful MMA command.\n\ 2. You can't put ", StyleBox["?", FontFamily->"Courier"], " on multiple lines.\n3. What's crucial here is that t wasn't assigned \ a value before v and x were defined. This wasn't true with the symbols \ g and vo. That's where immediate assignment comes into play. \n4. What do \ you think ", StyleBox["Global`", FontFamily->"Courier"], " means? " }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell[TextData[{ "Step 8 - The following is a replacement technique that allows t not to \ be permanently replaced as it was in the previous calculation. Look over the \ cell below and then activate it by pressing ", StyleBox["", FontFamily->"Helvetica"], "." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Clear[t] v /. t -> 1.0 x /. t -> 1.0\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Comment 8", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. Not surprisingly, the function ", StyleBox["Clear[] ", FontFamily->"Courier"], "removes previous values set for symbols. It can contain more than one \ symbol provided commas separates the symbols. You might want to use ", StyleBox["Clear[]", FontFamily->"Courier"], " before introducing any collection of symbols, even if it isn't necessary \ to do so.\n2. You can read ", StyleBox["/.", FontFamily->"Courier"], " as \"given\".\n3. In the rule ", StyleBox["t -> 1.0", FontFamily->"Courier"], ", you can read ", StyleBox["->", FontFamily->"Courier"], " as \"goes to\" or \"replaced by\"." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ " Step 9- Check to see what the current value is for the symbol ", StyleBox["t", FontFamily->"Courier"], " in two different ways." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Step 10 - Evaluate v and x for t = 1.5 in a cell below without \ permanently altering the value for t. \ \>", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell[TextData[{ "Step 11 - In your free fall problem, you have lots of values for v and \ x to calculate -- the values corresponding to the times 0.0, 0.5, 1.0, \ 1.5, 2.0, 2.5, and 3.0. You can automate such repeated calculations \ using the ", StyleBox["Table[]", FontFamily->"Courier"], " command. Type in the following and describe what happens." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\t Table[v, {t, 0.0, 3.0, 0.5}]", "Special3", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ "Step 12 - Get help on the command ", StyleBox["Table[]", FontFamily->"Courier"], " by entering ", StyleBox["?Table", FontFamily->"Courier"], " in the cell below." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Comment 12", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. Describe the structure of ", StyleBox["Table[]", FontFamily->"Courier"], ", including its arguments. Note that square brackets are used to contain \ the arguments of the function, and curly brackets are used to form lists. \ Lists are very important in MMA.\n2. What kind of object is the iterator?\n\ 3. Why does the iterator have so many different forms? \n4. The output is a \ list.\n5. The symbol ", StyleBox["expr", FontFamily->"Courier"], " can, itself, be a list. If ", StyleBox["expr", FontFamily->"Courier"], " were ", StyleBox["{t, v}", FontFamily->"Courier"], ", a list of values of both t and v would be produced. Try this out by \ editing the cell in Step 11.\n6. If you want the list in tabular form, place \ the ", StyleBox["Table[] ", FontFamily->"Courier"], "command as the argument in the MMA function ", StyleBox["TableForm[]", FontFamily->"Courier"], "." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[TextData[{ "Step 13 - Get help of the command ", StyleBox["TableForm[]", FontFamily->"Courier"], " and then produce a list of t, v, and x in tabular form over the \ desired time range." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Comment 13", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ 1. Note the usefulness of the nesting property of functions in MMA.\ \>", "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell[TextData[{ "Step 14 - You could assign a symbol to the table to begin with and then \ use ", StyleBox["TableForm[] ", FontFamily->"Courier"], "on that name. Examine the code in the following cell and then activate \ it. What do you find?" }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Clear[myTable] myTable = Table[{t, v, x}, {t, 0.0, 3.0, 0.5}]; TableForm[myTable]\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell["\<\ Step 15 - Suppose you want to repeat the series of time calculations for \ v[t] and x[t], but now with vo = 50 m/s. Further suppose you want to \ print the results in a table with columns corresponding to t, v[t], and \ x[t]. Do this in a single cell below. Remember to use semicolons to \ suppress unnecessary output.\ \>", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["Comments 15", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. If you had not defined vo before defining v and x, you could have \ used your old code. Note the importance of the order in which symbols are \ defined when one uses ", StyleBox["Set", FontFamily->"Helvetica"], "." