\mnb150<nb wrap="0" txtwidth="75" autowidth="0" pprint="1" typeset="1" xface="Times New Roman" xsize="360" xbold="0" xitalic="0" xcolor="#0000FF" slidents="1" sizemult="71" minsize="160" ifont="0 280 255 0 49 Lucida Sans Typewriter" ofont="0 280 16711680 0 49 Lucida Sans Typewriter" tfont="0 280 0 0 34 Trebuchet MS"></nb>ÿ{\rtf1\ansi\deff0\deftab720{\fonttbl{\f0\fswiss MS Sans Serif;}{\f1\froman\fcharset2 Symbol;}{\f2\fswiss\fprq2 System;}{\f3\fswiss\fprq2 Trebuchet MS;}{\f4\fmodern\fprq1 Lucida Sans Typewriter;}}
{\colortbl\red0\green0\blue0;\red255\green0\blue0;}
\deflang1033\pard\ri4\plain\f3\fs72\cf0 Chapter 3  Introduction to calculus 
\par of one variable\plain\f3\fs36\cf0 
\par 
\par 2.1 Declaring functions of one variable
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}g := x -> (x^3+x^2+x+1)/(x^2+x+1)
\par {\pntext\f1\'b7\tab}g := x -> nextprime(floor(x)):
\par {\pntext\f1\'b7\tab}g(21.98)
\par {\pntext\f1\'b7\tab}G := plot::Function2d(g(x), x=0..10, XSubmesh=10):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(G)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}F := plot::Function2d(floor(x), x=-5..5): plot(F)
\par {\pntext\f1\'b7\tab}H := plot::Function2d(x-floor(x), x=-4..4):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(H, Scaling=Constrained)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}g := n -> ithprime(n)-ithprime(n-1):
\par {\pntext\f1\'b7\tab}g(25)
\par {\pntext\f1\'b7\tab}g(2567)
\par {\pntext\f1\'b7\tab}f := x -> [ 2^n $ n=1..floor(x)]:
\par {\pntext\f1\'b7\tab}f(12)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}reset():
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 CharInt01 := x -> piecewise([x<0,0],[x<=1,1],[x>1,0]):
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}F1 := x -> piecewise([x< 1, 2], [x>-1, 1]):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F2 := x -> piecewise([x>-1, 1], [x< 1, 2]):
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc2d(F1(x), x=-3..3, YRange=0..2)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc2d(F2(x), x=-3..3, YRange=0..2)
\par {\pntext\f1\'b7\tab}CharInt01 := x -> piecewise(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1     [x < 0 or x > 1, 0],
\par     [x <= 1 and x >= 0, 1]
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}H := x -> piecewise(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   [x >= 0 and x < 1, sqrt(1-x^2)],
\par    [x >= 1 and x < 2, 1],
\par    [x =  3, 1]
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par >>>>>>>>>> Time for Exercise 3.1
\par 
\par 3.2 How does MuPAD plot graphs of functions?
