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Practice Exam I

Note: the actual exam will be significantly shorter than this.

1.
(Vectors) Page 714, #47.

\begin{displaymath}
\vec{v}_a=(500+25\sqrt{2})\vec{\imath}+(25\sqrt{2})\vec{\jmath}\end{displaymath}

2.
(Dot Product)
(a)
Page 724, #55.

\begin{displaymath}
\approx 7323.385 \text{~calories}\end{displaymath}

(b)
Page 724, #61.

\begin{displaymath}
\langle w_1, -\frac{7}{2}w_1, -2w_1 \rangle\end{displaymath}

will work for any non-zero w1.

3.
(Cross Product)
(a)
Page 792, #5. Use the cross product.

(b)
Page 732, #24.

\begin{displaymath}
A\approx 8864.925 \text{~square feet}\end{displaymath}

4.
(Lines)
(a)
Page 792, #6.

\begin{displaymath}
x=2t+1, y=-t-1, z=3t; \frac{x-2}{2}=\frac{y+1}{-1}=\frac{z}{3}\end{displaymath}

(b)
Page 792, #7.

\begin{displaymath}
x=1+2t, y=-1+3t, z=2-3t; \frac{x-1}{2}=\frac{y+1}{3}=\frac{-z+2}{3}\end{displaymath}

5.
(Planes)
(a)
Page 792, #8.

x+y+2=0

(b)
Page 792, #9.

-13x+22y+6z=-23

6.
(Parametric Equations)
(a)
Page 620, #39. The straight line y=x+1.

(b)
Page 620, #46.

x+y=2

7.
(Vector Differentiation) Page 754, #13.

\begin{displaymath}
\vec{v}=\vec{\imath} + 3e^t \vec{\jmath}+3t^2\vec{k};
\vec{a}=3e^t\vec\jmath+6t\vec{k}, v=\sqrt{1+4t^2+9t^4}\end{displaymath}

8.
(Vector Integration) Page 792, #19.

Integrate twice to get position vector; then check that it falls on x2+y2=1.

9.
(Curvature and Acceleration)
(a)
Page 792, #16.

x+2y+3z=6

(b)
Page 792, #23. Also find the arc-length parametrization when the starting point is t=1.

\begin{displaymath}
\kappa=\frac{1}{9}; a_T=2, a_N=1; x=\frac{1}{2}(-1+\sqrt{9+4...
 ...qrt{9+4s}))^2,
z=\frac{4}{3}(\frac{1}{2}(-1+\sqrt{9+4s}))^{3/2}\end{displaymath}

10.
(Cylinders and Quadric Surfaces) Page 793, #34 but ignore the part about cylindrical coordinates.

\begin{displaymath}
x^2+y^2+z^2=2\sqrt{x^2+y^2}\end{displaymath}

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Up: Math 103 Home Page
Joshua Holden
9/22/1999