Instructor Hentzel
Office Phone: 515-294-8141
E-mail:   hentzel@iastate.edu
Math Department Fax:      515-294-5454
http://www.math.iastate.edu/hentzel/class.166.08
Textbook: Calculus by Varberg, Purcell, Rigdon, eight edition. 

April 19   Chapter review 

p556 Sample Test Problems: (2),(6),(10),(14),(18),(22),(26),(30),(34),(38),(42)
Main Idea:  Chapter Review 
Key Words:  a, b, c, hyperbola, ellipse, parabola, limacon, cardioid, lemniscate,
spiral of Archimedes, hyperbolic spiral, circle 
Goal: Use knowledge of Conics to answer questions 
------------------------------------------------------------------
Previous assignment:    Page 556 Sample Test Problems: 
Page 556 Problem  4
Name the conic that has the given equation.
Find its vertices and foci and sketch its graph.

25 x^2 - 36 y^2 + 900 = 0

    x^2       y^2
 - ------ + --------- = 1
    36         25
a = 5  b = 6  c = Sqrt[61]
Vertices (0,{+/-} 5 )
Foci     {0, {+/-} Sqrt[61] }

Asymptotes     x/6 + y/5 = 0         x/6 - y/5 = 0
------------------------------------------------------
a = 5;
b = 6;
c = Sqrt[61];
h = b^2/a;
ans = Solve[25 x^2 - 36 y^2 + 900 == 0,y];
f[x_] = y /. ans[[1]];
g[x_] = y /. ans[[2]];
p1 = Plot[{f[x],g[x]},{x,-16,16}];
p2 = Plot[{ a/b x, -a/b x},{x,-16,16}, PlotStyle->{RGBColor[1,0,0]}];
p3 = Plot[{c,-c},{x,-h,h}, PlotStyle->{RGBColor[1,0,0]}];
p4 = ListPlot[{ {0,0},
               {0,a},{0,-a},
               {0,c},{0,-c},
               {h,c},{-h,c}},PlotStyle->{RGBColor[1,0,0]}];

p5 = Show[p1,p2,p3,p4,PlotLabel->"Page 556 Problem 4, 25 x^2 - 36 y^2 + 900 = 0"];
Display["4.ps",p5];
--------------------------------------------------------------------------
Name the conic that has the given equation.
Find its vertices and foci and sketch its graph.
Page 556 Problem  8

    2       2
  9x   + 9 y   - 225 = 0

Circle,   radius is 15/3.
---------------------------------------------
a = Plot[ Sqrt[(1/9)(225-9x^2)],{x,-15/3,15/3}];
b = Plot[-Sqrt[(1/9)(225-9x^2)],{x,-15/3,15/3}];
c = ParametricPlot[ (15/3+0.02) {Cos[t],Sin[t]},
    {t,0,2 Pi},PlotStyle->{RGBColor[1,0,0]}];
d = Show[a,b,c,PlotLabel->"Page 556 Problem 4: 9 x^2 + 9 y^2 -225 = 0",
    AspectRatio->Automatic];
Display["8.ps",d];
---------------------------------------------------------------------




Page 556 Problem 12
Find the Cartesian equation of the conic with the given properties.
Eccentricity 1,  focus (0,-3) and vertex (0,0)

parabola  p = 3
     x^2 = -12 y  
-------------------------------------------------------------------
p = 3
p1  = Plot[  -x^2/12,{x,-12,12}];
p2 = ListPlot[{{0,-p},{0,0},{-2p,-p},{2p,-p}},
     PlotStyle->{RGBColor[1,0,0],PointSize[0.04]}];
p3 = Plot[-p,{x,-2p,2p},PlotStyle->{RGBColor[1,0,0]}];
p4 = Plot[ p,{x,-12,12},PlotStyle->{RGBColor[1,0,0]}];
p5 = Show[p1,p2,p3,PlotLabel->" Page 556 Problem 12; x^2 = -12 y ",
        PlotRange->All,AspectRatio->Automatic];
Display["12.ps",p5];
-------------------------------------------------------------------
Page 556 Problem 16  Parabola with focus (3,2) and vertex (3,3)

           p = 1     (x-3)^2  = -4(y-3)
-------------------------------------------------------------------
vertex = {3,3};
p = 1;
p1 = Plot[ -(x-3)^2/4 + 3 ,{x,0,6}];
p2 = ListPlot[ { vertex+{  0,-p},
                 vertex+{ 2p,-p},
                 vertex+{-2p,-p}, 
                 vertex+{ 0, +p}}, 
     PlotStyle->{RGBColor[1,0,0],PointSize[0.03]}];
p3 = ParametricPlot[ vertex+{x,-p},{x,-2p,2p},PlotStyle->{RGBColor[1,0,0]}]; 
p4 = ParametricPlot[ vertex+{x,+p},{x, -6,6},PlotStyle->{RGBColor[1,0,0]}]; 
p5 = Show[p1,p2,p3,p4,PlotLabel->"Page 556 Problem 16; (x-3)^2 = -4 (y-3)",
            PlotRange->All,AspectRatio->Automatic];
Display["16.ps",p5];
-------------------------------------------------------------------
Page 556 Problem 20

