Math 53: Worksheet 6
October 12
1. Compute:
(a) ∇ffor f(x, y, z) = x2sin(yz).
(b) Directional derivative of g(x,y , z) = y2ex−cos(xz) at (0,−1, π/2) in the direction
of u= (1,5,−4).
(c) Equation of tangent plane to the surface xy +yz +zx + 3 = exyz at (−1,2,0).
2. Let u= (a, b) be a unit vector and let f(x, y) have continuous second-order partial
derivatives. Find an expression for Du(Duf(x, y)).
3. The temperature Tin an infinite ball is inversely proportional to the distance from
the center of the ball, which we take to be the origin. The temperature at the point
(2,1,2) is 60◦F.
(a) Find the rate of change of Tat (−2,1,0) in the direction toward the point
(4,1,−1).
(b) Show that at any point in the ball the direction of greatest increase in temperature
is given by a vector that points towards the origin.
4. Consider a swimming pool with the temperature of the water at (x, y, z) given by
T(x, y, z). A fish swims through the water with position at time tgiven by p(t).
(a) What is the directional derivative of Tin the direction that the fish is traveling
in at time 0?
(b) The fish feels a temperature at any point in time. How fast does the temperature
that the fish feels change, at time 0?
(c) At t= 1, the fish decides it is happy with its current temperature. Describe/specify
a set of directions (vectors) in which the fish should swim.
(d) The fish changes its mind instantaneously at time t= 1. It goes in the direction
such that the water gets colder, fastest. Give a vector pointing in this direction.
5. Let f(x, y, z, t) be a smooth function and let ∇f=hfx, fy, fzibe the gradient in the
space variables only. Let r(t) = hx(t), y(t), z(t)ibe a smooth curve and v(t) = r0(t).
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