{"text_format":"3","workbook":"xlsx","n_sheets":7,"sheets":["Riemann","TrapRule","MidPoint","Simpson","Sheet2","Defining Function","Sheet1"],"not_rendered":{"charts":5,"images":4,"macros":1,"external_links":1,"defined_names":19,"comments":4},"read":"whole","date_system":"1900"}

{"sheet":"Riemann","index":0,"n_rows":43,"n_cols":14,"rows_kept":43,"merged":[[36,2,36,3]]}
[null,"Riemann Sums and Integration"]
[]
[null,null,"This program uses both "]
[null,"         left- and right-hand Riemann Sums"]
[null,"         to approximate the definite integral"]
[null,"                       of any desired function"]
[{"$nulls":3},"y = f(x)"]
[null,"         on an interval from x = a  to x = b."]
[]
[]
["   Enter the function in terms of x:    f(x) = ",{"$nulls":4},"x^3"]
[]
[]
["Now enter the interval for x:  "]
[null,null,"xMin =",0]
[null,null,"xMax =",2]
[]
[]
[null,"How many subdivisions? (2-100)"]
[null,null,{"$f":"F20+2","v":10},null,null,8]
[]
[]
[]
[]
[]
[{"$nulls":8},{"$f":"\"For the Riemann Sums with \" & TEXT(n,\"0\") & \" subdivisions, \"","v":"For the Riemann Sums with 10 subdivisions, "}]
["Click each item below for suggestions and investigations"]
[null,"   Item 1",{"$nulls":6},{"$f":"\"        the left-hand sums = \" & TEXT(Sheet2!Z5,\"0.00000\")","v":"        the left-hand sums = 3.24000"}]
[null,"   Item 2"]
[null,"   Item 3",{"$nulls":6},{"$f":"\"       and  the right-hand sums = \" & TEXT(Sheet2!Z6,\"0.00000\")","v":"       and  the right-hand sums = 4.84000"}]
[null,"   Item 4"]
[null,"   Item 5",{"$nulls":6},"The actual area under the curve is about ",{"$nulls":4},{"$f":"Sheet2!Y3","v":{"v":3.999999999999915,"d":"4.00000"}}]
[]
[]
[]
["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":"TrapRule","index":1,"n_rows":40,"n_cols":14,"rows_kept":40,"merged":[[33,2,33,3]]}
[null,"       The Trapezoid Rule"]
[]
[null,"        This DIGMath program uses the "]
[null,"                     Trapezoid Rule"]
[null,"     to approximate the definite integral"]
[null,"                   of any desired function"]
[null,null,"  y = f(x)"]
[null,"     on an interval from x = a  to x = b."]
[]
["To change the function or the interval, "]
[null," click on the \"Riemann\" tab below "]
[]
["The function here is y =  f(x) = ",null,null,{"$f":"Riemann!F11","v":"x^3"}]
[]
[]
["The interval for x is:  ",{"$nulls":7},{"$f":"\"Using the Trapezoid Rule with \" & TEXT(nn,\"0\") & \" subdivisions, \"","v":"Using the Trapezoid Rule with 50 subdivisions, "}]
[null,null,"xMin =",{"$f":"xMin","v":0}]
[null,null,"xMax =",{"$f":"xMax","v":2},{"$nulls":4},{"$f":"\"        the approximate area  = \" & TEXT(Sheet2!Y11,\"0.00000\")","v":"        the approximate area  = 4.00160"}]
[]
[{"$nulls":8},"The actual area under the curve is about ",{"$nulls":4},{"$f":"Sheet2!Y3","v":{"v":3.999999999999915,"d":"4.00000"}}]
[null,"How many subdivisions? (2-50)"]
[null,null,{"$f":"F22+2","v":50},null,null,48]
[]
[]
["Click each item below for suggestions and investigations"]
[null,"   Item 1"]
[null,"   Item 2"]
[null,"   Item 3"]
[null,"   Item 4"]
[]
[]
[]
["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":"MidPoint","index":2,"n_rows":40,"n_cols":14,"rows_kept":40,"merged":[[33,2,33,3]]}
[null,"          The Midoint Rule"]
[]
[null,"        This DIGMath program uses the "]
[null,"                     MidPoint Rule"]
[null,"     to approximate the definite integral"]
[null,"                   of any desired function"]
[null,null,"  y = f(x)"]
[null,"       on an interval from x = a  to x = b."]