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Y:=plot::Function2d(sin(120*PI*x), x=-1..1):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(Y)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Y2 := plot::Function2d(sin(120*PI*x), x=-1/2..1/2):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(Y2)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Y3 := plot::Function2d(sin(120*PI*x), x=-1/30..1/30):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(Y3)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Y := plot::Function2d(sin(120*PI*x), x=-1..1,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   XMesh=242
\par ):
\par plot(Y)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Y := plot::Function2d(sin(120*PI*x), x=-1..1,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   XMesh=1001
\par ):
\par plot(Y)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}g := x -> (x^3+x^2+x+1)/(x^2-1):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plotfunc2d(g(x), x=0..2)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}g := x -> (x^3+x^2+x+1)/(x^2-1):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plotfunc2d(g(x), x=0..2,DiscontinuitySearch=FALSE)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par  >>>>>>>>>> Time for Exercise 3.2 
\par  
\par 3.3 Limits
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}limit(x^3+3*x-1, x=1)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := x -> (x^2-3*x-1)/(1-x)
\par {\pntext\f1\'b7\tab}limit(f(x), x=5)
\par {\pntext\f1\'b7\tab}limit(f(x), x=infinity)
\par {\pntext\f1\'b7\tab}limit(f(x),x=-infinity)
\par {\pntext\f1\'b7\tab}limit(f(x), x=1)
\par {\pntext\f1\'b7\tab}limit(f(x), x=1, Right)
\par {\pntext\f1\'b7\tab}limit(f(x), x=1, Left)
\par {\pntext\f1\'b7\tab}limit(1/x, x=0, Right)
\par {\pntext\f1\'b7\tab}limit(1/x, x=0, Left)
\par {\pntext\f1\'b7\tab}limit(sin(1/x), x=0)
\par {\pntext\f1\'b7\tab}limit(x*sin(1/x), x=0)
\par {\pntext\f1\'b7\tab}limit(sign(x), x=0)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}limit(sign(x), x=0, Right)
\par {\pntext\f1\'b7\tab}limit(sign(x), x=0, Left)
\par {\pntext\f1\'b7\tab}H := x -> piecewise(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    [x < -1 or x > 1, x^2],
\par    [x <= 1 and x >= -1, 1]
\par ):
\par plotfunc2d(H(x), x=-2..2, YRange=0..2)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}limit(H(x), x=1)
\par {\pntext\f1\'b7\tab}limit(H(x), x=1, Right)
\par {\pntext\f1\'b7\tab}limit(H(x), x=1, Left)
\par {\pntext\f1\'b7\tab}limit(floor(x), x=2)
\par {\pntext\f1\'b7\tab}limit(floor(x), x=2, Left)
\par {\pntext\f1\'b7\tab}limit(floor(x), x=2, Right)
\par {\pntext\f1\'b7\tab}an := n -> (1-1/n)
\par {\pntext\f1\'b7\tab}limit(an(n), n=+infinity)
\par {\pntext\f1\'b7\tab}bn := n -> (1-1/n)^n
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}limit(bn(n), n=+infinity)
\par {\pntext\f1\'b7\tab}cn := n -> (sin(1/n))^2+(cos(1/n))^2
\par {\pntext\f1\'b7\tab}limit(cn(n), n=+infinity)
\par {\pntext\f1\'b7\tab}dn := n -> (3^(2*n+1))^(1/n)
\par {\pntext\f1\'b7\tab}limit(dn(n), n=+infinity)
\par {\pntext\f1\'b7\tab}a := n -> sin(n*PI)
\par {\pntext\f1\'b7\tab}a(n) $ n=1..2
\par {\pntext\f1\'b7\tab}limit(a(n), n=infinity)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \plain\f3\fs36\cf0 
\par  >>>>>>>>>> Time for Exercise 3.3 
\par 
\par 3.4 The derivative
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := x -> x^3 + 3*x^2 + 5*x^(-3)
\par {\pntext\f1\'b7\tab}diff(f(x),x)
\par {\pntext\f1\'b7\tab}f'(x)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}diff(f(x),x,x)
\par {\pntext\f1\'b7\tab}diff(diff(f(x),x),x)
\par {\pntext\f1\'b7\tab}diff(f(x), x,x,x,x,x,x,x,x,x,x)
\par {\pntext\f1\'b7\tab}diff(f(x), x $ 10)
\par {\pntext\f1\'b7\tab}f  := x -> 3*x^4-11*x^2-5*x-5:
