{"text_format":"3","workbook":"xlsx","n_sheets":6,"sheets":["Linear","Quad","Cubic","Quartic","Quintic","Calcs"],"not_rendered":{"charts":5,"macros":1,"external_links":1,"defined_names":46,"comments":5},"read":"whole","date_system":"1900"}

{"sheet":"Linear","index":0,"n_rows":44,"n_cols":15,"rows_kept":44,"merged":[[37,2,37,3]]}
[null,"Building Linear Approximations"]
[null,"    to the Natural Log Function"]
[]
[null,"   This graphing utility lets you try to"]
[null,"  build the best linear approximation"]
[null,"         to the natural log function"]
[{"$nulls":3},"y = ln x"]
[null,null,"centered about x = 1"]
[]
[]
["Enter the interval for x:  "]
[null,null,"xMin =",0.5,{"$f":"IF(aa0<=0,\"This MUST be positive!\",\"\")","v":""}]
[null,null,"xMax =",1.5,{"$f":"IF(aa0<=0,\"This MUST be < 2!\",\"\")","v":""}]
[]
[]
["In the linear approximation ln x » a0 + b0(x -1)"]
[]
["What is the constant coefficient a0  (-1 to 1)?"]
[null,null,100,null,null,{"$f":"(C19-0)/100-1","v":0},null,null,"The largest deviation at 200 evenly spaced points is",{"$nulls":5},{"$f":"MAX(Calcs!E3:E203)","v":0.1931471805599453}]
[]
["What is the linear coefficient b0  (0 to 2)?",{"$nulls":7},"      The sum of the squares of the 200 deviations is",{"$nulls":5},{"$f":"Calcs!F205","v":0.8486023159982068}]
[null,null,100,null,null,{"$f":"(C22-0)/100","v":1}]
[]
["Your linear polynomial is y =",{"$nulls":3},{"$f":"TEXT(aa00,\"0.00\")&\" + \" &TEXT(bb00,\"0.00\")&\"(x - 1)  \"","v":"0.00 + 1.00(x - 1)  "}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[null,null,"   Item 1"]
[null,null,"   Item 2"]
[null,null,"   Item 3"]
[null,null,"   Item 4"]
[null,null,"   Item 5"]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Quad","index":1,"n_rows":48,"n_cols":15,"rows_kept":48,"merged":[[41,2,41,3]]}
[null,"Building Quadratic Approximations"]
[null,"      to the Natural Log Function"]
[]
[null,"   This graphing utility lets you try to"]
[null,"build the best quadratic approximation"]
[null,"         to the natural log function"]
[{"$nulls":3},"y = ln x"]
[null,null,"centered about x = 1"]
[]
[]
["Enter the interval for x:  "]
[null,null,"xMin =",0.5,{"$f":"IF(aa0<=0,\"This MUST be positive!\",\"\")","v":""}]
[null,null,"xMax =",1.6,{"$f":"IF(aa0<=0,\"This MUST be < 2!\",\"\")","v":""}]
[]
[]
["In the quadratic approximation ln x » a0 + b0(x - 1) + c0(x - 1)2"]
[]
["What is the constant coefficient a0  (-1 to 1)?"]
[null,null,100,null,null,{"$f":"(C19-0)/100-1","v":0},null,null,"The largest deviation at 200 evenly spaced points is",{"$nulls":5},{"$f":"MAX(Calcs!L3:L203)","v":0.07000362924574238}]
[]
["What is the linear coefficient b0  (0 to 2)?",{"$nulls":7},"      The sum of the squares of the 200 deviations is",{"$nulls":5},{"$f":"Calcs!M205","v":0.4323832057412932}]
[null,null,106,null,null,{"$f":"(C22-0)/100","v":1.06}]
[]
["What is the quadratic coefficient c0  (-1 to 1)?"]
