University	Auckland University of Technology (AUT)
Subject	ENGE601 Engineering Mathematics 2 and MATH605 Algebra and Calculus 2

ENGE601 Engineering Mathematics 2 MATH605 Algebra and Calculus 2 Assignment-1 Semester 2, 2025

Instructions

Use a pen, not a pencil to write your solutions.

You need to scan your solutions into a single pdf file and upload on Canvas in the assignment-1 section.

Only one pdf file should be uploaded.

One of the questions is on creating a MATLAB graph. You will need to provide the printouts from MATLAB (code and graph) and include them into your pdf file.

Write your full name and ID on the top right corner of the front page of your solutions.

Comply with the AUT Academic Integrity Declaration below.

AUT Academic Integrity Declaration

In submitting this work, I agree that:

The assessment has been produced by me and is solely my work, UNLESS the work is for a designated collaborative (group) task.
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I have properly acknowledged all sources used in my work.
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This work has not previously been submitted by me for any assessment task, OR I have received approval to resubmit this work.
I did not use any other unfair means to complete this work.
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Question 1

i) What does the equation 4x² + 4y² = 16 represent in 2D?
ii) What does the same equation represent in 3D?
iii) Use polar/cylindrical coordinates and MATLAB to produce a graph of this equation in 3D. Provide both the MATLAB code and graph.

12 marks

Question 2

Find the velocity and position of an object at any time t, given that its acceleration is a(t) = 6t, 12t + 2, e^t, its initial velocity is v(0) = < 2, 0, 1 > and its initial position is r(0) = < 0, 2, 8 >.

12 marks

Question 3

Find ∂z/∂x and ∂z/∂y given that F(x, y, z) = xy² + 4z² + sin(xy²) = 0.

10 marks

Question 4

The dimensions of a closed rectangular box are measured as 80 cm, 60 cm, and 30 cm, respectively, with a possible error of 0.1 cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box.

10 marks

Question 5

Find the volume of the solid under the surface z = 4x + y² and above the 2D region bounded by x = y² and x = √y. Sketch the 2D region.

12 marks

Question 6

Find the directional derivative of f(x, y, z) = xy + yz + xz at the point P(1, -1, 3) in the direction of the point Q(2, 4, 5).

8 marks

Question 7

Find the maximum rate of change of the function f(x, y) = sin(xy) at the point (1, 0) and the direction in which it occurs.

8 marks

Question 8

Find the local maximum and minimum values and saddle point(s) of the

function f(x,y)=x4+y4−4xy+2

8 marks

Question 9.

Find the absolute maximum and minimum values of the function f(x, y) = 1 + 4x − 5y on the closed triangular region with vertices (0, 0), (2, 0), and (0, 3).

10 marks

Question 10.

Use Lagrange multipliers to find the maximum and minimum values of the function f subject to the given constraint:
f(x, y, z) = 2x + 6y + 10z; x² + y² + z² = 35

10 marks

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ENGE601 Engineering Mathematics 2 and MATH605 Algebra and Calculus 2 Assignment-1: Applications of Multivariable Calculus, Vector Fields, and Optimization Techniques

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ENGE601 Engineering Mathematics 2 MATH605 Algebra and Calculus 2 Assignment-1 Semester 2, 2025

Instructions

AUT Academic Integrity Declaration

Stuck! Do not Know Assessment Answers?

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9.

Question 10.

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