DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2513

FINAL EXAMINATION
December 1997
TIME: 3 HOURS
 

CALCULATORS ARE NOT ALLOWED

MARKS
(10) 1. (a) Calculate the angle between the vectors
\begin{displaymath}\bar{u} = 3 \bar{i} + 4 \bar{j} + 5 \bar{k}\;\;\mbox{and}\;\;\bar{v} =
\bar{i} - 7 \bar{j} + 3 \bar{k}.
\end{displaymath}
(b) Given $\bar{u} = 2 \bar{i} - 3 \bar{j} + 4 \bar{k},\;\;\bar{v} =
-2 \bar{i} - 3 \bar{j} + 5 \bar{k}$, find the vector projection of $\bar{u}$ onto $\bar{v}$.
(c) Find the area of a triangle with vertices at $P = (1,3,-2),\;\;Q
= (2,1,4),\\ R = (-3,1,6)$.
(d) Find the work done by the force $\bar{F} = 5 \bar{i} - 3 \bar{j}
+ 2 \bar{k}$, as its point of application moves from the point A = (2,1,3) to B = (4,-1,5).
(20) 2. (a) Find the equation of the plane passing through the point A = (6,5,-2) and parallel to the plane x + y - z + 1 = 0.
(b) Find the (i) parametric and (ii) symmetric equations of the line through the point A = (1,-2,3) and perpendicular to the plane 2x - y + 3z = 5.
(c) Find the point at which the line $x = 2 + 3t,\;\;y = -4t,\;\;z =
5 + t$ intersects the plane 4x + 5y - 2z = 18.
(d) Find the equation of the plane that passes through the point (1,6,-4) and contains the line $x = 1 + 2t,\;\;y = 2 - 3t,\;\;z =
3-t$.
(e) Find the point of intersection of the lines L1 and L2, where
\begin{eqnarray*}L_1:x & = & 1 + 2t,\;\;y = t,\;\;z = 1 + 4t\\
L_2:x & = & x = s,\;\;y = 2s-2,\;\;z = 3s-2.
\end{eqnarray*}
(7) 3. (a) If z = f(x,y), where $x = r \cos \theta,\;\;y
= r \sin \theta$,
(i) find ${\displaystyle \frac{\partial z}{\partial r}}$ and ${\displaystyle \frac{\partial z}{\partial \theta}\;}$;
(ii) show that ${\displaystyle \left( \frac{\partial z}{\partial x}
\right)^2 + \left( \frac{\p...
...\right)^2 + \frac{1}{r^2}\;\left(
\frac{\partial z}{\partial \theta} \right)^2}$.
(b) If ${\displaystyle f(x,y) = \tan^{-1}\;\frac{y}{x}}$, show
\begin{displaymath}\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}
= 0.
\end{displaymath}
(9) 4. (a) Find the equation of the tangent plane and the equations of the normal line to the surface z = x2 + xy - y2 at the point where x = 2 and y = -1.
(b) Use differentials to find the approximate value of $\sqrt{(5.98)^2 + (8.01)^2}$.
(9) 5. (a) Given the function f(x,y,z) = xy + yz2 + xz3 and the point P = (2,0,3), find
(i) the directional derivative of the function at P in the direction of the vector $\bar{v} = \langle -2,-1,2 \rangle$;
(ii) the maximum rate of change at P and the direction in which it occurs.
(b) Given $w = xy + yz + zx,\;\;x = t \cos t,\;\;y = t \sin t,\;\;z
= t$, use the chain rule to find ${\displaystyle \frac{dw}{dt}}$ at ${\displaystyle t = \frac{\pi}{2}}$.
(10) 6. (a) Find the local maximum and minimum values and saddle points of f(x,y) = 2x3 - 24xy + 16y2.
(b) Use Lagrange multipliers to find the point on the plane 2x + 3y - z = 1, which is closest to the origin.
(18) 7. (a) Sketch the region of integration and evaluate
\begin{displaymath}\int_0^1 \int_{\sqrt{y}}^{1} \sqrt{x^3+1}\;dx\;dy
\end{displaymath}

by reversing the order of integration.
(b) Find the volume of the solid bounded by the planes $x = 0,\;\;y
= 0,\\ z = 0,\;\;x + y + z = 1$.
(c) Find the volume of the solid under the paraboloid z = x2 + y2 and above the region y = x2 and x = y2.
(17) 8. (a) Sketch the solid whose volume is given by the integral
\begin{displaymath}\int_0^{\frac{\pi}{3}} \int_0^{2 \pi} \int_0^{\sec \phi} p^2 \sin \phi\;\;d
\rho\;\;d \theta\;\;d \phi
\end{displaymath}
and evaluate the integral.
(b) Use cylindrical co-ordinates to find the volume of the region E bounded by the paraboloids z = x2 + y2 and z = 36 - 3x2 - 3y2.
(c) Use spherical co-ordinates to evaluate
${\displaystyle \int \int \int x^2\;dV,}$
E
where E lies between the spheres $\rho = 1$ and $\rho = 3$ and above the cone ${\displaystyle \phi = \frac{\pi}{4}}$.

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