MATH 23 – Midterm 2 Spring Semester 2008
Duration: 50 minutes
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 50.
1. (14 pts: 3, 3, 3, 5) z= 5 −3x+ 4yis the tangent plane to the graph of f(x, y)at (0,2).
(a) What is the value of f(0,2)?
(b) In which direction does f(x, y)increase the fastest at (0,2)?
(c) What is the directional derivative of f(x, y)at (0,2) in the direction ~v =−
~
i−~
j?
(d) If x(t) = sin tand y(t)=2et, calculate df
dt at t= 0.
2. (11 pts: 6, 5) Consider the function f(x, y) = x2−2x+y2.
(a) Find and classify all critical points of f(x, y ).
(b) Find the absolute maximum and absolute minimum values of f(x, y )over
D={(x, y)|x2+y2≤4}.
3. (8 pts) Consider the region in the xy–plane D={(x, y)|1≤x≤e, 0≤y≤ln x}. Set up, but
do not evaluate, two iterated integrals to find the area of D, one integrating xfirst and the
other integrating yfirst.
4. (12 pts: 6 each) A solid Elies in the first octant below the cone z=px2+y2and inside the
cylinder x2+y2= 1. The density of Eis given by d(x, y , z) = z. Set up, but do not evaluate,
iterated integrals to find the total mass of Ein the following coordinate systems.
(a) The cylindrical coordinates.
(b) The spherical coordinates.
5. (5 points) Show that the area of the region bounded by the ellipse x2
a2+y2
b2= 1 is πab.
1