MATH 2513

FINAL EXAMINATION December 1997 | TIME: 3 HOURS |

## CALCULATORS ARE NOT ALLOWED |

MARKS | ||||

(10) | 1. | (a) | Calculate the angle between the vectors | |

(b) | Given , find the vector projection of onto . | |||

(c) | Find the area of a triangle with vertices at . | |||

(d) | Find the work done by the force
,
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
intersects the plane
4x + 5y - 2z = 18.
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(d) | Find the equation of the plane that passes through the point (1,6,-4) and contains the line . | |||

(e) | Find the point of intersection of the lines L_{1} and L_{2},
where
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(7) | 3. | (a) | If
z = f(x,y), where
,
| |

find and ; | ||||

show that . | ||||

If , show | ||||

(9) | 4. | (a) | Find the equation of the tangent plane and the
equations of the normal line to the surface
z = x^{2} + xy - y^{2} at
the point where x = 2 and y = -1.
| |

Use differentials to find the approximate value of . | ||||

Given the function
f(x,y,z) = xy + yz^{2} + xz^{3}
and the point
P = (2,0,3), find
| ||||

(i) | the directional derivative of the function at P in the
direction of the vector
;
| |||

(ii) | the maximum rate of change at P and the direction in which it
occurs.
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(b) | Given , use the chain rule to find at . | |||

Find the local maximum and minimum values and
saddle points of
f(x,y) = 2x^{3} - 24xy + 16y^{2}.
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(b) | Use Lagrange multipliers to find the point on the plane
2x + 3y
- z = 1, which is closest to the origin.
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(18) | 7. | (a) | Sketch the region of integration and evaluate | |

| ||||

by reversing the order of integration. | ||||

(b) | Find the volume of the solid bounded by the planes . | |||

Find the volume of the solid under the paraboloid
z = x^{2} +
y^{2} and above the region y = x^{2} and x = y^{2}.
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(17) | 8. | (a) | Sketch the solid whose volume is given by the integral | |

and evaluate the integral. | ||||

(b) | Use cylindrical co-ordinates to find the volume of the region
E bounded by the paraboloids
z = x^{2} + y^{2} and
z = 36 - 3x^{2} -
3y^{2}.
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(c) | Use spherical co-ordinates to evaluate | |||

E
| ||||

where E lies between the spheres
and
and above
the cone
.
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(100) |