[null,null,48,null,null,{"$f":"(C25-0)/100-1","v":-0.52}]
[]
["Your quadratic polynomial is y =",{"$nulls":3},{"$f":"TEXT(a0,\"0.00\")&\" +\" &TEXT(b0,\"0.00\")&\"(x - 1)  + \"&TEXT(c0,\"0.00\")&\"(x - 1)^2\"","v":"0.00 +1.06(x - 1)  + -0.52(x - 1)^2"}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[{"$nulls":3},"   Item 6"]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Cubic","index":2,"n_rows":51,"n_cols":15,"rows_kept":51,"merged":[[44,2,44,3]]}
[null,"  Building Cubic Approximations"]
[null,"    to the Natural Log Function"]
[]
[null,"This graphing utility lets you try to"]
[null,"build the best cubic approximation"]
[null,"         to the natural log function"]
[{"$nulls":3},"y = ln x"]
[null,null,"centered about x = 1"]
[]
[]
["Enter the interval for x:  "]
[null,null,"xMin =",0.5,{"$f":"IF(aa0<=0,\"This MUST be positive!\",\"\")","v":""}]
[null,null,"xMax =",1.5,{"$f":"IF(aa0<=0,\"This MUST be < 2!\",\"\")","v":""}]
[]
[]
["In the cubic approximation ln x » a0 + b0(x - 1) + c0(x - 1)2 + d0(x-1)3"]
[]
["What is the constant coefficient a0  (-1 to 1)?"]
[null,null,100,null,null,{"$f":"(C19-0)/100-1","v":{"v":0,"d":"0.00"}},null,null,"The largest deviation at 200 evenly spaced points is",{"$nulls":5},{"$f":"MAX(Calcs!S3:S203)","v":0.0781471805599453}]
[]
["What is the linear coefficient b0  (0 to 2)?",{"$nulls":7},"      The sum of the squares of the 200 deviations is",{"$nulls":5},{"$f":"Calcs!T205","v":0.20700201274466484}]
[null,null,90,null,null,{"$f":"(C22-0)/100","v":{"v":0.9,"d":"0.90"}}]
[]
["What is the quadratic coefficient c0  (-1 to 1)?"]
[null,null,50,null,null,{"$f":"(C25-0)/100-1","v":{"v":-0.5,"d":"-0.50"}}]
[]
["What is the cubic coefficient d0  (-1 to 1)?"]
[null,null,132,null,null,{"$f":"(C28-0)/100-1","v":{"v":0.32000000000000006,"d":"0.32"}}]
[]
["Your cubic polynomial is y =",{"$nulls":3},{"$f":"TEXT(a00,\"0.00\")&\" + \"&TEXT(b00,\"0.00\")&\"(x - 1) + \"&TEXT(c00,\"0.00\")&\"(x - 1)^2 + \"&TEXT(d00,\"0.00\")&\"(x - 1)^3\"","v":"0.00 + 0.90(x - 1) + -0.50(x - 1)^2 + 0.32(x - 1)^3"}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[{"$nulls":3},"   Item 6"]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Quartic","index":3,"n_rows":54,"n_cols":15,"rows_kept":54,"merged":[[47,2,47,3]]}
[null,"Building Quartic Approximations"]
[null,"    to the Natural Log Function"]
[]
[null,"This graphing utility lets you try to"]
[null,"  build the best quartic approximation"]
[null,"         to the natural log function"]
[{"$nulls":3},"y = ln x"]
[null,null,"centered about x = 1"]
[]
[]
["Enter the interval for x:  "]
[null,null,"xMin =",0.3,{"$f":"IF(aa0<=0,\"This MUST be positive!\",\"\")","v":""}]
[null,null,"xMax =",1.7,{"$f":"IF(aa0<=0,\"This MUST be < 2!\",\"\")","v":""}]
[]
[]
["In the quartic approximation ln x » a0 + b0(x - 1) + c0(x - 1)2 + d0(x-1)3 + e0(x - 1)4"]
[]
["What is the constant coefficient a0  (-1 to 1)?"]
