MATH 23 – Final Exam Fall Semester 2007
Duration: 3 hours
Instructions: Answer all questions, without the use of notes, books or calculators. Partial credit
will be awarded for correct work, unless otherwise specified. The total number of points is 100.
1. (20 pts: 2 each)
(a) Write down two properties of the gradient ∇fof a function f(x, y, z).
(b) If you know that limx→0f(x, mx) = limx→0f(x, k x2), what can you conclude about the
continuity of f(x, y)at (0,0)?
(c) Give an example of a two–variable function f(x, y )(a formula, a sketch, or a word
description) that is continuous but not differentiable at the origin.
(d) Let fbe a scalar field and ~
Fa vector field. Which of the following expressions are
meaningful?
(i) grad ~
F(ii) curl ~
F(iii) div f
(iv) grad(div ~
F) (v) curl(grad f) (vi) curl(curl ~
F)
(e) Write down the formula for the average value of a scalar function f(x, y, z)over a solid
region Ein space.
(f) Given ~
F=−y~
i+x~
j
x2+y2, what condition do we have to impose on a smooth simple closed
curve Cso that we can use Green’s Theorem to calculate ZC
~
F·d~r?
(g) If a transformation Tis given by x=u−v,y=uv, what is the Jacobian of T?
(h) Does there exist a vector field ~
Fsuch that curl ~
F=x
~
i+~
j+z~
k? Why or why not?
(i) Given a vector function ~r(t), how do you check whether or not it is parametrized by arc
length?
(j) If all component functions of ~
Fhave continuous partial derivatives on R3, and if
IC
~
F·d~r = 0 for any closed space curve C, what can you say about ~
F?
2. (8 pts) Consider three points in space P(3,−1,1),Q(4,0,2), and R(5,−1,−1)
(a) Find the angle between the vectors −−→
P Q and −→
P R.
(b) Find an equation of the plane going through these three points.
3. (9 pts) The tangent plane to the graph z=f(x, y )at the point above (0,1) is given by z=
5 + x−3y.
(a) What is value of f(0,1)?
(b) What is the gradient f(0,1) of f(x, y )at (0,1)?
(c) What is the directional derivative of f(x, y)at (0,1) in the direction ~v =−
~
i−~
j?
(d) If x(t) = sin tand y(t) = e2t, find df
dt at t= 0.
continuted on the back −→
1
