Math 174  Exam II  3/31/2000  D. Gottlieb

(5) 1. Suppose that f(x, y) is a real valued function defined everywhere on the plane. Suppose its gradient ∇f^{⟶} is not defined at (0, 0).

a) Is it possible to determine whether or not f is continuous at (0, 0) given the above information? (if yes, is it continuous at (0, 0) or not)

b) Is it possible to determine whether or not f(x, y) is differentiable at (0, 0)? (if yes, is it differentiable at (0, 0))

HINT: If you don’t know the answers for a) or b), write down the definitions of continuity for a) or differentiability for b) for some partial credit.

(20) 2. Let f(x, y, z) = xyz and P be the point (2, −3, 4).

a) Calculate the gradient vector field ∇f^{⟶}.

b) Find the gradient vector of f at the point P (2, −3, 4).

c) Find the directional derivative (Du⃗f)_{P} at the point P (2, −3, 4) in the direction

u⃗ = \frac{1}{\sqrt{2}} i⃗ − \frac{1}{\sqrt{2}} k⃗.

d) In which direction v⃗ does the directional derivative (Dv⃗f)_{p} have its minimum value at the point P.

e) Give the equation of the level surface of f(x, y, z) through the point P.

f) Find the normal vector to the level surface at P.

g) Find the tangent plane to the level surface at P.

h) Give a critical point of f(x, y, z) and state whether or not it gives a maximum or minimum value of f(x, y, z).

(10) 3. Let w = f(x, y, z) be a differentiable function and x, y, z be differentiable functions of t.

a) Express \frac{dw}{dt} by the chain rule.

b) Express \frac{dw}{dt} as the dot product of a gradient vector field and a velocity vector field of a path.

c) What is the rate of change with respect to t of f(x, y, z) = x + y^{2} + z^{3} along the path ti⃗ + tj⃗ + 2k⃗ at the point (0, 0, 2)?

(10) 4. Consider the curve defined by the intersection of the surfaces x^{2} + y^{2} + z^{2} = 2 and x^{2} + 2y^{2} = 1. Find a tangent vector T⃗ for this curve at the point (1, 0, 1).

(5) 5.a) Write down the second order (quadratic) Taylor’s formula for the function f(x, y) about (0, 0).

b) Suppose f(0, 0) = 0, ∇f^{→}|_{(0,0)} = 0 and the second order partial derivatives are all equal to 3 at (0, 0). Write down the second order Taylor’s formula in this case.

(50) 6. Find the volume under the paraboloid z = x^{2} + y^{2} and above the triangle in the xy plane enclosed by the lines y = x and x + y = z and the y–axis.

a) Sketch the region of integration.

b) Set up the integral.

c) Evaluate the integral.