MTH316 TMA01:Analysis of Continuity, Derivatives, and Gradient Functions – Singapore Assignment Jan 2025

(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 𝑓𝑦(0, 0) both exist and find their values. (10 marks)

(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)

(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 hiker’s ascent rate (concerning time) 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)