\par {\pntext\f1\'b7\tab}F1 := plot::Function2d(f(x)):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F2 := plot::Function2d(f'(x)):
\par plot(F1,F2)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f  := x -> 3*x^4-11*x^2-5*x-5:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F1 := plot::Function2d(f(x), LineWidth=0.6):
\par F2 := plot::Function2d(f'(x)):
\par plot(F1,F2,ViewingBox=[-2.5..2.5,-25..25])
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := x -> 3*x^4-11*x^2-5*x-5:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F1 := plot::Function2d(f(x),
\par    LineWidth=0.8,
\par    LegendText="Function f(x)"
\par ):
\par  F2 := plot::Function2d(f'(x),
\par    LineColor=[0,0,0],
\par    LegendText="Derivative f'(x)"
\par ):
\par plot(F1,F2,
\par    ViewingBox=[-2.5..2.5,-25..25],
\par // parameters to show the grid
\par    XGridVisible=TRUE,
\par    YGridVisible=TRUE,
\par    XSubgridVisible=TRUE,
\par    YSubgridVisible=TRUE,
\par // parameters to define density of the grid
\par    XTicksDistance=1,
\par    YTicksDistance=10,
\par    XTicksBetween=9,
\par    YTicksBetween=3,
\par // show the legend
\par    LegendVisible=TRUE,
\par    plot::Canvas::BorderWidth=0.5,
\par // add the title on the canvas
\par    plot::Canvas::Header="Function f(x) and its derivative"
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par >>>>>>>>>> Time for Exercise 3.4   
\par 
\par 3.5 Curve-sketching with MuPAD
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := x -> 3*x^3/(9*x^2-25)
\par {\pntext\f1\'b7\tab}plotfunc2d(f(x), x=-10..10)
\par \pard\ri4\plain\f4\fs36\cf1 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}g:= x -> diff(f(x), x)
\par {\pntext\f1\'b7\tab}g(x)
\par {\pntext\f1\'b7\tab}solve(g(x)=0, x), float(solve(g(x)=0, x))
\par \pard\ri4\plain\f3\fs36\cf0 
\par Technical comment: 
\par =================  
\par In some situations, like the one above, it is convenient to obtain in the same 
\par line the results of two or more calculations. We achieve this by separating 
\par two or more commands by commas. In our example, we obtained the exact 
\par results and their decimal approximations. However, the situation described 
\par here is exceptional and in general we should avoid the placing two or more 
\par commands on the same input line since this type of code is harder to read. 
\par From a technical point of view, the comma cannot be used to separate two 
\par commands as we did with ":" and ";". The comma that we used here produces 
\par a sequence of results and we can apply it only when this makes a sense, 
\par in other words, when creating a sequence is possible.
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f(-5*sqrt(3)/3), float(f(-5*sqrt(3)/3))
\par {\pntext\f1\'b7\tab}f(5*sqrt(3)/3), float(f(5*sqrt(3)/3))
\par {\pntext\f1\'b7\tab}solve(9*x^2-25, x)
\par {\pntext\f1\'b7\tab}limit(f(x), x=-5/3, Left)
\par {\pntext\f1\'b7\tab}limit(f(x), x=-5/3, Right)
\par {\pntext\f1\'b7\tab}limit(f(x), x=5/3, Left)
\par {\pntext\f1\'b7\tab}limit(f(x), x=5/3, Right)
\par {\pntext\f1\'b7\tab}m := limit(f(x)/x, x=+infinity)
\par {\pntext\f1\'b7\tab}a := limit(f(x)-m*x, x=+infinity)
\par {\pntext\f1\'b7\tab}m := limit(f(x)/x, x=-infinity)
\par {\pntext\f1\'b7\tab}a := limit(f(x)-m*x,x=-infinity)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := x -> 3*x^3/(9*x^2-25):
\par {\pntext\f1\'b7\tab} // graph of the function
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 F := plot::Function2d(f(x),x=-6..6,
\par    LineWidth=0.50,
\par    LineColor=RGB::Red,
\par    LegendText="Function f(x)"
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // derivative plot
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 G := plot::Function2d(f'(x), x=-6..6,
