Function Festival Fall 2000

Rosemary Kunz, Math Artist

z = y^2 - x^2

Surface: z = y^2 - x^2
at points: A = (-1,-1,0), B = (-2,-1,-3), C = (-3,-2,-5)

z = y^2 - x^2 with points A, B, and C


The vector from pt. A to pt. B is <-1,0>.
The vector from pt. A to pt. C is <-2,-1>.


The slope in the direction of x, (fx(x,y)), is -2x.
The slope in the direction of y, (fy(x,y)), is 2y.


The gradient of f(x,y) = z is -2xi + 2yj.
The gradient of f(-1,-1) = z is 2i - 2j.


The unit vector from pt. A to pt.B is
u = -i / ((-1)^2 + 0^2)^2 = -i.


The unit vector from pt. A to pt. C is
u =( -2i - j ) / ((-2)^2 + (-1)^2)^(1/2)
= (-2 / (5)^(1/2))i + ( -1 / (5)^(1/2))j


The directional derivative of z at (-1,-1) in
the direction of u = -i is (-i) . (2i - 2j) = -2.


The directional derivative of z at (-1,-1)
in the direction of u is
u =( -2 / (5)^(1/2))i -(1 / (5)^(1/2))j
which is:
((-2/(5)^(1/2))i - (1/(5)^(1/2)j)) . (2i - 2j)
= -2/(5)^(1/2) = -0.894427191.


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