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Task Unit 1: Functions VV

1 .- Be n the number of their group. Given the role

z = g (x, y) = √ (x ^ 2 + ny ^ 2-n) + √ (nx ^ 2 + y ^ 2-n ^ 2)

a) Substitute the value of his group in the given function. Call z = g (x, y) the role of the group.

z = g (x, y) = √ (x ^ 2 +5 y ^ 2-5) + √ (5x ^ 2 + y ^ 2-5 ^ 2)

b) Determine algebraically Dom (g).

Dom (g) = {(x, y) ∈ R ^ 2 / x ^ 2 +5 y ^ 2 ≥ 5 ∩ 5x ^ 2 + y ^ 2 ≥ 25}, and that within the estate should be a real number greater than or equal to 0.

By intercepting both inequalities inequality is obtained that contains the other, therefore the general domain of the function is:

Dom (g) = {(x, y) ∈ R ^ 2 / 5x ^ 2 + y ^ 2 ≥ 25}

c) incorporate a graphic Dom (g) .

The Domain of the figure are all points Ellipse x ^ 2 / 5 + y ^ 2 / 25 = 1 and those around her (The blue).

x ^ 2 / 5 + y ^ 2 ≥ 1

x ^ 2 / 5 + y ^ 2 = 1 White Ellipse

x ^ 2 / 5 + y ^ 2 / 25 ≥ 1

x ^ 2 / 5 + y ^ 2 / 25 = 1 pink ellipse

2 .- Given the role z = h (x, y), whose graph of some contours in the " square " [-2.2] × [-2.2]

a) Calculate, approximately value:

assume for two different cases.

First case : h (1,1) = 16 (whichever is the representative of the intersection point on the graph)

n * h (-1, -1)-2n * h (1.1), n \u200b\u200b= 5

5 * (20) -10 * (16)

100-160 =- 60

second case : h (1.1) = x> 16; x ∈ R

n * h (-1, -1)-2n * h (1.1), n \u200b\u200b= 5

5 * (20) -10 * (x)

100-10x

This value depends on the value of x, assuming that the curve has a height greater than 16, as there may be a higher level at the point in question.

b) Make an outline of the contour, corresponding to level k , where:

n = 5;

n +0.5

5 +0.5

5.5 The graph shows the curve = 5.5 ( represented by the purple curve)