\par    LineColor=RGB::Blue,
\par    LegendText="Derivative f'(x)"
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // slant asymptote
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 A1:= plot::Function2d(x/3, x=-6..6,
\par    LineColor=RGB::Gray40,
\par    LineStyle=Dashed,
\par    LegendText="Asymptotes"
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab} // critical points
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 M1 := plot::Point2d([5*sqrt(3)/3,f(5*sqrt(3)/3)],
\par    PointSize=2,
\par    PointStyle=XCrosses):
\par M2 := plot::Point2d([-5*sqrt(3)/3,f(-5*sqrt(3)/3)],
\par    PointSize=2,
\par    PointStyle=XCrosses):
\par M0 := plot::Point2d([0,f(0)],
\par    PointSize=2,
\par    PointStyle=XCrosses
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}// plot of the complete scene
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(F,G,A1,M0,M1,M2,
\par    // grid parameters
\par    XGridVisible=TRUE,
\par    YGridVisible=TRUE,
\par    XSubgridVisible=TRUE,
\par    YSubgridVisible=TRUE,
\par    // annotation parameters
\par    LegendVisible=TRUE,
\par    Header="Function 3*x^3/(9*x^2-25)",
\par    // parameters of background and frame
\par    BackgroundColor=RGB::Gray80,
\par    BorderColor=RGB::Black,
\par    BorderWidth=0.25,
\par    // view parameters
\par    Scaling=Constrained,
\par    ViewingBox=[-6..6,-3.5..3]
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0    
\par >>>>>>>>>> Time for Exercise 3.5  
\par 
\par 3.6 Taylor polynomials
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}taylor(sin(x), x=0, 1)
\par {\pntext\f1\'b7\tab}taylor(sin(x), x=0, 2)
\par {\pntext\f1\'b7\tab}taylor(sin(x), x=0, 3)
\par {\pntext\f1\'b7\tab}taylor(sin(x), x=0, 5)
\par {\pntext\f1\'b7\tab}taylor(sin(x), x=0, 7)
\par \pard\ri4\plain\f3\fs36\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}f := x -> sin(x)+cos(x):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 T1 := taylor(f(x), x=0, 1):
\par T3 := taylor(f(x), x=0, 3):
\par T5 := taylor(f(x), x=0, 5):
\par T7 := taylor(f(x), x=0, 7):
\par T9 := taylor(f(x), x=0, 9):
\par T11 := taylor(f(x), x=0, 11):
\par plotfunc2d(f(x),T1,T3,T5, T7, T9, T11,
\par    x=-2*PI..2*PI, YRange=-3..3,
\par    LineWidth=0.3,
\par    XGridVisible=TRUE,
\par    YGridVisible=TRUE,
\par    XSubgridVisible=TRUE,
\par    YSubgridVisible=TRUE,
\par    Header="Function sin(x)+cos(x) and its
\par    Taylor polynomials",
\par    HeaderFont=["Arial", 12, Bold],
\par    LegendVisible=TRUE,
\par    BackgroundColor=RGB::Gray80,
\par    BorderColor=RGB::Black,
\par    BorderWidth=0.2
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par >>>>>>>>>> Time for Exercises 3.6
\par 
\par 3.7 Integration with MuPAD
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}int(x^2+3*x+5, x) //calculate antiderivative
\par {\pntext\f1\'b7\tab}int(x^2+3*x+5,x=0..10) //calculate antiderivative
\par {\pntext\f1\'b7\tab}int(x*ln(x),x)
\par {\pntext\f1\'b7\tab}int(x*ln(x),x=0..1)
\par {\pntext\f1\'b7\tab}int(sin(x)*cos(x),x)
\par {\pntext\f1\'b7\tab}int(sin(2*x)*cos(3*x), x)
\par {\pntext\f1\'b7\tab}int(sin(2*x)*cos(3*x), x=0..PI)
\par {\pntext\f1\'b7\tab} int(x*sin(2*x)*cos(x),x)
\par {\pntext\f1\'b7\tab}Simplify(%)
\par {\pntext\f1\'b7\tab}f := x -> x*exp(x)*cos(3*x)
\par {\pntext\f1\'b7\tab}int(f(x),x)
\par {\pntext\f1\'b7\tab}F:= x ->exp(-x^2)
\par {\pntext\f1\'b7\tab}plotfunc2d(F(x), x=-2..2, YRange=0..1)
\par {\pntext\f1\'b7\tab}int(F(x), x=-infinity..infinity)
\par {\pntext\f1\'b7\tab}G := x -> 1/(x^2+1)
\par {\pntext\f1\'b7\tab}int(G(x), x=-infinity..infinity)
\par {\pntext\f1\'b7\tab}g := x -> x^3-x^2-6*x:
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 G := plot::Function2d(g(x), x=-2.5..3.5):
\par H := plot::Hatch(G, -2..3,
\par    FillPattern=Solid,