[null,null,100,null,null,{"$f":"(C19-0)/100-1","v":{"v":0,"d":"0.00"}},null,null,"The largest deviation at 200 evenly spaced points is",{"$nulls":5},{"$f":"MAX(Calcs!S3:S203)","v":0.0781471805599453}]
[]
["What is the linear coefficient b0  (0 to 2)?",{"$nulls":7},"      The sum of the squares of the 200 deviations is",{"$nulls":5},{"$f":"Calcs!T205","v":0.20700201274466484}]
[null,null,100,null,null,{"$f":"(C22-0)/100","v":{"v":1,"d":"1.00"}}]
[]
["What is the quadratic coefficient c0  (-1 to 1)?"]
[null,null,48,null,null,{"$f":"(C25-0)/100-1","v":-0.52}]
[]
["What is the cubic coefficient d0  (-1 to 1)?"]
[null,null,133,null,null,{"$f":"(C28-0)/100-1","v":{"v":0.33000000000000007,"d":"0.33"}}]
[]
["What is the quartic coefficient e0  (-1 to 1)?"]
[null,null,75,null,null,{"$f":"(C31-0)/100-1","v":-0.25}]
[]
["Your quartic polynomial is y =",{"$nulls":3},{"$f":"TEXT(a000,\"0.00\")&\" + \"&TEXT(b000,\"0.00\")&\"(x - 1) + \"&TEXT(c000,\"0.00\")&\"(x - 1)^2 + \"&TEXT(d000,\"0.00\")&\"(x - 1)^3 +\"&TEXT(e000,\"0.00\")&\"(x - 1)^4\"","v":"0.00 + 1.00(x - 1) + -0.52(x - 1)^2 + 0.33(x - 1)^3 +-0.25(x - 1)^4"}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[null,null,"   Item 1"]
[null,null,"   Item 2"]
[null,null,"   Item 3"]
[null,null,"   Item 4"]
[null,null,"   Item 5"]
[null,null,"   Item 6"]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Quintic","index":4,"n_rows":58,"n_cols":15,"rows_kept":58,"merged":[[51,2,51,3]]}
[null,"Building Quintic Approximations"]
[null,"    to the Natural Log Function"]
[]
[null,"    This graphing utility lets you try to"]
[null,"  build the best quintic approximation"]
[null,"         to the natural log function"]
[{"$nulls":3},"y = ln x"]
[null,null,"centered about x = 1"]
[]
[]
["Enter the interval for x:  "]
[null,null,"xMin =",0.3,{"$f":"IF(aa0<=0,\"This MUST be positive!\",\"\")","v":""}]
[null,null,"xMax =",1.7,{"$f":"IF(aa0<=0,\"This MUST be < 2!\",\"\")","v":""}]
[]
[]
["In the quintic approximation ln x » a0 + b0(x - 1) + c0(x - 1)2 + d0(x - 1)3 + e0(x - 1)4 + f0(x - 1)5"]
[]
["What is the constant coefficient a0  (-1 to 1)?"]
[null,null,100,null,null,{"$f":"(C19-0)/100-1","v":{"v":0,"d":"0.00"}},null,null,"The largest deviation at 200 evenly spaced points is",{"$nulls":5},{"$f":"MAX(Calcs!AF3:AF203)","v":0.5801188043259361}]
[]
["What is the linear coefficient b0  (0 to 2)?",{"$nulls":7},"      The sum of the squares of the 200 deviations is",{"$nulls":5},{"$f":"Calcs!AH205","v":5.4665026216615935}]
[null,null,100,null,null,{"$f":"(C22-0)/100","v":{"v":1,"d":"1.00"}}]
[]
["What is the quadratic coefficient c0  (-1 to 1)?"]
[null,null,100,null,null,{"$f":"(C25-0)/100-1","v":{"v":0,"d":"0.00"}}]
[]
["What is the cubic coefficient d0  (-1 to 1)?"]
