Suppose that z is a linear function of x and y with slope 2 in the x-direction and slope 3 in the y-direction. (a) a change of 0.5 in x and - 0.1 in y produces what change in z?

Answer = 0.7

Solution -

Similar to linear functions of one variable, we have the Slopes - Intercept Formula for a linear function of two variables given by,

⇒ z = m₁x+m₂y+c

where

m₁ = slope in X direction = 2

m₂ = slope in Y direction = 3

c = c is the z-intercept of the plane defined by the equation.

Taking the derivatives of both sides

⇒ dz = m₁dx + m₂dy

dx = change in x = 0.5

dy = change in y = - 0.1

Putting all the values in the equation

⇒ dz = 2*0.5 + 3 (-0.1) = 0.7

Hence, change in z = 0.7