Please answer Problem1. Problem 2 is belowed is need for Problem 1 Problem 1. a)

Question

Please answer Problem1. Problem 2 is belowed is need for Problem 1

Problem 1. a) We consider again the contour plot of the function f(x, y) depicted in HW 4 Problem 2. Imagine that we move from the point (3, ) along a straight line whose direction is given by a unit vector(a, b, so x(t) = -at and y(t) +6t. and consider the value of f(x(t), y(t)) as a function of t (i) What are the possible direction(s) a,b for which df/dt 0 at t-0? (Either sketch a picture showing a portion of level curve and the vector a,b), or give approximate angle from the x-axis). For each of these directions, when travelling along the chosen line, does the value of f pass through a minimum or a minimum at t 0? (ii) What are the direction(s) (a, ó for which df/dt at t 0 is largest, resp. smallest? Measure the contour plot to estimate the values of df/dt at t-0 for those directions b) Recall that the contour plot depicted in HW 4 Problem 2 corresponds to f(x,y) (i) find the unit vectors(a, b)such that the derivative off along the line x(t) -at, y(t)- bt is zero at t-0 (ii) find the unit vectors a,b such that the derivative of f along the line x(t) = + at, y(t) = -bt is largest, resp. smallest at t = 0, and calculate the value of df/dt at t 0 for those directions

Explanation / Answer

1) x(t) = 3/2 + at

y(t) = 1/2 + bt

at t = 0,

x = 3/2

y = 1/2

tanA = y/x = 1/3

Angle from X axis = tan1(1/3)

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