The question posted below involves in mathematics unit called Functions of several variables. The unit includes Domains,

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The Question Posted Below Involves In Mathematics Unit Called Functions Of Several Variables The Unit Includes Domains 1 (37.6 KiB) Viewed 23 times

The question posted below involves in mathematics unit called Functions of several variables. The unit includes Domains, Partial Derivatives (High-order Partial Derivatives), Approximation to f(xQ.yo) (Total Increment Theorem), Chain Rule (simple case, General case), Direction derivative, Tangent Planes and normal to surfaces, Double Integrals (Type 1 region, Type 2 region) and Extreme values (Local Extreme values, Global Extreme Values). Data Set: A f(x,y) 7 - x2 + 2x - y2 + 2y B (2.01, 1.98) direction ENE (2,2) z The members of a surveying team want to estimate the slopes at various points on a hill and in various directions. The hill can be modelled by a surfaces with equation z =f(x, y) where 2 2 0, as given below. According to this model, the surface is positioned above a horizontal (x,y)-plane where x points eastward and y points northward. a) Using the point A given above, and the direction specified above, apply the following strategy to find the slope of the surface in the specified direction at the point P on the surface directly above A. (i) Find the gradient vector Of and evaluate it at the point A. (ii) Construct a vector u that points in the specified direction. (iii)Use your answers to parts(i) and (ii) to find the required slope. b) Does the hill slope upwards, downwards or neither at P in the specified direction?Explain your answer. c) What is the direction of the steepest uphill slope at L. and what is the steepest uphillslope? Justify your answer. d) What is the direction of the steepest downhill slope at L and what is the steepestdownhill slope? Justify your answer. e) In what direction is the slope horizontal at 22 Justify your answer. f) What shape is the surface S? Why? g) What is the maximum height of the hill, and above what point in the (x,y)-plane doesit occur? Justify your answer. h) At how many points above the ground on the hill does the steepest slope occur, and where are those points? Please answer question (h) only, use questions above as reference