How is the linear perspective in Detail from a Mural in Cubiculum M indicative of the Second Style of Roman painting?

It creates the illusion of the pattern.
It creates the illusion of depth.
It creates the illusion of texture.
It creates the illusion of nature.

Answer:

The equation of the tangent of g(x)^3 at x = 4 is y = 3 - x

Explanation:

The tangent of y = g(x) = -3·x + 11

Therefore, the slope of g(x) = 1/3

The value of y = -3*4 + 11 = -1

The equation of the line g(x) is given as follows;

y - 1 = 1/3*(x - 4)

y - 1 = 1/3x - 4/3

y = 1/3x - 4/3 + 1 = 1/3x - 1/3

g(x) = 1/3x - 1/3

g(x)^3 = (1/3x - 1/3)^3 = [tex]\dfrac{x^3 -3\cdot x^2 + 3 \cdot x - 1}{27}[/tex]

The slope is therefore;

[tex]\dfrac{\mathrm{d} g(x)^{3}}{\mathrm{d} x} = \dfrac{27 \cdot (3 \cdot x^2 -6\cdot x +3 )}{729}[/tex]

The slope of the tangent is the negative reciprocal of the slope of the line which gives;

[tex]Slope \ of \ tangent \ of \ g(x)^3= -\dfrac{729}{27 \cdot (3 \cdot x^2 -6\cdot x +3 )} = -\dfrac{9}{x^2 -2\cdot x + 1}[/tex]

The value of the slope at x = 4 is  [tex]-\dfrac{9}{4^2 -2\cdot 4 + 1} = \dfrac{-9}{9} = -1[/tex]

Therefore, we have;

y at x = 4

[tex]y = \dfrac{4^3 -3\cdot 4^2 + 3 \cdot 4 - 1}{27} = \dfrac{27}{27} = 1[/tex]

Therefore, the equation of the tangent is given as follows;

y - 1 =(-1) × (x - 4) = 4 - x

y = 4 - 1 - x = 3 - x

The equation of the tangent of g(x)^3 at x = 4 is y = 3 - x.