[null,null,136,null,null,{"$f":"(C28-0)/200-1","v":{"v":-0.31999999999999995,"d":"-0.32"}}]
[]
["What is the quartic coefficient e0  (-1 to 1)?"]
[null,null,100,null,null,{"$f":"(C31-0)/100-1","v":{"v":0,"d":"0.00"}}]
[]
["What is the quintic coefficient f0  (0 to 0.5)?"]
[null,null,40,null,null,{"$f":"(C34-0)/200","v":{"v":0.2,"d":"0.200"}}]
[]
["Your quintic polynomial is y =",{"$nulls":3},{"$f":"TEXT(a0000,\"0.00\")&\" + \"&TEXT(b0000,\"0.00\")&\"(x - 1) + \"&TEXT(c0000,\"0.00\")&\"(x - 1)^2 + \"&TEXT(d0000,\"0.00\")&\"(x - 1)^3 + \"&TEXT(e0000,\"0.00\")&\"(x - 1)^4 + \"&TEXT(f0000,\"0.00\")&\"(x - 1)^5  \"","v":"0.00 + 1.00(x - 1) + 0.00(x - 1)^2 + -0.32(x - 1)^3 + 0.00(x - 1)^4 + 0.20(x - 1)^5  "}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[null,null,"   Item 1"]
[null,null,"   Item 2"]
[null,null,"   Item 3"]
[null,null,"   Item 4"]
[null,null,"   Item 5"]
[null,null,"   Item 6"]
[]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Calcs","index":5,"n_rows":205,"n_cols":35,"rows_kept":205}
[null,"Linear",{"$nulls":6},"Quadatic",{"$nulls":6},"Cubic",{"$nulls":6},"Quartic",{"$nulls":6},"Quintic"]
["x","ln x","P1(x)","Difference","ABS","Squares",null,"x","ln x","P2(x)","Difference","ABS","Squares",null,"x","ln x","P3(x)","Difference","ABS","Squares",null,"x","ln x","P4(x)","Difference","ABS","Squares",null,"x","ln x","P5(x)","Difference","ABS","Squares"]
[{"$f":"aa0","v":{"v":0.5,"d":"1"}},{"$f":"LN(A3)","v":{"v":-0.6931471805599453,"d":"-0.69315"}},{"$f":"aa00+bb00*(A3-1)","v":{"v":-0.5,"d":"-0.50000"}},{"$f":"C3-B3","v":{"v":0.1931471805599453,"d":"0.19315"}},{"$f":"ABS(D3)","v":{"v":0.1931471805599453,"d":"0.19315"}},{"$f":"D3^2","v":{"v":0.03730583335825611,"d":"0.0373"}},null,{"$f":"a","v":0.5},{"$f":"LN(H3)","v":{"v":-0.6931471805599453,"d":"-0.69315"}},{"$f":"a0+b0*(A3-1)+c0*(A3-1)^2","v":{"v":-0.66,"d":"-0.66000"}},{"$f":"J3-I3","v":{"v":0.033147180559945255,"d":"0.03315"}},{"$f":"ABS(K3)","v":{"v":0.033147180559945255,"d":"0.03315"}},{"$f":"K3^2","v":{"v":1.0987355790736126E-3,"d":"0.0011"}},null,{"$f":"aa","v":{"v":0.5,"d":"0.50"}},{"$f":"LN(O3)","v":{"v":-0.6931471805599453,"d":"-0.69315"}},{"$f":"a00+b00*(O3-1)+c00*(O3-1)^2+d00*(O3-1)^3","v":{"v":-0.615,"d":"-0.61500"}},{"$f":"Q3-P3","v":{"v":0.0781471805599453,"d":"0.07815"}},{"$f":"ABS(R3)","v":{"v":0.0781471805599453,"d":"0.07815"}},{"$f":"R3^2","v":{"v":0.006