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\u{\LARGE {MA2160 \hs Sections 6 and 8 \hs Exam 1}}\\
\u{\Large {February 11, 2004}}\\
\u{\Large {Do all problems}}\\
\u{\Large {Show all work}}\\
\end{center}

{\large {\n 1. Suppose ${\overline u} = -1{\overline i}-2{\overline j}+
2{\overline k}$, ${\overline v} = {\overline i}+4{\overline j}$,
and ${\overline w} = 4{\overline i}+2{\overline j}-4{\overline k}$. Find the 
following.

\vso

(a) (4 points) $\parallel {\overline u} \parallel$,

\vso

(b) (4 points) a unit vector in the direction of ${\overline u}$,

\vso

(c) (8 points) write ${\overline v}$ as the sum of two vectors, one in the 
direction of ${\overline u}$ and the other perpendicular to ${\overline u}$,

\vso

(d) (8 points) ${\overline v}\times{\overline w}$,

\vso

(e) (8 points) the volume of the parallelepiped with adjacent sides 
${\overline u},{\overline v}$, and ${\overline w}$.

\vs

\n 2. (12 points) Let P, Q, and R be three points with Cartesian coordinates 
$(5,-2,0), (-2,2,-1),$ and $(3,2,-4)$. Find an equation of the plane containing 
P, Q, and R.

\vs

\n 3. (12 points) An airplane is heading west at an airspeed of $400$ km/hr, 
but there is a wind blowing from the southwest at $30$ km/hr. Set up a 
coordinate system so that the $x$-axis points east. Find the vector which 
represents the velocity of the plane relative to the ground.

\vs

\n 4. Evaluate the following integrals.

\vso

(a) (6 points) $\int (x^3+2x+sinx)\ dx$,

\vso

(b) (6 points) $\int xe^{2x}\ dx$,

\vso

(c) (6 points) $\int {{x+3}\over{x^3+3x^2+2x}}\ dx$,

\vso

(d) (6 points) $\int {1\over{4+x^2}}\ dx$. 

\pagebreak

\n 5. (8 points) Evaluate the definite integral $\int_0^1x^2e^{(x^3+5)}\ dx$ 
exactly.

\vs

\n 6. Consider the definite integral $\int_0^4x^2\ dx$.

\vso

(a) (6 points) Find LEFT(2), RIGHT(2), and TRAP(2).

\vso

(b) (4 points) Evaluate the integral exactly.

\vso

(c) (2 points) Which of the answers in part (a) is most accurate?

}}

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