MA2160: Exam 3: Section:_____
Name:_______________________ All questions are 10pts.  Show work for full credit.

1) ({{  }}) Match the slope fields and equations
 ({{f=(e^(-0.5t)), }, {f=(3-y) y, "           "}, {f=t, "           "}, {f=y |3-y|, "           "}})

Needs["VectorFieldPlots`"]
Clear[f,t,y]
f1[t_,y_]=y(3-y);
f2[t_,y_]= (E^(- t/2));
f3[t_,y_]=t ;
f4[t_,y_]= y Abs[3-y];
Grid[{
{f[t,y]==f2[t,y],},
{f[t,y]==f1[t,y],"           "},
{f[t,y]==f3[t,y],"           "},
{f[t,y]==f4[t,y],"           "}
},
Dividers->All,
Alignment->Left
]
TableForm[{{
VectorFieldPlot[
{1,f1[t,y]},
{t, 0,3},
{y,0,4},
PlotLabel->"Slope Field 1",
Frame->True,
ScaleFunction->Function[x,1],
AspectRatio->Automatic,
FrameLabel->{"t","y"}
],
VectorFieldPlot[
{1,f2[t,y]},
{t, 0.05,3},
{y,0.05,4},
PlotLabel->"Slope Field 2",
Frame->True,
FrameLabel->{"t","y"},
AspectRatio->Automatic,
ScaleFunction->Function[x,1]
]
},
{VectorFieldPlot[
{1,f3[t,y]},
{t,0.05,3},
{y,0.05,4},
PlotLabel->"Slope Field 3",
Frame->True,
AspectRatio->Automatic,
FrameLabel->{"t","y"},
ScaleFunction->Function[x,1]
],
VectorFieldPlot[
{1,f4[t,y]},
{t, 0.05,3},
{y,0.05,4},
PlotLabel->"Slope Field 4",
Frame->True,
FrameLabel->{"t","y"},
AspectRatio->Automatic,
ScaleFunction->Function[x,1]
]
}}
]

2) ({{  }}) Is 3+3 1/8 + 3 1/(8^2)+… bigger than or less than 3.429?

3) ({{  }}) Find the Taylor polynomial about θ=0 of degree 6 for ln(1+(θ^2)).

4) ({{  }}) Write down the Taylor series about x=0 (including the general term) for sin(x).  Give a reasonable bound for the error in using the Taylor polynomial of degree 3 about x=0 to approximate sin(x) on the interval -0.2≤x≤0.2.

5) ({{  }}) Compute the plane through the three points P =(3,5,2) , Q =(-2,0,4), and R = (6,1,3).

({{Q1, Q2, Q3, Q4, Q5, Q6, Q7, Q8, Q9, Q10, Total}, {    ,     ,     ,     ,     ,     ,     ,     ,     ,      ,  }})

6) ({{  }}) Compute ∫x cos(2x)dx.

7) ({{  }}) Does (∫_0^∞)x (e^(1-(x^2)))dx converge?  Give the value if it converges.

8) ({{  }}) Approximate (∫_1^2)(w/(1+(w^2)))dw using Simpson's rule with 2 subdivisions. Compute an exact value for the integral.  How much smaller would your error be if you used Simpson's rule with 8 subdivisions.  Show your work!

9) ({{  }}) Sketch the region ℛ  in the xy plane bounded by y=(((x+1))^2) and y=9.  Set up (but do not evaluate) an integral for the volume of the solid with square cross-sections perpendicular to the x-axis on the region ℛ .

Plot[{0},{x,-5,5},
PlotRange->{0,10},
AxesLabel->{"x","y"}]

10) ({{  }}) A circular cone of height 12ft and radius 4ft is filled with oil weighing 50 lbs/(ft^3). Compute the work required to pump all the oil out of the top.

Plot3D[
12- 3(Sqrt[(x^2)+(y^2)]),
{x, -4,4},
{y,-4,4},
PlotRange->{0,12},
Axes->Automatic,
BoxRatios->Automatic
]