\par    FillColor=[0.5, 0.67, 0.87]
\par ):
\par plot(H, G, AxesInFront=TRUE)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}area := int(g(x),x=-2..0)+abs(int(g(x),x=0..3))
\par \pard\ri4\plain\f3\fs36\cf0 
\par Technical comment:
\par =================   
\par When plotting Hatch in the final plot statement, we used a specific order of 
\par objects to be plotted. First we placed H and then G. It is important to note 
\par that if we plot Hatch as the last object, its area will cover at least partially 
\par the curve and coordinate axes. Therefore, to avoid this unwanted effect, we 
\par plot Hatch first and then the other objects. Finally, we have to move the 
\par coordinate axes to the front. For this purpose, we use the parameter 
\par AxesInFront=TRUE.
\par 
\par >>>>>>>>>> Time for Exercise 3.7 
\par 
\par 3.8 Numerical integration
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}int(sin(x)/x, x)
\par {\pntext\f1\'b7\tab}int(sin(x)/x, x=0..PI)
\par {\pntext\f1\'b7\tab}numeric::int(sin(x)/x, x=0..PI)
\par {\pntext\f1\'b7\tab}F:= x ->exp(-x^2):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 int(F(x), x);
\par int(F(x), x=0..PI)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}numeric::int(F(x), x=0..PI)
\par \pard\ri4\plain\f3\fs36\cf0 
\par Technical comment:
\par =================   
\par The functions erf(x) and Si(x) obtained in our examples belong 
\par to a group of functions known as special functions. Such functions occur 
\par quite frequently in integration or when solving integral equations.
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A := student::plotRiemann(exp(-x^2), x = -1..1, 5):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(A)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A := student::plotRiemann(exp(-x^2),x=-1..1,10):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(A)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A := student::plotRiemann(exp(-x^2),x =-1..1,50):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(A)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}B := student::plotTrapezoid(exp(-x^2),x=-1..1, 10):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(B)
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}B := student::plotSimpson(exp(-x^2), x = -1..1, 10):
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 plot(B)
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0 
\par >>>>>>>>>> Time for Exercise 3.8   
\par 
\par 3.9 Solving differential equations
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}diff_eq := ode(
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1    y'(x) - 2*y(x)*cos(x) = cos(x)+sin(2*x), y(x)
\par ):
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}A := solve(diff_eq)
\par \pard\ri4\plain\f3\fs36\cf0 
\par 
\par Technical comment:
\par =================   
\par In many situations similar to the one above, MuPAD creates new parameters 
\par such as C2, C34, C76, etc. Try, for example, to execute the statement 
\par solve(diff_eq)  several times. Each time, a new parameter with a different 
\par name is created. Therefore, there is a great likelihood that the parameters 
\par created by MuPAD in my examples differ from those produced in your experiments.
\par 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}Eq := op(A); // remove the curly brace
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1 Functions := Eq $ C2=-10..10: // create functions
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f4\fs36\cf1 {\pntext\f1\'b7\tab}plotfunc2d(Functions, x=-1.5..4.5,
\par \pard\li600\ri1\fi-300\plain\f4\fs36\cf1   LegendVisible=FALSE
\par )
\par \pard\ri4\plain\f4\fs36\cf1  \plain\f3\fs36\cf0   
\par >>>>>>>>>> Time for Exercise 3.9
\par }