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

ENGE601 MATH605 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.

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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, et, 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)