MTH316 TMA01 Questions on Continuity, Derivatives, and Terrain Gradient Analysis Jan2025, Singapore

Tutor-Marked Assignment 01 (TMA01)

This assignment is worth 12% of the final mark for MTH316 Multivariable Calculus. The cut-off date for this assignment is 27 February 2025, 2355hrs.

Note to Students: You are to include the following particulars in your submission: Course Code, Title of the TMA, SUSS PI No., Your Name, and Submission Date. For example, ABC123_TMA01_Sally001_TanMeiMeiSally (omit D/O, S/O). Use underscore and not space.

Question 1

Define a function 𝑓: ℝ² ⟶ ℝ² by

(a) Determine the value of k for the function 𝑓 to be continuous at (0, 0), if any. (Use this value of k for the following sub-questions.) (10 marks)

(b) Show that 𝑓ₓ(0, 0) and 𝑓ᵧ(0, 0) both exist and find their values. (10 marks)

(c) Define 𝑓ₓ(𝑥, 𝑦) and 𝑓ᵧ(𝑥, 𝑦) each as a piecewise function in the entire domain of ℝ². (12 marks)

(d) Hence or otherwise, explain whether the function 𝑓 is differentiable at (0, 0). (18 marks)

Question 2

A GPS application designed for hikers measures the altitude, z, at various positions on a terrain and performs a regression, yielding the altitude function below, where x, y, and z are measured in km.

(a) State an equation that describes the level set of the function and use it to plot the contours of the terrain at the altitudes of 1.3 km, 1.5 km, and 2 km with a graph plotter. Attach a screenshot of the graph and label each contour with its equation clearly. (8 marks)

(b) A hiker crosses the position (0, 1) while walking along a contour. Determine the contour equation and calculate the gradient vector of the terrain function at the position (0, 1). Show that this gradient vector is orthogonal to the contour at the position (0, 1). (18 marks)

(c) A hiker is walking along a path given by 𝑦 = 1 − 𝑥 − 𝑥³. Sketch the path on your graph in part (a) and compute the directional derivative along the path at (0, 1) in the direction of increasing altitude. Explain briefly the meaning of this value of the directional derivative. (10 marks)

(d) The GPS carried by the hiker in part (c) shows that his ground speed is 0.8 m/s. Compute the rate of ascent (with respect to time) of the hiker as he crosses (0, 1). What is the degree of slope and how fast is he walking in the upslope direction? (7 marks)

(e) Another hiker is planning to walk from (5/2, 5/4) to (4, 2) in a straight line. From the contours plotted in part (a), explain the terrain he will experience along this path (without using calculations). (7 marks)