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