# Sample Maple Commands for each Project 
# This notebook contains sample Maple commands which you may cut-and-paste from and use in your current notebook.  The projects below are in alphabetic order for ease of use.
# Analytic Geometry
> d1 := sqrt( (x+c)^2 + y^2 ):\
d2 := sqrt( (x-c)^2 + y^2 ):\
( d1^2 - d2^2 );
> q := c^2 - k^2 * c^2 + x^2 -a^2 * x^2 + y^2 - b^2 * y^2:\
collect( q, [x,y,c] );
> with(plots):\
implicitplot(((x-1)/3)^2 + y^2 = 8, x = -10..10, y = -10..10, \
   scaling = CONSTRAINED);
# Note: the scaling = CONSTRAINED options prevents Maple from rescaling and essentially drawing a circle.
# Arclength
> ds := (y,x) -> sqrt(1 + D(y)(x)^2);\
y := x -> x^2:\
dsParabola := ds(y,x);\
sParabola := int(dsParabola, x);\
subs(x=1, sParabola) - subs(x=0, sParabola);
# Area
> f := x -> sqrt(1 - x^2):\
a := 0:  b := 1.0:\
 n := 5:  delx := (b - a)/n:\
Rsum := delx * sum( f(a + (i+.5)*delx), i = 0..n-1):\
approxPi := 4 * Rsum;
# (The) Asteroid Problem
> for x from 0 by 0.2 to 1.0 do\
   lprint(x, cos(x) -x)\
od;
# (The) Binomial Theorem
> sum := 0:\
for k from 1 by 2 to 7 do\
   sum := sum + k^2\
od;
# Chaos
# See Newton Project below for the Newton loop.
# Center of Mass
> a := 'a':\
volume := int( int( int( 1, z = 0..a-x-y ), y = 0..a-x), x = 0..a);\
cmassX := int( int( int(x, z = 0..a-x-y ), y = 0..a-x ), x = 0..a )\
   / volume;
# Curve Fitting for Discrete Data Sets
> xdata := [.2, 1.1, 2.0, 3.1, 3.9, 5.1]:\
ydata := [-2, 0, 1, 2.8, 5, 7]:\
data := zip( (x,y) -> [x,y], xdata, ydata);\
with(stats):\
cof := linregress(ydata, xdata);\
y := cof[1] + cof[2] * x;
# Derivatives, Slopes and Tangents
# After defining f, we can get the tangent:
> b := 2.1:\
slope := (f(b) - f(2)) / (b - 2.0):\
secant := f(2.0) + slope*(x - 2.0):\
print(`slope is `, slope);\
plot({secant, f(x)}, x = 1.5..3.0);
# Derivative in Maple
> y := 1/(1 + sqrt(1 + sqrt(x)));\
diff(y, x);
# Finding Roots with Maple
> quadeqn := a*x^2 + b*x + c = 0;\
solutions := solve(quadeqn, x);\
subs(x = solutions[1], quadeqn);\
Digits := 10:\
lhs := x^3 - 3*x^2 + 1:\
solutions := fsolve(lhs = 0, x);
# Force Applications in 3D
# Here is a typical 3D integration in rectangular coordinates.
> z := 'z': x := 'x': y := 'y':\
h := 1:  R := 3:\
int( int( int( z+h+R, \
          x = -sqrt(R^2 - y^2 - z^2)..sqrt(R^2 - y^2 - z^2) ),\
          y = -sqrt(R^2 - z^2)..sqrt(R^2 - z^2) ),\
          z = -R..R);        
# Fourier Frequency Decomposition
> f := x -> x*(Pi - x)*(Pi + x):\
nmax := 2: npi := evalf(Pi):\
b := seq( 2/npi * int( f(x)*sin(k*x), x=0..npi), k=1..nmax);\
fn := x -> sum( b[n]*sin(n*x), n = 1..nmax):\
plot({f, fn}, 0..Pi);  
# Functions Defined by Integrals
> myLog := int( 1/x, x = 1..z);\
P := sqrt(1/Pi) * int( exp(-x^2/2), x = 0..z);\
z := 'z':\
MyErf := 2/sqrt(Pi) * int( exp(-t^2), t = 0..z);
# Graphical Equation Solving
> plot(2*a^3 + a - 2, a = 0.7..1.1);
# Integration in Maple
# A numerical integration:
> x := 'x':\
evalf( int( arcsin(x), x = 0..1/2) ); 
# Inverse Square Law Problems
> q := a + eps:\
R := sqrt( q^2 + r^2 - 2*q*r*cos(phi) ):\
F := -3*G*M*m/(4*Pi*a^3) * \
      int( int( int( (q - r*cos(phi))/R^3*r^2*sin(phi),\
           phi = 0..Pi ), r = 0..a ), theta = 0..2*Pi );\
q := 'q':\
subs( eps = q-a, F);
# Limits and Continuity
> f := x -> (sqrt(25 + 3*x) - sqrt(25 - 2*x))/x:\
plot(f(x), x = -1..1);\
for x from -1.0 by 0.2 to 1.0 do\
   print(x, `   `, f(x))\
od; 
# Limits with Maple
> limit(sin(x)/x, x=0);
# Line Integrals and Work
> x := t -> 2*t:\
y := t -> t:\
FT := t -> D(x)(t) + D(y)(t)*exp(c*x(t) - y(t)):
