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\deflang1033\pard\ri4\plain\f2\fs36\cf0 Chapter 3  Introduction to calculus of one variable\plain\f2\fs28\cf0 \
\
2.1 Declaring functions of one variable\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}g := x -> (x^3+x^2+x+1)/(x^2+x+1)\
g := x -> nextprime(floor(x)):\
g(21.98)\
G := plot::Function2d(g(x), x=0..10, XSubmesh=10):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(G)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}F := plot::Function2d(floor(x), x=-5..5): plot(F)\
H := plot::Function2d(x-floor(x), x=-4..4):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(H, Scaling=Constrained)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}g := n -> ithprime(n)-ithprime(n-1):\
g(25)\
g(2567)\
f := x -> [ 2^n $ n=1..floor(x)]:\
f(12)\
\pard\ri4\plain\f2\fs28\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}reset():
\par \pard\li600\ri1\fi-300\plain\f3\fs28\cf1 CharInt01 := x -> piecewise([x<0,0],[x<=1,1],[x>1,0]):\
\pard\ri4\plain\f3\fs28\cf1  \plain\f2\fs28\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}F1 := x -> piecewise([x< 1, 2], [x>-1, 1]):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 F2 := x -> piecewise([x>-1, 1], [x< 1, 2]):\
\pard\ri4\plain\f2\fs28\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}plotfunc2d(F1(x), x=-3..3, YRange=0..2)\
\pard\ri4\plain\f2\fs28\cf0 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}plotfunc2d(F2(x), x=-3..3, YRange=0..2)\
CharInt01 := x -> piecewise(\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1     [x < 0 or x > 1, 0],\
    [x <= 1 and x >= 0, 1]\
)\
\pard\ri4\plain\f2\fs28\cf0    
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}H := x -> piecewise(\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1   [x >= 0 and x < 1, sqrt(1-x^2)],\
   [x >= 1 and x < 2, 1],\
   [x =  3, 1]\
)\
\pard\ri4\plain\f2\fs28\cf0    
\par >>>>>>>>>> Time for Exercise 3.1
\par \
3.2 How does MuPAD plot graphs of functions?\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}Y:=plot::Function2d(sin(120*PI*x), x=-1..1):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(Y)\
\pard\ri4\plain\f2\fs28\cf0     \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}Y2 := plot::Function2d(sin(120*PI*x), x=-1/2..1/2):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(Y2)\
\pard\ri4\plain\f2\fs28\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}Y3 := plot::Function2d(sin(120*PI*x), x=-1/30..1/30):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(Y3)\
\pard\ri4\plain\f2\fs28\cf0 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}Y := plot::Function2d(sin(120*PI*x), x=-1..1,\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1   XMesh=242\
):\
plot(Y)\
\pard\ri4\plain\f2\fs28\cf0     
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}Y := plot::Function2d(sin(120*PI*x), x=-1..1,\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1   XMesh=1001\
):\
plot(Y)\
\pard\ri4\plain\f2\fs28\cf0     \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}g := x -> (x^3+x^2+x+1)/(x^2-1):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plotfunc2d(g(x), x=0..2)\
\pard\ri4\plain\f2\fs28\cf0     
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}g := x -> (x^3+x^2+x+1)/(x^2-1):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plotfunc2d(g(x), x=0..2,DiscontinuitySearch=FALSE)\
\pard\ri4\plain\f2\fs28\cf0     \
 >>>>>>>>>> Time for Exercise 3.2 \
 \
3.3 Limits\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}limit(x^3+3*x-1, x=1)\
\pard\ri4\plain\f2\fs28\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f := x -> (x^2-3*x-1)/(1-x)\
limit(f(x), x=5)\
limit(f(x), x=infinity)\
limit(f(x),x=-infinity)\
limit(f(x), x=1)\
limit(f(x), x=1, Right)\
limit(f(x), x=1, Left)\
limit(1/x, x=0, Right)\
limit(1/x, x=0, Left)\
limit(sin(1/x), x=0)\
limit(x*sin(1/x), x=0)\
limit(sign(x), x=0)\
\pard\ri4\plain\f2\fs28\cf0 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}limit(sign(x), x=0, Right)\
limit(sign(x), x=0, Left)\
H := x -> piecewise(\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1    [x < -1 or x > 1, x^2],\
   [x <= 1 and x >= -1, 1]\
):
\par plotfunc2d(H(x), x=-2..2, YRange=0..2)\
\pard\ri4\plain\f2\fs28\cf0     
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}limit(H(x), x=1)\
limit(H(x), x=1, Right)\
limit(H(x), x=1, Left)\
limit(floor(x), x=2)\
limit(floor(x), x=2, Left)\
limit(floor(x), x=2, Right)\
