MAT2615 Assignment 3 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED

MAT2615 Assignment 3 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references.. 1. (Sections 10.1, 10.2) Consider the R2 − R function f defined by f (x, y) = x2 − 6x + 3y2 − y3. (a) Find all the critical points of f . (The function has two critical points.) (5) (b) Use Theorem 10.2.9 to determine the local extreme values and minimax values of f . Also determine (by inspection) whether any of the local extrema are global extrema. (5) [10] 2. (Sections 2.6, and Chapter 10) Let L be the line with equation y = x − 1. Find the minimum distance between L and the point (4, 5) by using (a) Theorem 10.2.4 (5) (b) The Method of Lagrange. (5) Hints. • Minimize the square of the distance between the point (x, y) and the point (4, 5), under the constraint that the point (x, y) lies on the line L. (The required distance is a minimum at the same point where its square is a minimum.) • In order to use Theorem 10.2.4 you need to write the function that you wish to minimize as a function of x alone. (Eliminate y by using the given constraint.) • In order to use the Method of Lagrange, write the function that you need to minimize as a function of x and y and also define an R2 −R function g such that the given constraint is equivalent to the equation g (x, y) = 1. [10] 3. (Sections 13.1–13.6) Let R be the region in the sketch below. (a) Describe the region R as a union of two Type 1 regions. (Use set–builder notation.) Hints: 2 Downloaded by Edge Tutor () lOMoARcPSD| MAT2615/AS3/0/2026 • Read the description of a Type 1 Region on p. 18 of Guide 3 and study Fig. 13.9 carefully. • Shade the region R by means of vertical lines and highlight the curves which form lower and upper boundaries of R. Write the equations of these curves in the form y = g(x). • The lower boundary of R is formed by two different curves. (3) (b) Describe the region R as a union of two Type 2 regions. (Use set–builder notation.) Hints: • Read the description of a Type 2 Region on p. 21 and study Fig. 13.11 carefully. • Shade the region R by means of horizontal lines and highlight the curves which form left and right boundaries of R. Write the equations of these curves in the form x = h (y) . • The right boundary of R is formed by two different curves. (3) (c) Describe the region R in terms of polar coordinates. Hints: • Shade the region R by drawing rays from the origin and highlight the curves on which these rays enter and exit the region. • All points on the given circle lie at the same distance from the origin, but this is not the case for points on a straight line. • Remember that θ is measured from the positive X–axis. (4) [10] 4. p(Section 13.3 and Chapter 14) Let D be the region in R3 that lies inside the cone z = x2 + y2 above the plane z = 1 and below the hemisphere z = p 4 − x2 − y2. (a) Sketch the region D in R3. Hint: The given cone intersects the given hemisphere in a circle and also intersects the given plane in a circle. Determine the radius of each of these circles and show the circles on you sketch. (2) (b) Express the volume of D as a sum of triple integrals, using cylindrical coordinates. The main aim of this problem is to test whether you understand the geometric meaning of a triple integral and whether you are able to obtain the correct limits of integration. You do not need to evaluate the integral. Hints: 3 Downloaded by Edge Tutor () lOMoARcPSD| • First study the geometric meaning of a double integral in Section 13.3 and then study the geometric meaning of a triple integral in Section 14.3 Also study Section 14.7. • Study the steps given in Example 14.6.4. Also study Example 14.7.1 • Note that r represents the horizontal distance from the Z-axis. (5) (c) Express the volume of D as a triple integral using spherical coordinates. Do not evaluate the integral. Hints: • Study the steps described in Example 14.6.7. • Remember that ρ represents the distance from the origin. • All points on the given sphere lie at the same distance from the origin, but this is not the case for the points on a plane. Thus the upper limit for ρ will be constant, but the lower limit will depend on ϕ. • Remember that ϕ is measured downward from the positive Z-axis. (4) [11] 5. (Sections 13.1 - 13.6) Let R be the region given in Question 1. Evaluate the integral ZZ R y dA by using (a) rectangular coordinates and the order of integration dx dy. (4) (b) polar coordinates. (4) If your answers to (a) and (b) do not agree, then please try to find your mistake and correct it. [8] 6. (Pages 66-70 in Guide 1 and Chapter 15) Consider the directed line segments C =  (x, y) ∈ R2 | (x, y) = t (1, 3) ; t ∈ [0, 1] and D =  (x, y) ∈ R2 | (x, y) = (1, 3) − t (1, 3) ; t ∈ [0, 1] . Evaluate the following line integrals. (a) R R C (x − y) ds (2) (b) R R D (x − y) ds (2) (c) R R C (x − y) dy (2) 4 Downloaded by Edge Tutor () lOMoARcPSD| MAT2615/AS3/0/2026 (d) R R D (x − y) dy (2) Comment on your answers.