Sec. 1.2

#7a) xz plane: parabolas of the form z = x^2 + constant, yz plane: parabolas of the form z = 2y^2 + constant

#7c) xz plane: parabolas of the form z = -x^2 + constant, yz plane: parabolas of the form z = 3y - constant.

Sec. 4.3

#1c) df/dx = cos(x)cos(y), df/dy = -sin(x)sin(y)

#1d) df/dx = 2x/(x^2+y), df/dy = 1/(x^2+y)

#3a) df/dx = yz+2x, df/dy = xz, df/dz = xy

#3c) df/dx = -y*e^z*sin(xy), df/dy = -x*e^z*sin(xy), dy/dz = e^z*cos(xy)

Sec. 4.2

#2a) Du f = u1+2*u2. So the rate of change is zero along u1+2*u2=0. Below or left of the line f is decreasing, and above or right f is increasing.

#2b) Du f = -u1 + u2. Rate of change is zero along -u1+u2 = 0. To the right or below the line, f is decreasing, to the left or above the line f is increasing.

#2c) Du f = 6*u1. Rate of change is zero along u2 = 0. In the left half plane f is decreasing and in the right-half plane f is increasing.

#5a) Du f = 2*x0*u1 + u2.

#5b) At x0 = (1,1), Du f = 2*u1 + u2. u is maximized when it points in the same direction as (2,1).

#5c) Du f is maximized when u is in the direction of grad f(x0,y0).

Sec. 4.5

#1a) grad f(2,-1) = (9,-4)

#1f) grad f(1,pi/2,0) = (0,0,0)

#3a) grad f(1,-1) = (4,-3) |grad f(1,-1)| = 5

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Last modified: Mon Oct 28 16:44:48 1996