MATH 2013

FINAL EXAMINATION April 1998 | TIME: 3 HOURS |

**CALCULATORS PERMITTED**

MARKS | |||

Evaluate
,
where R is the region in
between the curves y = x and
y = x^{4}, for
.
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Show that
,
where D_{a} is the disc of radius a > 0,
centred at the origin.
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Evaluate
where E is the region in
bounded by the
spheres of radii a and b (b > a > 0), centred at the origin.
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Show that the integral
satisfies
,
where S is the unit square
.
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(b) | Compute an approximate value for I by partitioning S by the
lines
and then using double sum with the
integrand evaluated at the mid-points of the sub-regions defined by
the partition.
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Consider the vector field defined on . | |||

(a) | Compute div and curl. | ||

(b) | Find a potential function for . | ||

Evaluate the line integral
where
and
is any smooth curve joining A to B.
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Compute the flux of
across the ellipsoid
a^{2}x^{2} +
b^{2}y^{2} + c^{2}z^{2} = 1.
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Show that where | |||

(10) | 4. | Use the change of coordinates in
defined by
,
to
show that
where T is the triangular region in the (x,y)
plane whose vertices are
and (0,1).
| |

(15) | 5. | In this problem you may use the standard
formulas for area of a disc and surface area of a sphere:
and ,
respectively, where R is the radius.
| |

Evaluate
where
is the circle of radius 3 centred at the origin, oriented
clockwise.
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Compute the value of
where
is the sphere
x^{2} + y^{2} + z^{2} = a^{2}, oriented by its
outward normal
and
,
where e is a constant and
is the unit normal. Express e in terms of Q
and a.
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Suppose that two vector fields
and ,
defined
and smooth on ,
are related by curl
.
Show that the flux
of J through a surface
is equal to the
circulation
of
around ,
where
is the boundary of .
[Assume that
and
are smooth and compatibly
oriented.]
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(10) | 6. | (a) | Determine which of the following sequences converges and obtain the limit in the convergent cases. |

(i) | |||

(b) | Determine whether each of the following series is absolutely convergent , conditionally convergent or divergent. | ||

(i) | |||

Find either the general (n
)
term,
or the terms of degree ,
in the
Taylor series expansion about x = 0 (Maclaurin series) for
| |||

(i) | |||

and determine the interval of convergence in each case. | |||

(b) | Set x = 1/3 in (a) (ii) and compute the approximation to
given by the sum of the first three non-zero terms of the Taylor series.
Compute the error, if the exact value is
.
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Compute an approximation to
by using the first three non-zero terms in the Taylor expansion for
f(t) about t = 0. Use this result to approximate F(1/2).
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(100) |