{"text_format":"3","workbook":"xlsx","n_sheets":4,"sheets":["Sheet1","Sheet2","Defining Function","Sheet3"],"not_rendered":{"charts":1,"images":6,"macros":1,"defined_names":13,"comments":1},"read":"whole","date_system":"1900"}

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[null,"            Euler's Method"]
[null,null,"      y' = f(x, y)"]
[]
[null,"          This graphing utility illustrates "]
[null,"Euler's Method to approximate the solution   "]
[null,"     of any first order differential equation"]
[{"$nulls":3},"y' = f(x, y)"]
[null,"        with initial conditions x0 and y0"]
[null,"        extending from x = a  to x = b."]
[]
["Enter the function in terms of both x and y:"]
[{"$nulls":4}," f(x, y) =","x*sin(y)"]
[]
["Now enter the initial conditions:"]
[null,"The initial value x0 of  x is",null,null,1]
[null,"The initial value y0 of  y is",null,null,5]
[]
["To what value of x should the solutions extend?",{"$nulls":8},"The solution of the D.E.  y' =",null,null,{"$f":"F12","v":"x*sin(y)"}]
[null,null,"xMax =",8]
[{"$nulls":8},"Now select all the Euler approximations you want to see  "]
[]
[{"$nulls":9},true]
[]
[{"$nulls":9},true]
["Click each item below for suggestions and investigations"]
[null,"   Item 1",{"$nulls":7},true]
[null,"   Item 2"]
[null,"   Item 3"]
[null,"   Item 4",{"$nulls":7},false]
[]
[]
["To access/download all the DIGMath Apps for Calculus",{"$nulls":8},false]
["     Click this link:",null,"DIGMath for Calculus"]
[{"$nulls":9},false]
["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."]

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[null,null,"x","y","y' = f(x,y)","y"]
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["                                                     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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