DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2513

FINAL EXAMINATION
April 1997
TIME: 3 HOURS
 

ANSWER ALL QUESTIONS WITHOUT THE USE OF A CALCULATOR. SHOW ALL WORK AS CREDIT WILL BE GIVEN TO PRESENTATION OF SOLUTIONS. NO CREDIT WILL BE GIVEN FOR ILLEGIBLE WORK.

MARKS
(12) 1. Given ${\bf u} = 3{\bf i} - {\bf j} + {\bf k},
\;\;{\bf v} = {\bf i} - 3 {\bf j} + 2 {\bf k},\;\;{\bf w} = 2{\bf i}
- {\bf j}$, find:
(a) ${\bf u} \cdot {\bf v}$ and ${\bf u} \times {\bf v}$;
(b) the vector projection of ${\bf u}$ on ${\bf w}$;
(c) the volume of the parallelipiped for which ${\bf u},{\bf v},{\bf w}$ are concurrent edges;
(d) the cosine of the angle $\theta$ between ${\bf v}$ and ${\bf w}$;
(e) a vector in the plane of ${\bf u}$ and ${\bf w}$ that is orthogonal to ${\bf w}$.
(12) 2. (a) Determine the symmetric equations of the line through the two points, A(1,-1,2) and B(3,-2,1).
(b) Determine the scalar equation of the plane through the three points $P(0,-1,1),\;\;Q(2,-3,4)$ and R(2,-1,-1).
(c) Find the point of intersection of the line, ${\displaystyle
\frac{x-2}{1} = \frac{y-3}{5} = \frac{z-1}{-2}}$ with the plane 3x + y - 2z + 5 = 0.
(d) Identify and neatly sketch the surface given by 4x2 - y2 - z2 + 8x + 2z + 7 = 0.
(13) 3. (a) Find ${\displaystyle \frac{\partial^2 z}{\partial x
\partial y}}$ for the function, z = (x2 + y2)exy.
(b) Given ${\displaystyle w = \frac{x}{3y-2z}}$ where $x = 3r^2 +
2s^2,\;\;y = r \sin (r^2 + s^2)$ and $z = s \cos(r^2 + s^2)$; find ${\displaystyle \frac{\partial w}{\partial r}}$ by means of the chain rule.
(c) Let z be given implicitly as a function of x and y by the relationship, x2 + 2y2 + z3 - 6xy + 2yz + 6 = 0. Determine the derivative, ${\displaystyle \frac{\partial z}{\partial x}}$.
(13) 4. (a) Compute the directional derivative $D_{\bf u}f$ of the function
f(x,y,z) = x2 + 2y2 - z2 + 4xyz
in the direction of ${\bf u} = 2{\bf i} - {\bf j} + 2{\bf k}$ at the point (-1,2,1).
(b) Determine the equation of the tangent plane to the surface
z = x2 - 4xy + 2y2
at the point (1,1,-1).
(c) Use differentials to find an approximate value for the number $\sqrt{(6.02)^2 + (2.97)^2 + (2.01)^2}$.
(11) 5. Find all critical points of the function
f(x,y) = y3 + 3x2y - 3y2 - 3x2 + 2
and determine the nature of each critical point.
(12) 6. (i) Draw a neat sketch of the region, R, in the first quadrant of the x-y plane, that is bounded below by the line y = x and above by the parabola, y = -x2 + 4x. Hence evaluate the double integral,
${\displaystyle \int \int (1-x)dA}$.
R
(ii) Change the order of integration in the double integral,
\begin{displaymath}\int_4^5 \int_0^{\sqrt{25-x^2}} f(x,y)dy\;dx.
\end{displaymath}
(10) 7. Let R denote the region in the space of the 3-dimensional coordinates (x,y,z) that is bounded on its sides by the planes $x = 0,\;y = 0$ and x + y + z = 2, and bounded from below by the plane z = 1. Calculate, the triple integral,
${\displaystyle \int \int \int x\;dV}$
R
(17) 8. (i) Use double integration in polar coordinates to determine the area of the region in the x-y plane that lies outside the circle x2 + y2 = 1 and inside the circle x2 + y2 - 2y = 0.
(ii) Use triple integration in spherical coordinates to determine the volume of the region in the space $z \geq 0$ of the x,y,z-coordinate system, which is bounded below by the cone x2 + y2 = z2 and above by the sphere x2 + y2 + z2 = 1.

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