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(MATH102)[2005](s)final~1659^_10420.pdf
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HKUST MATH 102
Final Examination
Multivariable Calculus
Answer ALL 8 questions
Time allowed C 3 hours
Problem 1
3
(a)
If f (x, y)=(x+ y2 ) 3 , .nd fx(0, 0) and fy (0, 0).
(b)
Let z = f (x, y), where x = g(t) and y = h(t).
(i) Show that
d . .z . .2 zdx .2 zdy d . .z . .2 zdx .2 z dy
=+ and =+ .
dt.x .x2 dt .y.x dt dt .y .x.y dt .y2 dt
d2 z
(ii) Use the formulas in part (i) to help .nd a formula for .
dt2
Problem 2
(a) Find the equation of the tangent plane at the point (.1, 1, 0) to the surface
2 23 .z
x . 2y + z = .e.
(b) Find the absolute extrema of the function
5
z = f (x, y)= xy . x . 3y
3
on the closed and bounded set R, where R is the triangular region with vertices (.1, 0), (.1, 4), and (5, 0).
Problem 3
. 1 .
2 22
Let f : R. R be de.ned by f (x, y)= x 2 sin + y for x .0 and f (0,y)= y.
=
x
(a)
Show that f is continuous at (0, 0).
(b)
Find the partial derivatives of f at (0, 0).
(c)
Show that f is di.erentiable at (0, 0).
(d)
Is the function .f /.x continuous everywhere in the xy-plane?
C1C
Problem 4
(a) Find the volume of the solid that lies under the cone z = .x2 + y2 , above the xy-plane, and
22
inside the cylinder x+ y=2x.
(b) Rewrite the integ ral . 1 .1 . 1 x2 . 1.y 0 f (x, y, z) dz dy dx as an iterated integral in the order
(i) dx dy dz,
(ii) dy dz dx.
Problem 5
(a)
Let r = x i + y j + z k and r = r. Find
(i)
. r
(ii)
.r
(iii) . (rr)
(iv) .2 r.
(b)
Prove .. F = .(. F) ..2 F.
(c)
Find F n.dS if F(x, y, z)=(x + yz) i + (2y + tan x2 ) j + xy2 k, S is the surface of the
3
S
222 222
solid bounded by the sphere x+ y+ z= 4 and x+ y+ z= 9.
Problem 6
Evaluate the line integral F dr, where
C
2
F(x, y, z) =(2x + y) i +(x + z 3 ) j + (3yz + 1) k
and C is the curve given by t
r(t)= e t sin t i + cos t j + k for 0 . t . ..
Problem 7
(a) State the Greens Theorem and the general version of the Greens Theorem (i.e. apply Greens Theorem with holes, that is, region that are not simply-connected).
.y i + x j .
(b) Use part (a) or otherwise, show that if F(x, y) = , then F dr = a., for every
x2 + y2
C
simple closed path that encloses the origin. Find the constant a.
Problem 8
2
Evaluate (y + sin x) dx +(z + cos y) dy + x 3 dz, where C is the curve given by
C
r(t) = (1+sin t, 2 cos t, 4 cos t + 2sin2t), where 0 . t . 2..
C2C