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(MATH101)[2004](f)midterm~1659^_10417.pdf
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HKUST MATH 101
Midterm Examination

Multivariable Calculus
14 October 2004
Answer ALL 5 questions
Time allowed C 120 minutes
Directions C This is a closed book examination. No talking or whispering are allowed. Work must be shown to receive points. An answer alone is not enough. Please write neatly. Answers which are illegible for the grader cannot be given credit.
Note that you can work on both sides of the paper and do not detach pages from this exam packet or unstaple the packet.
Student Name:
Student Number:
Tutorial Session: Problem 1
Question No. Marks
1 /20
2 /20
3 /20
4 /20
5 /20
Total /100


(a)
If e is any unit vector and a an arbitrary vector show that a =(a e)e + e (a e).

This shows that a can be resolved into a component parallel to and one perpendicular to an arbitrary direction e.


(b)
Show that the two lines r = a + vt, r = b + ut


where t is a parameter and u and v are two unit vectors, will intersect if
a (u v)= b (u v).


Problem 2
222

(a) Describe (sketch) the intersection curve C of the sphere x + y + z = 1 and the elliptic
2
cylinder y 2 +2z = 1 in the .rst octant.

(b) Find the parametric equation of the curve C in the .rst octant.
. . .
1 21

(c) Find the vector equation of the tangent line L of C at the point ,, .
222

(d) Find the equation of the plane through the point (1, 1, 1) and parallel with the tangent line L obtained in part (iii).
Problem 3
3
. x 3 . y

Let f (x, y) = . x2 + y2 , if (x, y) .= (0, 0)
.
. 0, if (x, y) = (0, 0).

Calculate fx(x, y) and fy (x, y) at all points (x, y)(include the point (0, 0)) in the xy-plane. Are fx and fy continuous at (0, 0) (why)? Is f continuous at (0, 0) (why)?
Problem 4
(a) Show that
..
b(x)
d .
f (t) dt = f (b(x))b.(x) . f (a(x))a (x)
dx
a(x) v
[Hint : Let u = a(x), v = b(x), and F (u, v)= f (t) dt.
u

(b) Show that if z = f (x, y) is di.erentiable at x0 =(x0 ,y0 ), then f (x) . f (x0 ) ..f (x0) (x . x0 )
lim =0
x.x0 x . x0
[Hint : Use the de.nition on di.erentiability.]
C1C


Problem 5
2

(a)
The temperature T (x, y) at points of the xy-plane is given by T (x, y)= x 2 . 2y .

(i)
Draw a contour diagram for T showing some isotherms (curves of constant temperature).

(ii)
In what direction should an ant at position (2, .1) move if it wishes to cool o. as quickly as possible?


(iii) If an ant moves in that direction at speed k (units distance per unit time), at what rate does it experience the decrease of temperature?
(iv)
At what rate would the ant experience the decrease of temperature if it moves from (2, .1) at speed k in the direction of the vector . i . 2 j?

(v)
Along what curve through (2, .1) should the ant move in order to continue to experience maximum rate of cooling?



(b)
Find and classify the critical points of the given function f