Download Exam 1 Math 205 Fall 1998

Exam 3
Calculus 3
Winter 2013
Name: ___________________ This is a closed book exam. You may use a calculator and the formulas handed out
with the exam. You may find that your calculator can do some of the problems. If this is so, you still need to show
how to do the problem by hand, even if you use a calculator to check your work. This is particularly true for
derivatives and integrals. On all problems, show all work and explain any reasoning which is not clear from the
computations. (This is particularly important if I am to be able to give part credit.) Turn in this exam with your
answers. However, don't write your answers on the exam itself; leave them on the pages with your work. Also turn
in the formulas; put them on the formula pile.
1.
2.
(14 points) The equation xz + y ln(x) – x2 = - 4 can be used to define x as a function of y and z.
x
Find
when x = 1, y = -1 and z = -3.
z
1
a = 150  ½ ft
(14 points) The area of a triangle is A = ab sin  where a and b are
2
 = 60  2
the lengths of two sides of the triangle and  is the angle whose sides
b = 200  ½ ft
have lengths a and b. In surveying a triangular plot, you have
measured a to be 150 ft, b to be 200 ft and  to be 60. The corresponding area would be
A = 12 (150)(200) sin(60)  12990.4. However your values of a and b might be off by as much as
half a foot and your value of  might be off by as much as 2. Use linearization to estimate the
possible error in the value of A  12990.4.
3.
(20 points) Suppose z = f(x,y) is some function of x and y and we make the change of variables
z
 2z
 z ,  z ,  2z ,  2z
 2z
x = r cos  and y = r sin . Find formulas for
and
and 2 .
2 in terms of
2
r
r
 x  y  x  x y
y
4.
Suppose that you are climbing a hill whose shape is given by the equation
x2
y2
z = 1000 100 50
and you are standing at the point with coordinates x = 60, y = 100, z = 764.
5.
a.
(7 points) In which direction should you proceed so that your rate of ascent is greatest
initially.
b.
(7 points) If you climb in the direction in part a, what is your rate of ascent?
Let z = 2x2 + 2xy + 5y2 + 4x.
a.
(14 points) Find the critical points of the function z = 2x2 + 2xy + 5y2 + 4x. Use the second
derivative test to classify each as to whether they are a local maximum, local minimum,
saddle point, or can’t tell from the second derivative test.
b.
(20 points) Consider the triangle with vertices (0, 0), (- 1, 0) and (0, 1). Let S be the set of
points which are on the three sides of the triangle, i.e. the boundary of the triangle. Find the
absolute maximum and absolute minimum of z = 2x2 +2xy + 5y2 + 4x as x and y vary over S.
You need to look at each side separately. On each side it is not enough to check the value of
z at the ends. You have to look at the interior of the sides also.
c.
(4 points) Find the absolute maximum and absolute minimum of z = 2x2 +2xy + 5y2 + 4x as
x and y vary over the inside and boundary of the triangle with vertices (0, 0), (- 1,0) and
(0, 1).