# Now integrate this function to get work.
# Linearization of Nonlinear Data
> ddata := [8.6, 10.7, 11., 11.4, 12.0, 13.3, 14.5, 16.0,17.3,18.0,20.6]:\
vdata := [10.3, 18.8, 18.2, 21.4, 19.1, 27.4, 36.3, 38.3,55.4,51.5,77.]:\
with(plots):\
data := zip( (x,y) -> [x,y], ddata, vdata);\
plot(data, style = POINT);\
with(stats):\
cof := linregress( vdata, ddata);\
linearFit := cof[1] + cof[2]*x;
# Newton's Method
# f := x -> x^2 - 5: 
# xn := 6.0: 
# for k to 5 do 
#    xn := xn - f(xn)/ D(f)(xn); 
# od;
# Newton's Method in 2D
> p := 4:  q := 1:\
r := 0:\
f := (a,b) -> 2*a^3 + 8*a*b^2 + (1 - 2*r)*a - p:\
g := (a,b) -> 32*b^3 + 8*b*a^2 + (1 - 8*r)*b - q:\
dfx := (a,b) -> D[1](f)(a,b):\
dfy := (a,b) -> D[2](f)(a,b):\
dgx := (a,b) -> D[1](g)(a,b):\
dgy := (a,b) -> D[2](g)(a,b):\
printlevel := 0:\
an := 1.1:  bn := .1:\
for n to 8 do\
   an := an - ( f(an,bn)*dgy(an,bn) - g(an,bn)*dfy(an,bn) ) /\
              ( dfx(an,bn)*dgy(an,bn) - dgx(an,bn)*dfy(an,bn) ):\
   bn := bn - ( g(an,bn)*dfx(an,bn) - f(an,bn)*dgx(an,bn) ) /\
              ( dfx(an,bn)*dgy(an,bn) - dgx(an,bn)*dfy(an,bn) ):\
   lprint( an, bn )\
od;
# Numerical Integration I
> f := x -> x^4 + 1:\
a := 0:  b := 5:\
n := 5:  delx := (b - a)/n:\
p := 0.0:\
Rsum :=  delx * sum(f(a + (i+p)*delx), i = 0..n-1);
# Numerical Integration II
# Here are the basic integration procedures:
> Mid := proc(f, a, b, n)\
   Rsum(f, a, b, n, 0.5)\
end;\
Trap := proc(f, a, b, n)\
    0.5 * ( Rsum(f, a, b, n, 0) + Rsum(f, a, b, n, 1) );\
    evalf(") \
end;\
Simp := proc(f, a, b, n)\
    ( Trap(f, a, b, n, 0) + 2*Mid(f, a, b, n, 1) ) / 3.0\
end;
# Parametric Curves in 2D
> x := t -> cos(t):\
y := t -> sin(t):\
plot( [x, y, 0..2*Pi] );
# (The) Piston Problem
> q := (x,L) -> sin(2*t) + sqrt(L^2 -cos(2*t)^2):\
L := 'L':\
qpr := diff(q(t,L),t);
# Plotting in Polar Coordinates
> with(plots, polarplot):\
k := 2:\
r := t -> sin(k*t):\
polarplot( r(t), t = 0..2*Pi);
# (The) Probability Integral
> p := x -> exp(-x^2/2) / sqrt(2*Pi);\
int( p(x), x = -infinity.. infinity);
# Simple Harmonic Motion
> f := x -> cos(x) - sin(x):\
plot(f, -Pi..Pi);
# Snell's Law
# f := x -> 3*x^4 - 24*x^3 + 51*x^2 - 32*x + 64; 
# plot(f, -10..10);
# Solving Differential Equations
> f := x -> 3*cos(2*x):\
kMax := 11:\
Pn := taylor( f(x), x=0, kMax );\
for k from 0 to kMax do\
   b[k] := coeff( Pn, x, k)\
od:\
a[0] := 2:  a[1] := 0:\
for k from 0 to kMax do\
   a[k+2] := (b[k] - a[k]) / ((k+1)*(k+2))\
od; \
yn := sum( a[n]*x^n, n = 0..kMax+2); 
# (The) Tank Problem
> 4*r* int( sqrt(1 - (x/r)^2), x = 0..r );\
4*a* int( sqrt(1 - (y/b)^2), y = 0..b );
# Taylor Polynomials
> f := x -> x * cos(2*x):\
a := 0:  n := 6:\
ourSeries := taylor( f(x), x = 0, n);\
Pn := convert(ourSeries, polynom);
# Tutorial Exercises
> evalf(sin(Pi/3)); \
f := x^2 + 1;\
g := x^3 - x^2 - 9*x + 9;\
plot({f,g}, x = -5..5); 
# Twisting Space Curves
> n := 'n':\
x := cos(2*n*t)*sin(t):\
y := sin(2*n*t)*sin(t):\
z := 3*cos(t):\
ds := simplify( sqrt( diff(x,t)^2 + diff(y,t)^2 + diff(z,t)^2 ) );\
n := 1:\
plots[spacecurve]( [x, y, z], t = 0..2*Pi, axes = BOXED,\
       orientation=[45, 60] );
# Vectors and Work
> with(linalg):\
v := [2, 3, 4]:  w := [1, 1, 1]:\
u := add(9*v, -29*w);
> u := [1, 2, 3]:\
norm2 := (sqrt( dotprod(u,u) ) ):\
uhat := u / norm2;
# Work Along an Arc
> sqrt((3-1)^2 + (3-2)^2) +\
	sqrt((4-3)^2 + (5-3)^2) +\
		sqrt ((5-4)^2 + (5-5)^2) +\
			sqrt ((6-5)^2 + (8-5)^2):\
evalf(");
> h := x -> sqrt(4 - x^2):\
a := -2:  b := 2:\
ArcLength := int( sqrt(1 + D(h)(x)^2), x = a..b);