MAT2615 Assignment 3
(COMPLETE ANSWERS) 2026
- DUE June 2026.
Kenneth Githaiga
, MAT2615 Assignment 3 (COMPLETE ANSWERS) 2026 - DUE June 2026.
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
(COMPLETE ANSWERS) 2026
- DUE June 2026.
Kenneth Githaiga
, MAT2615 Assignment 3 (COMPLETE ANSWERS) 2026 - DUE June 2026.
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
