8/19/2019 Vector Calc Quiz

1/1

UNIVERSITY OF NEW SOUTH WALES

MATH2069 MATHEMATICS 2A

VECTOR CALCULUS SAMPLE QUIZ 1

This sheet must be filled in and stapled to the front of your answers

Student’s Surname Initials Student Number

Tutorial Code Tutor’s Surname

Time allowed: 30 minutes 5 questions Attempt all questions

UNSW Approved Calculators are permitted

1. [3 marks] Find the tangent line to the curve

r(t) = (t2 + 7)i + (t3) j + (4t)k

at the point  P   where   t = 1.

2. [3 marks] Sketch the surface

−

x2

4  +

 y2

9  −

 z 2

25 = 1

and write down the point(s) on the surface which are closest to the origin.

3. [3 marks] Find and classify the critical points of 

f (x, y) = 6x2 + 3y2 − 2x3 + 6xy.

4. [3 marks]

The volume  V   of a cylinder with radius  r  and perpendicular height  h  is givenby V    = πr2h.  Find the maximum error in calculating  V    given that  r = 5 cmand  h = 6 cm to the nearest millimetre.

5. [3 marks] The pressure in a region of space is given by   φ(x,y,z  ) =   x4y3z 2.Calculate the rate of change of the pressure with respect to distance at thepoint (1,1,1) in the direction 2i + j + 2k.

Answers 1)

x

y

z

=

814

+

234

t   2)   ±

030

  3)(0, 0)min, (1,−1)saddlepnt.   4)13.35 5) 5