[]
["To change the function or the interval, "]
[null," click on the \"Riemann\" tab below "]
[]
["The function here is y =  f(x) = ",null,null,{"$f":"Riemann!F11","v":"x^3"}]
[{"$nulls":8},{"$f":"\"Using the MidPoint Rule with \" & TEXT(nnnn,\"0\") & \" subdivisions, \"","v":"Using the MidPoint Rule with 10 subdivisions, "}]
[]
["The interval for x is:  ",{"$nulls":7},{"$f":"\"        the approximate area  = \" & TEXT(Sheet2!Y19,\"0.00000\")","v":"        the approximate area  = 3.98000"}]
[null,null,"xMin =",{"$f":"xMin","v":0}]
[null,null,"xMax =",{"$f":"xMax","v":2},{"$nulls":4},"The actual area under the curve is about ",{"$nulls":4},{"$f":"Sheet2!Y3","v":{"v":3.999999999999915,"d":"4.00000"}}]
[]
[]
[null,"How many subdivisions? (2-50)"]
[null,null,{"$f":"F22+2","v":10},null,null,8]
[]
[]
["Click each item below for suggestions and investigations"]
[null,"   Item 1"]
[null,"   Item 2"]
[null,"   Item 3"]
[null,"   Item 4"]
[]
[]
[]
["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":"Simpson","index":3,"n_rows":41,"n_cols":14,"rows_kept":41,"merged":[[34,2,34,3]]}
[null,"           Simpson's Rule"]
[]
[null,"           This DIGMath program uses "]
[null,"                     Simpson's Rule"]
[null,"     to approximate the definite integral"]
[null,"                   of any desired function"]
[null,null,"  y = f(x)"]
[null,"    on an interval from x = a  to x = b."]
[]
["To change the function or the interval, "]
[null," click on the \"Riemann\" tab below "]
[]
["The function here is y =  f(x) = ",null,null,{"$f":"Riemann!F11","v":"x^3"}]
[]
[]
["The interval for x is:  ",{"$nulls":7},{"$f":"\"Using Simpson's Rule with \" & TEXT(nnn,\"0\") & \" subdivisions, \"","v":"Using Simpson's Rule with 16 subdivisions, "}]
[null,null,"xMin =",{"$f":"xMin","v":0}]
[null,null,"xMax =",{"$f":"xMax","v":2},{"$nulls":4},{"$f":"\"        the approximate area  = \" & TEXT(Sheet2!Y16,\"0.00000\")","v":"        the approximate area  = 4.00000"}]
[]
[{"$nulls":8},"The actual area under the curve is about ",{"$nulls":4},{"$f":"Sheet2!Y3","v":{"v":3.999999999999915,"d":"4.00000"}}]
[null,"How many subdivisions? (2-30)"]
[null,null,{"$f":"2*F22+2","v":16},null,null,7]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[null,"   Item 1"]
[null,"   Item 2"]
[null,"   Item 3"]
[null,"   Item 4"]
[]
[]
[]
["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":"Sheet2","index":4,"n_rows":856,"n_cols":26,"rows_kept":856}
["fmin =",{"$f":"MIN(E3:E403)","v":0},"i","x","y = f(x)",null,"Partition Valuess",{"$nulls":3},"Left-Hand Sums",null,"RHS","      Trapezoids",null,"     MidPoint Rule   ",null,"Simpson's Rule",null,null,"Simpson Calcs",null,null,"Simpson"]
["fmax =",{"$f":"MAX(E3:E403)","v":7.999999999999752},{"$nulls":5},"x","f(x)",null,"x","f(x)","f(x)","x","f(x)","x","f(x)","x","f(x)","n","f(x)","mult","product",null,"Simp ="]
["Graph values",null,1,{"$f":"Riemann!D15","v":0},{"$f":"x^3","v":0},{"$nulls":3},{"$f":"E3","v":0},null,{"$f":"xMin","v":0},{"$f":"0","v":0},{"$f":"0","v":0},{"$f":"xMin","v":0},{"$f":"0","v":0},{"$f":"xMin","v":0},{"$f":"0","v":0},{"$f":"xMin","v":0},{"$f":"0","v":0},0,{"$f":"E655","v":0},1,{"$f":"U3*V3","v":0},{"$f":"E3","v":0},{"$f":"SUM(X3:X403)*B5/3","v":3.999999999999915}]
[null,null,2,{"$f":"D3+Sheet2!B$5","v":0.005},1.2500000000000002E-7,null,0,{"$f":"xMin","v":0},null,0,{"$f":"xMin","v":0},{"$f":"E706","v":0},{"$f":"E707","v":0.008000000000000002},{"$f":"xMin","v":0},{"$f":"E404","v":0},{"$f":"xMin","v":0},{"$f":"E807","v":1.0000000000000002E-3},{"$f":"xMin","v":0},{"$f":"E655","v":0},1,{"$f":"E656","v":1.953125E-3},4,{"$f":"U4*V4","v":7.8125E-3},{"$f":"4*E4","v":5.0000000000000008E-7}]