(a)  write the equation in standard form
(b)  name the curve.

4 x^2 + 9 y^2 - 24 x - 36 y + 36 = 0

4( x^2 - 6 x     )  + 9 ( y^2 - 4 y    ) = -36

4( x^2 - 6 x + 9 )  + 9 ( y^2 - 4 y + 4 ) = -36 + 36 + 36

  4 (x-3)^2   + 9 (y-2)^2 = 36

    (x-3)^2      (y-2)^2 
  ---------- + ----------- = 1         It is an ellipse.
     9              4
------------------------------------------------------------------------
center = {3,2}; a=3; b=2; c=Sqrt[5];
h = b^2/a;
ans = Solve[4 x^2 + 9 y^2 - 24 x - 36 y + 36 == 0,y];
f[x_] = y /. ans[[1]];
g[x_] = y /. ans[[2]];
p1 = Plot[{f[x]+0.02,g[x]+0.02},{x, 0,6}];
p2 = Plot[  2 + 2 Sqrt[ 1- (x-3)^2/9],{x, 0, 6},PlotStyle->{RGBColor[1,0,0]}];
p3 = Plot[  2 - 2 Sqrt[ 1- (x-3)^2/9],{x, 0, 6},PlotStyle->{RGBColor[1,0,0]}];
p4 = ListPlot[ { center+{0,0}, center+{-a, 0},center+{ a, 0}, center+{ 0, b},
   center+{ 0,-b}, center+{-c, 0},center+{ c, 0}, center+{-c,-h},center+{-c, h}, 
   center+{ c,-h},center+{ c, h}}, PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}];
p5 = ParametricPlot[center+{-c,y},{y,-h,h},PlotStyle->{RGBColor[1,0,0]}];
p6 = ParametricPlot[center+{ c,y},{y,-h,h},PlotStyle->{RGBColor[1,0,0]}];
p7 = Show[p1,p2,p3,p4,p5,p6,PlotLabel->"Page 556 Problem 20", 
      AspectRatio->Automatic,PlotRange->All];
Display["20.ps",p7];
---------------------------------------------------------------------

Page 556 Problem 24
Determine the rotation angle theta needed to eliminate the
cross product term in  

        7 x^2 + 8 xy + y^2 = 9.

Then obtain the corresponding uv-equation and identity the 
conic that it represents.

                  6
Ctn[2 theta] = -------- = 3/4
                  8

Sin[2 theta] = 4/5    Sin[theta] = Sqrt[ (1-3/5)/2 ] = Sqrt[1/5] = 1/Sqrt[5]
Cos[2 theta] = 3/5    Cos[theta] = Sqrt[ (1+3/5)/2 ] = Sqrt[4/5] = 2/Sqrt[5]