106981829468692,"d":"0.0061"}},null,{"$f":"aaa","v":{"v":0.3,"d":"0.30"}},{"$f":"LN(V3)","v":{"v":-1.2039728043259361,"d":"-1.20397"}},{"$f":"a000+b000*(V3-1)+c000*(V3-1)^2+d000*(V3-1)^3+e000*(V3-1)^4","v":{"v":-1.1280149999999998,"d":"-1.12802"}},{"$f":"X3-W3","v":{"v":0.07595780432593635,"d":"0.07596"}},{"$f":"ABS(Y3)","v":{"v":0.07595780432593635,"d":"0.07596"}},{"$f":"Y3^2","v":{"v":0.005769588038017235,"d":"0.0058"}},null,{"$f":"aaaa","v":{"v":0.3,"d":"0.30"}},{"$f":"LN(AC3)","v":{"v":-1.2039728043259361,"d":"-1.20397"}},{"$f":"a0000+b0000*(AC3-1)+c0000*(AC3-1)^2+d0000*(AC3-1)^3+e0000*(AC3-1)^4+f0000*(AC3-1)^5","v":{"v":-0.623854,"d":"-0.62385"}},{"$f":"AE3-AD3","v":{"v":0.5801188043259361,"d":"0.58012"}},{"$f":"ABS(AF3)","v":{"v":0.5801188043259361,"d":"0.58012"}},{"$f":"AF3^2","v":{"v":0.33653782713255376,"d":"0.3365"}}]
[{"$f":"A3+(bb0-aa0)/200","v":0.505},{"$f":"LN(A4)","v":{"v":-0.6831968497067772,"d":"-0.68320"}},{"v":-0.495,"d":"-0.49500"},{"$f":"C4-B4","v":{"v":0.1881968497067772,"d":"0.18820"}},{"$f":"ABS(D4)","v":{"v":0.1881968497067772,"d":"0.18820"}},{"$f":"D4^2","v":{"v":0.03541805423955528,"d":"0.0354"}},null,{"$f":"H3+(b-a)/200","v":0.5055},{"v":-0.6822072405216111,"d":"-0.68221"},{"v":-0.652113,"d":"-0.65211"},{"$f":"J4-I4","v":{"v":0.030094240521611026,"d":"0.03009"}},{"$f":"ABS(K4)","v":{"v":0.030094240521611026,"d":"0.03009"}},{"$f":"K4^2","v":{"v":9.0566331257257516E-4,"d":"0.0009"}},null,{"$f":"O3+(bb-aa)/200","v":0.505},{"v":-0.6831968497067772,"d":"-0.68320"},{"v":-0.60682446,"d":"-0.60682"},{"$f":"Q4-P4","v":{"v":0.07637238970677718,"d":"0.07637"}},{"$f":"ABS(R4)","v":{"v":0.07637238970677718,"d":"0.07637"}},{"$f":"R4^2","v":{"v":0.005832741909523845,"d":"0.0058"}},null,{"$f":"V3+(bbb-aaa)/200","v":0.307},{"v":-1.18090753139494,"d":"-1.18091"},{"v":-1.1102173993102502,"d":"-1.11022"},{"$f":"X4-W4","v":{"v":0.07069013208468977,"d":"0.07069"}},{"$f":"ABS(Y4)","v":{"v":0.07069013208468977,"d":"0.07069"}},{"$f":"Y4^2","v":{"v":0.004997094774150886,"d":"0.0050"}},null,{"$f":"AC3+(bbbb-aaaa)/200","v":0.307},{"v":-1.18090753139494,"d":"-1.18091"},{"v":-0.6184665612973387,"d":"-0.61847"},{"$f":"AE4-AD4","v":{"v":0.5624409700976013,"d":"0.56244"}},{"$f":"ABS(AF4)","v":{"v":0.5624409700976013,"d":"0.56244"}},{"$f":"AF4^2","v":{"v":0.31633984484433086,"d":"0.3163"}}]
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