["delta x = ",{"$f":"(Riemann!D16-Riemann!D15)/400","v":0.005},3,{"$f":"D4+Sheet2!B$5","v":0.01},1.0000000000000002E-6,null,1,{"$f":"H4+dx","v":0.2},null,null,{"$f":"K4+dx","v":0.2},{"$f":"L4","v":0},{"$f":"E707","v":0.008000000000000002},{"$f":"N4+dxx","v":0.04},{"$f":"E405","v":6.4000000000000011E-5},{"$f":"P4+dxxxx","v":0.2},{"$f":"Q4","v":1.0000000000000002E-3},{"$f":"R4+dxxx","v":0.125},0,2,0.015625,{"$f":"IF(2*nnn>T5,2,IF(2*nnn=T5,1,0))","v":2},0.03125,{"$f":"2*E5","v":2.0000000000000003E-6},"LHS=",{"$f":"dx*(SUM(L3:L302)/2)-LOOKUP(n,J4:J315,L4:L315)*dx","v":3.2399999999999993}]
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{"sheet":"Defining Function","index":5,"n_rows":12,"n_cols":1,"rows_kept":12}
["                                                     Defining a Function in Excel"]
[]
["Computers have very specific rules you must follow when defining a function.  If you don't, the program will either not work at all or will work incorrectly."]
[]
["1. The Variable used should be the one named in the instruction, usually x, though sometimes y, t, Q (for an angle theta), "]
[" or even x and y (for a function of two variables). "]
[]
["2.  Order of Operations:  There five arithmetic operations, addition (+), subtraction (-), multiplication (*), division (/), and  exponentiation or power (^).  When computers see any of these operation symbols, they perform the indicated steps in a very particular order.  First, the power is done.  Second, products and quotients of adjacent terms is done.  Third, and last, sums and differences of adjacent terms are done.  So, in 10*x^3 + 5*x – 7, say with x = 2, the first pass only calculates 2^3 and ignores the multiplications and the sums and differences to calculate 2^3 = 8.  The second pass looks at 10*8 + 5*2 – 7 and only does the multiplications, so it produces 80 + 10 -  7.  The third pass looks at the +’s and –‘s to produce 80 + 10 – 7 = 83.  "]
[]
["3.    Parentheses  If you need an operation involving more than two adjacent terms, as in Ö4^3 + 6^2, the computer would take the square root of  4 to produce 2^3 + 6^2;  it would then cube the 2 and square the 6 to produce 8 + 36; and finally it would add the 8 and the 36 to get 44.  If you actually wanted the square root of the entire expression, you would need to use parentheses, as in Ö(4^3 + 6^2) to tell the computer to do all the operations inside the parentheses before taking the square root.  Therefore, it would calculate (4^3 + 6^2) = 64 + 36 = 100 and then take the square root of Ö100  =  10."]
[]
["4.  Built-In Functions  Computer programs have built-in functions for most common algebraic functions.  In Excel, the square root function is denoted by SQRT(…), so SQRT(100) = 10.  Alternatively, you could enter it as 100^.5 or 100^(1/2); notice that parentheses are needed to have the ½ done first.  The common trig functions are SIN (…), COS (…), and TAN (…), where the arguments are automatically interpreted as being in radians, not degrees.  The logarithm to the base 10 is represented by LOG (…) and the logarithm with the base e, known as the natural logarithm, is LN (…). The corresponding exponential function with base e is EXP (x).  If x is the variable, then each of these would be SQRT (x), SIN (x), COS (x), TAN (x), LOG (x) (for log with base 10 ), LN (x) (for log with base e ), and EXP (x)."]

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