Worksheet 2
Consider the function z = f (x, y) whose graph contains the point (2, 3, 7)
and which satisfies fx (2, 3) = 2, fy (2, 3) = 5.
(1) Find the equation of the tangent line to the trace of f at y = 3.
What is a direction vector for the tangent line
Department of Mathematics
Bucknell University
MATH 202
Supplementary Handout
Revised Spring 2011
Graphing Polar Functions
The purpose of this handout is to provide some further practice at sketching the graphs of
polar functions. The problems aim to devel
Math 211, Exam I
September 13th, 2016
1
2
3
4
5
6
7
total
14
10
12
18
16
16
14
100
Show all work and write clearly! Do as much as you can and do not assume
the problems are in order of difficulty or importance.
1. Match the descriptions of the geometric o
Non-book problems for Section 14.1
The level set of a function w = f (x, y, z) at c = where 0 is given by
1
(x, y, z) : x + y + z =
1 + 2
2
2
2
.
(1) Can you tell if f attains a maximum or a minimum value?
(2) As , what is happening to the level sets?
(3)
Non-book problems for Section 14.1
The level set of a function w = f (x, y, z) at c = where 0 is given by
(x, y, z) : x2 + y 2 + z 2 =
1
1 + 2
.
Assume the level set for f at c for any c < 0 is the empty set.
(1) Can you tell if f attains a maximum or a m
Non-book problems
(Turn in what you can on Monday, September 5th)
(1) Find the perpendicular distance between the parallel planes
2x 3y + 5z = 1 and 2x 3y + 5z = 4
(2) Describe all lines through the origin that make an angle of
3
with
the xy-plane.
(3) Fi
Worksheet 3
(1) Consider the function z = f (x, y) whose graph contains the point
(2, 3, 7) and which satisfies fx (2, 3) = 2, fy (2, 3) = 5.
(a) Find the gradient of f at (2, 3)
Solution. f (2, 3) =< fx (2, 3), fy (2, 3) >=< 2, 5 >
(b) In what direction
Solutions to the Non-book problems.
(1) Find the perpendicular distance between the parallel planes
2x 3y + 5z = 1 and 2x 3y + 5z = 4
Solution: Pick a point P , such as P = (2, 0, 1) that is in the first
plane and a point Q, such as Q = (2, 0, 0) that is