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curve that bounds the surface. (b) Consider the surface
S =
(x, y, z) | z = x
2 + y
2
; z ≤ 9
.
Sketch the surface S in R
3 and show the curve that bounds S on your sketch. Then use Stokes’ Theorem
to evaluate the flux integral
Z Z
S
(curl F)
Z
C
(xy + z) ds
where C is the line segment in R
3 with initial point (1, 0 − 1) and endpoint (2, 1, 1). [8]
9. Let D be the region in R
3
that lies inside the sphere x
2 + y
2 + z
2 = 2 and above the plane z = 1.
(a) Sketch the region D in R
3
. (3)
(b) Express the volume of D in terms of a triple integral, using spherical coordinates. DO NOT
EVALUATE THE INTEGRAL. (7)
[10]
10. Let S be that part of the surface z =
p
x
2 + y
2 that lies between the plane z = 1 and the plane z = 3.
(a) Sketch the surface S, together with its XY-projection. (3)
(b) Use a surface integral to determine the area of S
2 â R function f defined by
f (x, y) = â
xy
and the R â R
2
function r defined by
r (t) = ï ¿ ¾
e
2t
, cost
.
Use the General Chain Rule to determine the value of (f â ¦ r)
0
(0). [7]
6. Consider the R
2 â R function f defined by
f (x, y) = 2x + 3y
2 â x
2 + y
3
.
Determine how many saddle points the function f has. (Make use of the Second Order Partial Derivatives
Test for Local Extrema.) [9]
7. Use the Method of Lagrange to determine the largest possible volume that a rectangular block with a square
base can have if the height of the block plus the perimeter of its base equals 30cm
3
. A parametric equation for L1 is
(x, y, z) = (1, 0, 2) + t(2, 4, −1); t ∈ R.
L2 is parallel to L1 and contains the point (3, 4, 3). V is the plane that contains both the lines L1 and L2. Find
an equation for V . [7]
2. Prove from first principles that
lim
(x,y)→(2,2)
(x − y) = 0.
[8]
3. Prove that lim
(x,y)→(0,0)
x
2 + y
x − y
does not exist. [8]
4. Consider the R
2 − R function f defined by
f (x, y) = x
2 − 2xy + y
2
.
Let V be the plane that is tangent to the graph of f at the point (x, y, z) = (1, −1, 4).
(a) Find a vector that is perpendicular to the plane V . (4)
(b) Find an equation for the plane V .
A closed rectangular box whose length is double its width has a total surface area of 600 cm². Find the dimensions of the box with maximum volume.
A bee flies on a trajectory such that its polar coordinate at time t are given by "r=bt\/T"2(2T-t) "\\theta=t\/T" (0<t<2T) where b and T are positive constants. Find the velocity vector of the bee at time t. Show that the least speed achieved by the bee is b/T. Find the acceleration of the bee at this instant.
A body oscillates with simple harmonic motion according to the equation x(t)=5sin(4pie.t+pie/4
(x in meters). At time t = 5 seconds what are : /(6 x
2.5mrks)
(a) the displacement (b) the velocity
(c) the acceleration (d) the phase of the motion
(e) the frequency (f) the period of the motion.
Suppose that the profit P obtained in selling x units of a certain item each week is
given by
p=50 √ 3 - 0.5-500(0<x<8000)
.
Find the rate of change of P with respect to x when x=1600.
#340153 on Dec 2023