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// This file will help you learn to use CC.
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// Lines following // are called COMMENTS and
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// are ignored by CC.  Comments can be used to
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// annotate your calculations.
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// As you read this file, place the cursor on
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// each line and push <Enter>.  That will cause
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// the line to be executed, so you will be able
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// to see what it does.
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// CC runs best off a hard disk.  If you are   
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  using a floppy disk, then remove the program
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  disk and put a blank, formatted disk in the 
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  disk drive.  This disk may be necessary for 
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  storing graphics images for recall.
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2+3   // execute this line
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// If you executed 2+3, you saw 5 as the ANSwer
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// and the Last Result.  Enter your own formula
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// on the blank line below and execute it.
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// Anytime you want to interrupt this lesson to
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// try a function of your own, just push 
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// function key F3 a few times to open up empty
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// lines in the window.  Try pushing F3 now.
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// To erase a line, put the cursor on it and
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// press Shift-F4.  Altering the screen will 
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// not change the file DEMO.CC on your disk.
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// You can use transcendental functions with CC
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// Try the following lines:
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3.6^2.1   // ^ means exponentiation
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Sin(2.4)
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Exp(-30)  // very large and small numbers are  
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           represented in exponential notation
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ln(1.3)   // natural logarithm
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sin(3.4)-4*ln(2.7)   //* means multiply
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sqrt(18.27)   // sqrt means square root
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// CC includes an on-line help facility.  To 
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// read about CC's functions, press function 
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// key F1 and choose topic H.
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// Push <ESC> to leave Help and return here.
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// CC can calculate with complex numbers.  If
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// you are interested in complex math, try the
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// following lines.
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(1+Œ)/(3-4Œ)  // enter Œ=sqrt(-1) with alt-i
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exp(3+4Œ)
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Asin(3)    // real argument, complex result
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Ln(-1)
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// You can assign the result of a calculation
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// to a variable.  Watch the Output window as
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// you execute these lines:
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a = 4
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b = (3.2 - 5.4)/2.1
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c = a-3 //variables can be used in calculations
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// Test yourself -- answers at end of file
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// 1.  Find the hypotenuse of a right triangle 
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      with legs of length 3.8 and 7.2.
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// 2.  What is the final balance if $5000 is   
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      invested at 7% for 8 years
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// You can also define functions with CC, which
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// can be evaluated, graphed, integrated,
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// differentiated, and solved for roots.
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f(x) = x^2 + sin(x)  // execute this line to   
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                       define f(x)
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y = f(2)
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z = 3+f(sin(y))
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// There are now more variables defined than
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// can be shown.  Use function keys F5 and F6
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// to scroll the output window.
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// Now we will graph f(x).  Push <Enter> when  
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  you wish to stop viewing the graph and      
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  return to this screen.
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graph(f(x),x)
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// To see more or less of the curve, you can 
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// alter the viewing window with the WINDOW
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// command.  The syntax is 
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//    WINDOW(xmin, xmax, ymin, ymax)
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Window(-3,3,-.5,10)
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Graph(f(x),x)
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// Compare to curve y = x^2
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Graph(x^2,x)
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// You can graph any expression like f(x) or
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// x^2 or even x^2*f(x).  
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// You can display as many curves as you want 
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// on one graph.  To erase the graph and start
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// over, enter the command ERASE or resize the
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// window with WINDOW.
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// Read more about graphing in the help file.
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// To differentiate the function f(x) at x=3,  
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  i.e. to calculate f'(3), enter the command
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d1 = dif(f(x),x=3)
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// You can also differentiate an expression
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d2 = dif(x^2+sin(x),x=3)   // will equal d1
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// You can define one function to be the       
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  derivative of another
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g(x) = dif(f(x),x)
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window(-1,1,-1,3)
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graph(f(x),x)
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graph(g(x),x)
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Write@(0.6,0.85,'F(x)')  // graphs can be 
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Write@(0.5,1.75,'dF/dx') // labeled
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// To see the symbolic derivative of f(x),     
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  differentiate f(x) with respect to an       
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  undefined variable name.
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df = dif(f(w),w)
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// To find where the graph of x^2+sin(x)       
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  crosses the x-axis, we solve the equation   
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  f(p) = 0
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Solve(f(p)=0, p=-1)   // p=-1 is first guess
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// Recall what the graph of f(x) looks like by 
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  pressing function key F9 to review the last 
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  graph.  To find the area between the graph  
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  of f(x) and the x-axis, we integrate f(x)   
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  between p (where the graph crosses the x-   
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  axis) and 0. 
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// This would not be possible without first    
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  solving for p.
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Area := -IN(f(x),x=p to 0)
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// CC includes a version of the integration    
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  operator, INTEG, that shows a graph of the  
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  area being integrated.  It returns the same 
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  answer as the ordinary integration operator 
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  IN.
