UNIVERSITY OF THE FREE STATE
BLOEMFONTEIN/QWAQWA CAMPUS
MATM2614
DEPARTMENT OF MATHEMATICS AND APPLIED
MATHEMATICS
CONTACT NUMBER: 051 401 9935
EXAMINATION: ADDITIONAL MID-YEAR 2020
ASSESSOR: Dr E.C.M. Maritz & Dr U. Koumba
MODERATOR: Prof J.H. Meyer
TIME: 3 HOURS MARKS: 100
Answer all the following questions.
1. Consider the space curve represented by 𝒓(𝑡) = 5𝑡𝒊 + 3 sin 𝑡 𝒋 + 3 cos 𝑡 𝒌.
(a) Give parametric equations for the tangent line to the curve at the point
𝑃(5𝜋, 0, −3). [4]
(b) Reparametrise the curve with respect to arc length from the point where
𝑡 = −1 in the direction of increasing 𝑡. [6]
2. Consider the function 𝑓(𝑥, 𝑦) = ln(𝑥 2 + 1) − 𝑥 2 𝑦.
(a) Find an equation for the tangent plane to the surface in the form
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 where 𝑥 = 1 and 𝑦 = 0. [3]
(b) For which points in the domain do we have a horizontal tangent plane? [4]
𝑥−𝑦 4
3. Find the limit or show that it does not exist: lim . [4]
(𝑥,𝑦)→(1,1) 𝑥 2 −𝑦 4
1
𝑥−2
4. Consider the function 𝑓(𝑥, 𝑦) = (𝑥+2) ln(4 − 𝑦 2 ).
(a) Find and sketch the domain of 𝑓. [3]
𝜕𝑓
(b) If 𝑥 = 𝑠 − 𝑡 and 𝑦 = 𝑠𝑡 2 , find where 𝑠 = 𝑡 = 1. [4]
𝜕𝑡
(c) Find all vectors 𝒖 for which the directional derivative 𝐷𝒖 𝑓(1, −1) = 0. [4]
(d) What is the minimum value of 𝐷𝒖 𝑓(1, −1)and in which direction does it
occur? [2]
5. Find the absolute maximum value of the function 𝑓(𝑥, 𝑦) = 4𝑥 2 − 𝑥𝑦 defined over
the trapezoidal region with vertices (1,1), (2,2), (1, −1) and (2, −2). [6]
6. Evaluate the following double integrals:
(a) ∬𝐷 𝑥 sin 𝑦𝑑𝐴where 𝐷is the region bounded by 𝑦 = 0, 𝑥 = 1 and 𝑦 = 𝑥 2 . [5]
(b) ∬𝑅(𝑥 − 𝑦) 𝑑𝐴 where 𝑅 is the region that lies to the left of the 𝑦-axis between
the circles 𝑥 2 + 𝑦 2 = 1 and 𝑥 2 + 𝑦 2 = 9. [5]
(c) Use a suitable substitution to evaluate the double integral
𝑥−𝑦 2
∬𝐷 (𝑥+𝑦) 𝑑𝐴where 𝐷 is the trapezoidal region with vertices
(−1,0), (−2,0), (0, −1) and (0, −2). [7]
7. Consider the solid 𝐸 that is bounded above by 𝑧 = 16 − 𝑥 2 − 𝑦 2 and below by
3 9
𝑧 = 9 − 𝑥 2 − 𝑦 2 with domain 𝐷in the 𝑥𝑦-plane given by 𝑥 2 + (𝑦 − 2)2 ≤ 4 and
𝑥 ≤ 𝑦 in the first quadrant. Calculate the volume of the solid 𝐸 over the domain
given. [13]
8. (a) Find the work done by the vector field 𝑭(𝑥, 𝑦) =< 𝑥 2 , 𝑥 − 𝑦 > in moving an object
along the straight line from (−2, −1) to (−8, 2). [5]
2
(b) Use the Fundamental Theorem of Line Integrals to evaluate
∫𝐶(2𝑥 sin 𝑦)𝑑𝑥 + (𝑥 2 cos 𝑦 − 3𝑦 2 )𝑑𝑦, where 𝐶 is the curve represented by
𝒓(𝑡) =< 𝑡 3 − 2𝑡 − 1, cos(𝑡 2 − 𝑡) > from 𝑡 = 0 to 𝑡 = 1. [7]
(c) True or false: There exists a vector field 𝑮(𝑥, 𝑦, 𝑧) such that
𝑐𝑢𝑟𝑙𝑮 =< 𝑥𝑦𝑧, −𝑦 2 𝑧, 𝑦𝑧 2 >. Motivate your answer. [3]
(d) A particle starts at the point (−2,0), moves along the 𝑥-axis to (2,0), and then
along the semicircle 𝑦 = √4 − 𝑥 2 to the starting point. Use Green’s Theorem to find
the work done on this particle by the force field 𝑭(𝑥, 𝑦) =< 𝑥, 𝑥 3 + 3𝑥𝑦 2 >. [5]
9. (a) Set up, but do not evaluate, the surface integral ∬𝑆 𝑥𝑦𝑑𝑆 where 𝑆 is the surface
𝑧 = 4 + 𝑥 2 + 𝑦 2 over the domain bounded by 𝑟 = 2in the first octant. [5]
(b) Find the area of the part of the surface 𝑧 = 𝑥 2 + 𝑦that lies above the triangle with
vertices (0,0), (1,0) and (1,2). [5]