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell[TextData[{ "Step 16 - The delayed assignment, ", StyleBox[":=", FontFamily->"Courier"], ", (also called ", StyleBox["SetDelayed", FontFamily->"Helvetica"], ") doesn't assign a value until the symbol is called. It can be used to \ avoid some predefinition problems arising from ", StyleBox["Set", FontFamily->"Helvetica"], ". Verify that the following code produces two different tables if ", StyleBox["SetDelayed", FontFamily->"Helvetica"], " is used in the definitions of v and x, and two identical tables if \ ", StyleBox["Set", FontFamily->"Helvetica"], " is used. " }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Clear[g, vo, v, x, t] g = 9.8; vo = 20.0; v := vo - g t x := vo t - (1/2) g t^2 Table[{t, v, x}, {t, 0.0, 3.0, 0.5}] vo = -20.0; Table[{t, v, x}, {t, 0.0, 3.0, 0.5}]\ \>", "Input", CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Clear[g, vo, v, x, t] g = 9.8; vo = 20.0; v = vo - g t; x = vo t - (1/2) g t^2; Table[{t, v, x}, {t, 0.0, 3.0, 0.5}] vo = -20.0; Table[{t, v, x}, {t, 0.0, 3.0, 0.5}]\ \>", "Input", CellMargins->{{45, Inherited}, {Inherited, Inherited}}, CellLabelMargins->{{6, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Comment 16", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. Some people prefer to use ", StyleBox["Set", FontFamily->"Helvetica"], " and proper sequencing of symbol definitions over ", StyleBox["SetDelayed", FontFamily->"Helvetica"], ". The choice is yours." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Practice Problems", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[CellGroupData[{Cell["\<\ 1. A block of mass m = 2.0 kg is projected at an initial velocity of 20 \ m/s along a horizontal surface. Air resistance is present and modeled with \ a force proportional to the block's velocity, i.e., -cv, where c is a \ positive constant. Assume that c = 0.50 kg/s. Under these circumstances, \ Newton would predict that\ \>", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox[" ", FontWeight->"Bold"], StyleBox["v[t] = v[0]e", FontFamily->"Courier", FontWeight->"Bold"], StyleBox["-ct/m", FontFamily->"Courier", FontWeight->"Bold", FontVariations->{"CompatibilityType"->"Superscript"}] }], "Special2", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ StyleBox["Using ", FontWeight->"Bold"], StyleBox["Set", FontFamily->"Helvetica", FontWeight->"Bold"], StyleBox[", produce a table of values of t and v for t going from \ 0.0 s to 5.0 s in steps of 1.0 s. You'll need to know that MMA's symbol \ for the exponential constant e is ", FontWeight->"Bold"], StyleBox["E", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[". Arrange your code so that you could easily vary m, c, and \ vo, and that the only output that occurs is the final table. Be sure your \ start your code with ", FontWeight->"Bold"], StyleBox["Clear[vo, c, m, v]", FontFamily->"Courier", FontWeight->"Bold"], StyleBox[" .", FontWeight->"Bold"] }], "Special2", CellMargins->{{22, Inherited}, {Inherited, Inherited}}, ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell["Comment", "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "1. Find out information about the symbol ", StyleBox["E", FontFamily->"Courier"], " using ", StyleBox[" ?", FontFamily->"Courier"], "\n2. Describe how you did the calculation.\n3. Remember to use indents \ and spaces to improve the readability of your code." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]], Cell[CellGroupData[{Cell[TextData[{ "2. Consider the following code. What are the values of x1 and x2 at \ the appropriate places? Verify your conclusion by entering the code and then \ using ", StyleBox["?", FontFamily->"Courier"], " to find out information about x1 and x2." }], "Subsection", ImageRegion->{{0, 1}, {0, 1}}], Cell["\<\ Clear[vel, time, x0, x1, x2] vel = 10.0; time = 20.0; x1 = x0 + vel*time; x2 := x0 + vel*time x1 x2 x0 = 20.0; vel = 5.0; time = 10.0; x1 x2\ \>", "Input", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]], Cell[CellGroupData[{Cell["Save Your Notebook", "Section", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "Remove any output cells from this notebook and then use ", StyleBox["Save As", FontFamily->"Helvetica"], " to store your notebook under the name ", StyleBox["myEx03", FontFamily->"Helvetica"], "." }], "Special1", ImageRegion->{{0, 1}, {0, 1}}]}, Open]]}, Open]] }, FrontEndVersion->"4.1 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, WindowToolbars->{}, CellGrouping->Manual, WindowSize->{499, 599}, WindowMargins->{{147, Automatic}, {23, Automatic}}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False} ] (******************************************************************* Cached data follows. 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