Review Problems for Second Midterm
Math 234, Spring 2008
This exam will be worth a total of 100 points. The actual exam will consist of 8-10 questions, and you will have 50
minutes. You are allowed to use your calculator and 1 page of notes (8 1/2 by 11, handwritten, one-sided).
These review problems are a guide to help you study, not an exhaustive list of all possible exam questions. Anything
we have covered so far this semester, whether or not it appears here, is fair game. However, these questions are a good
representation of the difficulty and breadth of the questions which might be asked on the exam, and many exam questions
will be similar to those here. You should use these review problems to identify the areas where you need more work, and
then turn to those sections of the text for more practice. Answers, but not detailed solutions, are provided so you can check
your work.
Review Problems
1. Say that z=f(x, y) is defined implicitly by the equation xyz = cos(x+y+z). Find ∂z/∂x and ∂z/∂y at the p oint
(0, π/4, π/4).
2. Find the directional derivative of f(x, y) = ln(x2+y2) at the point (2,1) in the direction of the vector ~v =h−1,2i.
3. The temperature at a point (x, y, z) is given by T(x, y, z) = 200e−x2
−3y2
−9z2, where Tis measured in degrees Celsius,
and x, y, z in meters.
(a) Find the rate of change of temperature at the point P(2,−1,2) in the direction toward the point (3, -3, 3).
(b) In which direction does the temperature increase fastest at P?
(c) Find the maximum rate of increase at P.
4. Find the equation of the tangent plane to the surface x−z= 4 arctan(yz) at the point (1 + π, 1,1).
5. Consider the function f(x, y)=2x3+xy2+ 5x2+y2.
(a) Find all the critical points of this function.
(b) Classify each critical point as a local maximum, local minimum or saddle point.
(c) Find the absolute maximum and minimum values of the function on the triangle with vertices (0,0),(−2,0),(−2,−2).
6. Find three positive numbers whose sum is 100 and whose product is a maximum.
7. Find the maximum and minimum values of f(x, y, z)=2x+ 6y+ 10z, subject to the constraint x2+y2+z2= 35.
8. The base of an (open-topped) aquarium with given volume Vis made of slate and the sides are made of glass. If slate
costs five times as much (per unit area) as glass, find the dimensions of the aquarium that minimize the cost of the
materials.
9. Find the point on the plane x−y+z= 4 that is closest to the point (1,2,3).
10. If R= [−1,3] ×[0,2], use a Riemann sum with m= 4, n= 2 to estimate the value of RRR(y2−2x2)dA. Use the
midpoints of the subrectangles as your sample points.
11. Set up an iterated integral to compute RRDf(x, y)dA over each of the following regions D. You may split the region
into pieces or change to polar coordinates at your discretion.
(a) Dis the quadrilateral with vertices (0,0),(2,1),(1,3),(2,3).
(b) Dis the region bounded by the curve y+ 1 = x2and the line y=x+ 1.
(c) Dis the region bounded by the polar curve r= sin(3θ) in the first quadrant.
(d) Dis the region inside both the circle x2+y2= 1 and the circle (x−2)2+y2= 4.