an := n -> (1-1/n)\
limit(an(n), n=+infinity)\
bn := n -> (1-1/n)^n\
\pard\ri4\plain\f2\fs28\cf0 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}limit(bn(n), n=+infinity)\
cn := n -> (sin(1/n))^2+(cos(1/n))^2\
limit(cn(n), n=+infinity)\
dn := n -> (3^(2*n+1))^(1/n)\
limit(dn(n), n=+infinity)\
a := n -> sin(n*PI)\
a(n) $ n=1..2\
limit(a(n), n=infinity)\
\pard\ri4\plain\f2\fs28\cf0 \
\
 >>>>>>>>>> Time for Exercise 3.3 \
\
3.4 The derivative\
\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f := x -> x^3 + 3*x^2 + 5*x^(-3)\
diff(f(x),x)\
f'(x)\
\pard\ri4\plain\f2\fs28\cf0 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}diff(f(x),x,x)\
diff(diff(f(x),x),x)\
diff(f(x), x,x,x,x,x,x,x,x,x,x)\
diff(f(x), x $ 10)\
f  := x -> 3*x^4-11*x^2-5*x-5:\
F1 := plot::Function2d(f(x)):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 F2 := plot::Function2d(f'(x)):\
plot(F1,F2)\
\pard\ri4\plain\f2\fs28\cf0     
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f  := x -> 3*x^4-11*x^2-5*x-5:\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 F1 := plot::Function2d(f(x), LineWidth=0.6):\
F2 := plot::Function2d(f'(x)):\
plot(F1,F2,ViewingBox=[-2.5..2.5,-25..25])\
\pard\ri4\plain\f2\fs28\cf0  
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f := x -> 3*x^4-11*x^2-5*x-5:\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 F1 := plot::Function2d(f(x),\
   LineWidth=0.8,\
   LegendText="Function f(x)"\
):\
 F2 := plot::Function2d(f'(x),\
   LineColor=[0,0,0],\
   LegendText="Derivative f'(x)"\
):\
plot(F1,F2,\
   ViewingBox=[-2.5..2.5,-25..25],\
// parameters to show the grid\
   XGridVisible=TRUE,\
   YGridVisible=TRUE,\
   XSubgridVisible=TRUE,\
   YSubgridVisible=TRUE,\
// parameters to define density of the grid\
   XTicksDistance=1,\
   YTicksDistance=10,\
   XTicksBetween=9,\
   YTicksBetween=3,\
// show the legend\
   LegendVisible=TRUE,\
   plot::Canvas::BorderWidth=0.5,\
// add the title on the canvas\
   plot::Canvas::Header="Function f(x) and its derivative"\
)\
\pard\ri4\plain\f2\fs28\cf0     \
>>>>>>>>>> Time for Exercise 3.4   
\par \
3.5 Curve-sketching with MuPAD\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f := x -> 3*x^3/(9*x^2-25)\
plotfunc2d(f(x), x=-10..10)\
\pard\ri4\plain\f2\fs28\cf0 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}g:= x -> diff(f(x), x)\
g(x)\
solve(g(x)=0, x), float(solve(g(x)=0, x))\
\pard\ri4\plain\f2\fs28\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.\
\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f(-5*sqrt(3)/3), float(f(-5*sqrt(3)/3))\
f(5*sqrt(3)/3), float(f(5*sqrt(3)/3))\
solve(9*x^2-25, x)\
limit(f(x), x=-5/3, Left)\
limit(f(x), x=-5/3, Right)\
limit(f(x), x=5/3, Left)\
limit(f(x), x=5/3, Right)\
m := limit(f(x)/x, x=+infinity)\
a := limit(f(x)-m*x, x=+infinity)\
m := limit(f(x)/x, x=-infinity)\
a := limit(f(x)-m*x,x=-infinity)
\par \pard\ri4\plain\f2\fs28\cf0 \
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f := x -> 3*x^3/(9*x^2-25):\
 // graph of the function\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 F := plot::Function2d(f(x),x=-6..6,\
   LineWidth=0.50,\
   LineColor=RGB::Red,\
   LegendText="Function f(x)"\
):\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab} // derivative plot\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 G := plot::Function2d(f'(x), x=-6..6,\
   LineColor=RGB::Blue,\
   LegendText="Derivative f'(x)"\
):\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab} // slant asymptote\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 A1:= plot::Function2d(x/3, x=-6..6,\
   LineColor=RGB::Gray40,\
   LineStyle=Dashed,\
   LegendText="Asymptotes"\
):\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab} // critical points\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 M1 := plot::Point2d([5*sqrt(3)/3,f(5*sqrt(3)/3)],\
   PointSize=2,\
   PointStyle=XCrosses):\
M2 := plot::Point2d([-5*sqrt(3)/3,f(-5*sqrt(3)/3)],\
   PointSize=2,\
   PointStyle=XCrosses):\
M0 := plot::Point2d([0,f(0)],\
   PointSize=2,\
   PointStyle=XCrosses\
):\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}// plot of the complete scene\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(F,G,A1,M0,M1,M2,\
   // grid parameters\
   XGridVisible=TRUE,\
   YGridVisible=TRUE,\
   XSubgridVisible=TRUE,\
   YSubgridVisible=TRUE,\
   // annotation parameters\
   LegendVisible=TRUE,\
   Header="Function 3*x^3/(9*x^2-25)",\
   // parameters of background and frame\