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Area = -INTEG(f(x),x=p to 0)
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// CC will graph parametric and polar equations
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  It will also graph surfaces in three dimen- 
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  sions.  You can read about all these opera- 
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  tions in the help file.  Here's an example  
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  of 3D graphing.
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Graph3d(2x^2-y^2, x=-1,1, y=-1,1)
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// You can change the shading of 3d graphs by  
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  pressing the keys 1 (opaque--what you see   
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  first), 2 (striped), 3 (striped the other   
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  way), 4 (transparent).  Press F9 to recall  
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  the last graph, then press the number keys  
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  to change its shading.
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// You can also rotate the 3d graph on the     
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  screen by pressing the keys x, X, y, Y, z, Z
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// Test yourself
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// 3.  Graph the curve cos(x)-x.  Find the     
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      intersections of this curve with the x- 
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      axis, the area between the curve and the
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      x-axis, and the slope of the curve where
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      it crosses the x-axis.
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// 4.  Find the first three points to the right
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      of the y-axis where the line y = x meets
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      the graph of y = tan(x).
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// If you aren't interested in vectors, skip   
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  the next section.  CC3 will do vector arith-
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  metic and other operations with arrays and  
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  lists.  Note that if you define
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v = (3, 5, 6)   // execute this line
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// v is displayed in the output window as three
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  parts: v[1], v[1], v[3].
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// Let's do some vector arithmetic.
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w = (4, 6, 3)
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u = v+w
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u = 5*v
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u = v^2
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u = v*w  
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// CC is not just a calculator.  It is also a  
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  programming environment.   Push function key
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  F10 to view the "Scratchpad", where CC's    
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  programs are written, then push F10 again to
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  return to this screen.
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// Now that you have viewed the Scratchpad,    
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  execute the function PC(x,n)
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a1=PC(1,10)    // approximate solution
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a2 = sin(1)    // exact solution
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a3 = PC(1,20)  // better approximation
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a4 = PC(1,100) // excellent approximation
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// The algorithm fails for x > 1.5 because CC
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// always computes positive square roots.
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// The manual has many more examples of using  
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  programming with CC.
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// These notes have not mentioned the          
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  constructors SUM, PRODUCT, and LIST; CC's   
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  statistical features; symbolic calculations;
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  saving work with disk and printer; or       
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  advanced programming techniques.
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// All these points are explained in the manual
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// *** Answers to Test Yourself ***
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Ans1 = sqrt(3.8^2 + 7.2^2)
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Ans2 = 5000*(1.07)^8
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// Ans3
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f(x) = cos(x)-x
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window(0,1,-1,1)
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graph(f(x),x)
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solve(f(p)=0,p=1)       // curve crosses x-axis
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Ans3_1 = in(f(x),x=0,p)  // area
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Ans3_2 = dif(f(x),x=p)     // slope
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// Ans4
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window(0,12,0,12)
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graph(tan(x),x)
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graph(x,x)
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// using the crosshairs, we see that the x-    
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  coordinates of the intersections are approx-
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  imately 4.504, 7.735, 10.893
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solve(Tan(x)=x, x=4.505)
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Ans4_1 = x
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solve(tan(x)=x, x=7.735)
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Ans4_2 = x
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solve(tan(x)=x, x=10.893)
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Ans4_3 = x
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È

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// This is CC's Scratchpad, where you can write programs using CC's features.
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// One warning.  If you push <Enter> while in the Scratchpad, you will split
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// the line you are on at the cursor and push the lines below the cursor down
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// one line.  To avoid rearranging the lines in the Scratchpad, use the up-
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// and down-arrow keys to move the cursor around the Scratchpad.
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// Below is a sample program that uses the predictor-corrector method to
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// compute the solution to the differential equation:
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//              y'(t) = sqrt(1-y(t)^2)
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//              y(0) = 0
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// Of course, the exact solution is y(t) = sin(t).  To execute this program,
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// push function key F10 to return to the calculator window and follow the
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// instructions there.
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Function PC(x,n)
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// use n-step predictor corrector method to solve the initial value problem  
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  above and return the estimated value for y(x).
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  dt = x/n             // use n steps between 0 and x
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  t = 0                // starting value for t
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  y = 0                // initial value for y
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  f(y) = sqrt(1-y^2)   // f(y) is slope of y(t) 
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  for j = 1 to n do
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      slope1 = f(y)    
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      y1 = y + slope1*dt   // first estimate for next value of y
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      slope2 = f(y1)
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      avgslope = (slope1+slope2)/2
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      y = y+avgslope*dt    // corrected estimate for next value of y
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    end
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  return(y)
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end      
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