DEPARTMENT OF MATHEMATICS & STATISTICS

MATH 2013

FINAL EXAMINATION
April 1997
TIME: 3 HOURS

MARKS
(5)1. Show ${\bf F}(x,y,z) = 2xy{\bf i} + (x^2 + 2yz){\bf j} + y^2{\bf
k}$ is a conservative vector field and find a potential function for ${\bf F}$.
(5)2. Evaluate: ${\displaystyle \int_0^1 \int_x^1
e^{x/y}dy\;dx}$.
(5)3. Find the volume of the solid bounded above by the cone $z = \sqrt{x^2 + y^2}$ and below by z = x2 + y2.
(5)4. Evaluate: ${\displaystyle
\int_0^1 \int_0^{\sqrt{1-x^{2}}} \int_0^{\sqrt{1-x^{2}-y^{2}}} (x^2 +
y^2 + z^2)^{1/2}dz\;dy\;dx}$.
 

You need not evaluate any integrals for the questions on the remainder of the exam.

(5)5. Set up an integral for the area of that part of the hyperbolic paraboloid z = y2 - x2 that lies between the cylinders x2 + y2 = 1 and x2 + y2 = 4.
(8)6. Let C be the closed curve consisting of the arc of y = x2 from (0,0) to (1,1) and then the straight line from (1,1) to (0,0).
(a) Set up ${\displaystyle \oint_C \sin y^2dx - \sin x^2dy}$ as a single integral.
(b) Using Green's Theorem, set this up as a double integral.
(8)7. Set up the two integrals of Stokes Theorem for the vector field
${\bf F}(x,y,z) = y^2{\bf i} + x^2{\bf j} + z^2{\bf k}$ and the surface consisting of that part of
z = 1 - x2 - y2 that lies above the xy plane.
(8)8. Set up the two integrals of the Divergence Theorem for the vector field $F(x,y,z) = x{\bf i} + y{\bf j} + z{\bf
k}$ and the unit ball $x^2 + y^2 + z^2 \leq 1$.
(12)9. Find power series representations of the following (either present the general term or at least 5 non-zero terms):
(a) ${\displaystyle f(x) = \frac{1}{1 + x^4}}$
(b) ${\displaystyle \int \frac{1}{1 + x^4}\;dx}$
(c) ${\displaystyle f(x) = x^2 \cos(x^2)}$
(d) ${\displaystyle f(x) = \frac{1}{(1+x)^{3}}}$
(6)10. (a) Find the radius of convergence for ${\displaystyle
\sum_{n=0}^{\infty}\;\frac{x^n}{n^2}}$.
(b) What is the sum of ${\displaystyle
\sum_{n=0}^{\infty}\;\frac{2^n}{3^{n-1}}}$?
(10)11. Use power series to solve the following differential equation. Give at least 5 non-zero terms or state the general term.
\begin{displaymath}y'' + 3xy' - 3y = 0,\;\;\;y(0) = 1,\;\;\;y'(0) = 0
\end{displaymath}

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