   BackgroundColor=RGB::Gray80,\
   BorderColor=RGB::Black,\
   BorderWidth=0.25,\
   // view parameters\
   Scaling=Constrained,\
   ViewingBox=[-6..6,-3.5..3]\
)\
\pard\ri4\plain\f2\fs28\cf0     \
>>>>>>>>>> Time for Exercise 3.5  
\par \
3.6 Taylor polynomials\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}taylor(sin(x), x=0, 1)\
taylor(sin(x), x=0, 2)
\par {\pntext\f1\'b7\tab}taylor(sin(x), x=0, 3)\
taylor(sin(x), x=0, 5)\
taylor(sin(x), x=0, 7)\
\pard\ri4\plain\f2\fs28\cf0 
\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}f := x -> sin(x)+cos(x):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 T1 := taylor(f(x), x=0, 1):\
T3 := taylor(f(x), x=0, 3):\
T5 := taylor(f(x), x=0, 5):\
T7 := taylor(f(x), x=0, 7):\
T9 := taylor(f(x), x=0, 9):\
T11 := taylor(f(x), x=0, 11):\
plotfunc2d(f(x),T1,T3,T5, T7, T9, T11,\
   x=-2*PI..2*PI, YRange=-3..3,\
   LineWidth=0.3,\
   XGridVisible=TRUE,\
   YGridVisible=TRUE,\
   XSubgridVisible=TRUE,\
   YSubgridVisible=TRUE,\
   Header="Function sin(x)+cos(x) and its\
   Taylor polynomials",\
   HeaderFont=["Arial", 12, Bold],\
   LegendVisible=TRUE,\
   BackgroundColor=RGB::Gray80,\
   BorderColor=RGB::Black,\
   BorderWidth=0.2\
)\
\pard\ri4\plain\f2\fs28\cf0    
\par >>>>>>>>>> Time for Exercises 3.6
\par 
\par 3.7 Integration with MuPAD\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}int(x^2+3*x+5, x) //calculate antiderivative\
int(x^2+3*x+5,x=0..10) //calculate antiderivative\
int(x*ln(x),x)\
int(x*ln(x),x=0..1)\
int(sin(x)*cos(x),x)\
int(sin(2*x)*cos(3*x), x)\
int(sin(2*x)*cos(3*x), x=0..PI)\
 int(x*sin(2*x)*cos(x),x)\
Simplify(%)\
f := x -> x*exp(x)*cos(3*x)
\par {\pntext\f1\'b7\tab}int(f(x),x)\
F:= x ->exp(-x^2)\
plotfunc2d(F(x), x=-2..2, YRange=0..1)\
int(F(x), x=-infinity..infinity)\
G := x -> 1/(x^2+1)\
int(G(x), x=-infinity..infinity)\
g := x -> x^3-x^2-6*x:\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 G := plot::Function2d(g(x), x=-2.5..3.5):\
H := plot::Hatch(G, -2..3,\
   FillPattern=Solid,\
   FillColor=[0.5, 0.67, 0.87]\
):\
plot(H, G, AxesInFront=TRUE)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}area := int(g(x),x=-2..0)+abs(int(g(x),x=0..3))\
\pard\ri4\plain\f2\fs28\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 \
>>>>>>>>>> Time for Exercise 3.7 
\par \
3.8 Numerical integration\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}int(sin(x)/x, x)\
int(sin(x)/x, x=0..PI)\
numeric::int(sin(x)/x, x=0..PI)\
F:= x ->exp(-x^2):
\par \pard\li600\ri1\fi-300\plain\f3\fs28\cf1 int(F(x), x);\
int(F(x), x=0..PI)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}numeric::int(F(x), x=0..PI)\
\pard\ri4\plain\f2\fs28\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 \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}A := student::plotRiemann(exp(-x^2), x = -1..1, 5):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(A)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}A := student::plotRiemann(exp(-x^2),x=-1..1,10):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(A)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}A := student::plotRiemann(exp(-x^2),x =-1..1,50):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(A)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}B := student::plotTrapezoid(exp(-x^2),x=-1..1, 10):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(B)\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}B := student::plotSimpson(exp(-x^2), x = -1..1, 10):\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 plot(B)\
\pard\ri4\plain\f2\fs28\cf0  
\par >>>>>>>>>> Time for Exercise 3.8   
\par \
3.9 Solving differential equations\

\par \pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}diff_eq := ode(\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1    y'(x) - 2*y(x)*cos(x) = cos(x)+sin(2*x), y(x)\
):\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}A := solve(diff_eq)\
\pard\ri4\plain\f2\fs28\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.\
\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}Eq := op(A); // remove the curly brace\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1 Functions := Eq $ C2=-10..10: // create functions\
\pard\li300\ri5\fi-300{\*\pn\pnlvlblt\pnf1\pnindent300{\pntxtb\'b7}}\plain\f3\fs28\cf1 {\pntext\f1\'b7\tab}plotfunc2d(Functions, x=-1.5..4.5,\
\pard\li600\ri1\fi-300\plain\f3\fs28\cf1   LegendVisible=FALSE\
)\
\pard\ri4\plain\f2\fs28\cf0    
\par >>>>>>>>>> Time for Exercise 3.9
\par }