Erik Bennett
MA 5901
3160  Exam

#) Match the following graphs with their respective equations below.

SetOptions[
          Plot3D,  
DisplayFunction -> 
            Identity, 
Shading -> 
            False, 
Ticks -> 
            False, 
ClipFill -> 
            None
];

f1[x_, y_] = -x*y*
          Exp[-(x^2 + y^2)];
f2[x_, y_] = 
        Cos[x];
f3[x_, 
          y_] = (x^2 + y^2)^(-1);
f4[
          x_, y_] = 
        Sin[(x^2 + y^2)]/(x^2 + y^2);
f5[
          x_, y_] = -(x^2 + y^2)^(1/
                2);
f6[x_, 
          y_] = -(x^2 + y^2);
Pic1  = 
        Plot3D[f1[x, y], {x, -2, 2}, {y, -2, 
            2}, 
PlotLabel -> "A"];
Pic2  = 
        Plot3D[f2[x, y], {x, -2 Pi, 2  Pi}, {y, -2 Pi, 
            2  Pi}, 
PlotLabel -> "B"];
\(Pic3\  = Plot3D[
          f3[x, y], {x, \(-1\), 1}, {y, \(-1\), 
            1}, 
PlotLabel -> "C"];\)
Pic4 = 
        Plot3D[f4[x, y], {x, -Pi, Pi}, {y, -Pi, 
            Pi}, 
PlotLabel -> "D"];
Pic5 = 
        Plot3D[f5[x, y], {x, -1, 1}, {y, -1, 
            1}, 
PlotLabel -> "E"];
Pic6 = 
          Plot3D[f6[x, y], {x, -1, 1}, {y, -1, 
              1}, 
PlotLabel -> "F"];

Show[
GraphicsArray[
{{Pic1, Pic2, 
              Pic3}, {Pic4, Pic5, 
              Pic6}}, 
GraphicsSpacing -> 
            0], 
Frame -> 
          True];
({{f[x, y], "Graph"}, {f5[x, y], "_____"}, {f2[x, y], "_____"}, {f6[x, y], "_____"}, {f1[x, y], "_____"}, {f4[x, y], "_____"}, {f3[x, y], "_____"}})

Solutions:(({{f(x, y), "Graph"}, {-(Sqrt[(x^2) + (y^2)]), "E"}, {cos(x), "B"}, {-(x^2) - (y^2), "F"}, {-(e^(-(x^2) - (y^2))) x y, "A"}, {((sin((x^2) + (y^2)))/((x^2) + (y^2))), "D"}, {(1/((x^2) + (y^2))), "C"}}))

Grading scheme: Each correct match is worth 2 points for a maximum of 10 points.

#) Given z = 2  (x^3) y, x = (e^(-t)), y = cos(t).  Find dz/dt using the chain rule. Your solution should be written in terms of t only.

Solution:	1)(dz/dt) =  ∂(z/∂)x 
          (dx/dt)+∂(z/∂)y (dy/dt)					(correct expression of chair rule (in either general form or as used in work to solve,i.e. either line 1 or line 2) worth 3 of 10 points)
		2)(dz/dt) =  
        6  (x^2) y(-(e^(-t)))  +  
          2  (x^3)(-sin(t))			(each correct derivative worth 1 point)
		3)(dz/dt) =  
        6 ((((e^(-t))))^2) cos(t) (-(e^(-t)))  + 
          2 ((((e^(-t))))^3) (-sin(t)) 	(correct answer in terms of t worth 3 points)
		4)(dz/dt)=  -2  (e^(-3 t))(
          3  cos(t)  + sin(t))				(simplified)

Simplify[D[2  Exp[-t]^3 Cos[t], t]]