x =  2/Sqrt[5] u - 1/Sqrt[5] v
y =  1/Sqrt[5] u + 2/Sqrt[5] v

 7 (2u-v)^2 + 8 (2u- v)(u+2v)+ ( u+2v)^2 = 9*5

             2      2
         45 u  - 5 v   = 45

           u^2     v^2   
        ------- - ------ = 1
            1       9
------------------------------------------------------------
U =  { 2/Sqrt[5],  +1/Sqrt[5] };
V =  {-1/Sqrt[5],  +2/Sqrt[5] };
a=1;
b=3;
c=Sqrt[10];
h = b^2/a;
ans = Solve[ 7 x^2 + 8 x y + y^2 == 9,y];
f[x_] = y /. ans[[1]];
g[x_] = y /. ans[[2]];
p1 = Plot[f[x],{x,-8, 1}];
p2 = Plot[g[x],{x,-1, 6}];
v[u_] :=  3 Sqrt[ u^2-1 ]; 
p3 = ParametricPlot[ u U +v[u] V,{u,-5,-a},
          PlotStyle->{RGBColor[0,1,0],Thickness[0.02]}];
p4 = ParametricPlot[ u U +v[u] V,{u, a, 5},
          PlotStyle->{RGBColor[0,1,0],Thickness[0.02]}];
p5 = ParametricPlot[ u U -v[u] V,{u,-5,-a},
          PlotStyle->{RGBColor[0,1,0],Thickness[0.02]}];
p6 = ParametricPlot[ u U -v[u] V,{u, a, 5},
          PlotStyle->{RGBColor[0,1,0],Thickness[0.02]}];
p7 = ListPlot[ {
     -a U, a U, -b V, b V,  -c U, c U, -c U - h V, -c U + h V,
     c U - h V, c U + h V},PlotStyle->{RGBColor[1,0,0],PointSize[0.02]}];
p8 = ParametricPlot[ x U + b/a x V, { x,-5,5},PlotStyle->{RGBColor[1,0,0]}];
p9 = ParametricPlot[ x U - b/a x V, { x,-5,5},PlotStyle->{RGBColor[1,0,0]}];
p10 = ParametricPlot[-c U + y V,{y,-h,h},PlotStyle->{RGBColor[1,0,0]}];
p11 = ParametricPlot[ c U + y V,{y,-h,h},PlotStyle->{RGBColor[1,0,0]}];
p12 = ParametricPlot[ x U + b V,{x,-a,a},PlotStyle->{RGBColor[1,0,0]}];
p13 = ParametricPlot[ x U - b V,{x,-a,a},PlotStyle->{RGBColor[1,0,0]}];
p14 = ParametricPlot[-a U + y V,{y,-b,b},PlotStyle->{RGBColor[1,0,0]}];
p15 = ParametricPlot[ a U - y V,{y,-b,b},PlotStyle->{RGBColor[1,0,0]}];
p16 = Show[p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12,p13,p14,p15,
     PlotLabel->"Page 556 Problem 24, 7 x^2 + 8 xy + y^2 = 9 ",
AspectRatio->Automatic];
Display["24.ps",p16];
----------------------------------------------------------------
Page 556 Problem 28
        3
r = ------------             r Cos[t] = 3          x = 3
      Cos[t]
----------------------------------------------------------

Page 556 Problem 32

r = 2 - 3 Cos[t]

Limacon  with an inner loop.

-------------------------------
p1 = ParametricPlot[ (2-3 Cos[t]){Cos[t],Sin[t]},{t,0,2 Pi}];
p2 = Show[p1,PlotLabel->"P 556 P 32  r=2-3 Cos[t]",AspectRatio->Automatic];
Display["32.ps",p2];
-----------------------------------------------------

Page 556 Problem 36

r = -theta    theta >= 0

Spiral of Archimedes.

-----------------------------------------------------
p1 = ParametricPlot[ -t {Cos[t],Sin[t]},{t,0,8 Pi}];
p2 = Show[p1,PlotLabel->"P 556 P 36  r = -theta",
      AspectRatio->Automatic,PlotRange->All];
Display["36.ps",p2];
-------------------------------------------------------

Page 556 Problem 40

Sketch the graphs of r = 5 Sin[t] and r = 2 + Sin[t]
--------------------------------------------------------
f[t_] := 5 Sin[t];
g[t_] := 2 + Sin[t];
a = ParametricPlot[ {f[t] Cos[t], f[t] Sin[t]},{t,0,2 Pi}];
b = ParametricPlot[ {g[t] Cos[t], g[t] Sin[t]},{t,0,2 Pi}];
c = ListPlot[ {f[Pi/6] {Cos[Pi/6],Sin[Pi/6]}, f[5Pi/6] {Cos[5Pi/6],Sin[5Pi/6]}},
    PlotStyle->{RGBColor[1,0,0],PointSize[0.09]}];
d = Show[a,b,c,AspectRatio->Automatic,
      PlotLabel->"Page 556 Problem:40 r=5 Sin[t],r=2+Sin[t]"]
Display["40.ps",d];
--------------------------------------------------------------

5 Sin[t] = 2 + Sin[t]

4 Sin[t] = 2

Sin[t] = 1/2

t = Pi/6   or t = 5 Pi/6
r = 5/2 
---------------------------------------------------------
Page 556 Problem 44 Match each polar equation with its graph.

r = 1 - 2 Sin[theta]   Limacon with an inner loop  along y axis

r = 1 + Sin[theta]/2   Limacon with no inner loop  along y axis

r = 1 + 2 Cos[theta]   Limacon with an inner loop  along x axis

r = 1 + Cos[theta]/2   Limacon with no inner loop  along x-axis
--------------------------------------------------------------------