F1.8CD1 Sample exam
There is a total of 100 marks in the paper
1. (10 marks) Let (an ) be a sequence. Define what is meant by lim an = L.
n→∞
Determine the limit
5n + (−1)n e−n
lim
n→∞ 2n2 + n sin(2n)
and prove it using the definition.
2. (8 marks)) Consider a function f : R2 → R defined as
f (x, y) =
2x2 + 3y 2
,
x2 − y 2
|x| =
6 |y|
Demonstrate that the limit
lim
f (x, y)
(x,y)→(0,0)
does not exist.
3. (10 marks) Suppose f : R → R is a function of one variable. Consider
g : R2 → R defined as
g(x, y) = xf 2 (xey )
Find the expressions for partial derivatives gx (x, y), gy (x, y) in terms of the
function f and its derivative f 0 .
4. (10 marks) Find the stationary points of the following function, and
determine their nature
f (x, y) = x3 − 3x + yx +
y2
.
9
5. (12 marks) Use Lagrange multipliers to find the minimum and maximum
of function
f (x, y) = xy
in the region 4x2 + 9y 2 ≤ 16.
6. (8 marks) Find the Taylor expansion of
f (x, y) = xey − 2 sin(yx)
about the point (x, y) = (−2, 0) up to the second order.
7. (10 marks) Interchange the order of limits and then evaluate the following
iterated integral
1−e
Z −1
0
Z5
ex/5 dxdy .
−5 ln(1−y)
CONTINUED OVER
8. (12 marks) Use polar coordinates to evaluate the double integral
ZZ
xdxdy
D
where D = {(x, y)|(x − 1)2 + y 2 ≥ 1 , (x − 2)2 + y 2 ≤ 4}.
9. (10 marks) Find the volume of the solid paraboloid of revolution
y = 10 − x2 − z 2 ,
y ≥ 0.
10. (10 marks) Use convergence tests to determine whether the following
integral converges
Z∞
sin(x)
x3/2
0
State which tests you are using.
END OF PAPER
Answers and hints
2. Hint: it suffices to find two sequences of points on the plane: (xn , yn ),
(x0n , yn0 ) for which the limits of f (xn , yn ), f (x0n , yn0 ) are different.
3.
gx = f 2 (xey ) + 2xey f (xey )f 0 (xey ) , gy = 2x2 ey f (xey )f 0 (xey ) .
4. (2,-9) local minimum; (− 21 , 94 ) a saddle point.
√
√ √
√
5. minimum fmin = −4/3 achieved
at
(
2,
−2
2/3),
(−
2, 2 2/3) ; maxi√ √
√
√
mum fmax = 4/3 achieved at ( 2, 2 2/3), (− 2, −2 2/3).
6.
f (x, y) = −2 + (x + 2) + 2y − y(x + 2) − y 2 + R3 (x, y)
7. I = 5e − 10
8. I = 7π
9. V = 50π
10. Converges. (Hints: The convergence for x → 0 and for x → ∞ has to
be shown. For x → 0 one needs to use Taylor’s formula)