MATH 16B SECOND MIDTERM
Instructions
(1) Do not begin this exam until instructed to start. (2) This exam will end promptly at 5:00. Stop writing when time has been called. (3) Write your name and student identification number on each and every page. (4) W
MATH 16B FINAL EXAMINATION
Name: Student Identification Number: Section Number:
Instructions
(1) Do not begin this exam until instructed to start. (2) This exam will end promptly by 3:30. Stop writing when time has been called. (3) Write your name and stu
MATH 16B FINAL EXAM
Name: Student Identification Number: Section Number: Instructions This exam begins at 5:00pm and ends promptly at 8:00pm. Wait for the start of the exam to be announced. Do not write after time is called. A list of potential answers is
Math 16B, Day 1
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
Math 16B, Lecture 1
Preliminaries
Functions of several variables
Graphs
Level curves
Welcome
Math 16B
John Lott
Telephone: (510) 642-1299
email: [email protected]
Ofce: 897 Evans Hall
Math 16B, Day 4
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
True or false?
1. If f has a relative minimum when x = a and y = b, then for
all x near a and y near b, we have f (a, b) f (x , y ).
2. If x = a and y = b is a candidate from the rst deriv
Math 16B, Day 5
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
True or false?
1. When we use Lagrange multipliers, we determine whether
we want to nd a maximum or a minimum from the wording of
the problem.
2. To maximize x + y + z , under the constrai
Math 16B, Day 6
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
True or false?
1.
b
d
b
d
f (x , y ) dydx =
a
2.
R
c
f (x , y ) dxdy .
a
c
f (x , y ) dxdy can be interpreted as an area.
3. If R is a box in three-dimensional space, and f (x , y , z ) is
Math 16B, Day 7
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
True or false?
1. If sin() = 0 then is an integer multiple of 2 .
2. If cos(2x ) = 1 then x is a multiple of .
3.
d2
sin(x ) = sin(x ).
dx 2
What would you like to see more of in class?
Wh
Math 16B, Day 8
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
Exercise
Find the area under the curve y = t + sin(t ) from t = 0 to t = .
2
Exercise
Find the area under the curve y = t + sin(t ) from t = 0 to t = .
2
Find the average value of sin(x )
Math 16B, Day 9
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
True or false?
1.
2.
d
F (g (x ) = F (x ) g (x )
dx
2x
dx = ln(x 2 + 1) + C .
+1
x2
3. When we nd an integral by substitution, theres always a
unique choice for the change of variable u =
Math 16B, Day 10
John Lott
UC-Berkeley
http:/math.berkeley.edu/lott
News
1. The rst midterm is on Thursday next week, Feb. 28, in
class. It covers everything up to and including Section 9.3,
which is what we cover today.
2. A sample midterm will be posted
MATH 16B MIDTERM II
Name: Student Identification Number: Section Number: Instructions This exam begins at 2:10pm and ends promptly at 3:00pm. Wait for the start of the exam to be announced. Do not write after time is called. A list of potential answers is
Math 16B Final Review
May 13, 2008 1.) For each of the following functions, find their relative maxima, minima and saddle points: a.) f (x, y) = x4 + y 4 - 4x - 64y b.) f (x, y) = yex + 4y 2 - 20x c.) f (x, y) = x2 + 6xy + 3y 2 + 2x 2.) Lagrange multiplie
MATH 16B - SPRING 2009 - FINAL EXAM REVIEW PROBLEMS
(1) Let f (x, y) = 3x2 + 6xy + 4y 2 + 4y. Find all maxima, minima, and saddle points of f (x, y). (2) Let R = cfw_(x, y)|0 y ln(x), 1 x e. Compute y dx dy
R
(3) Use two iterations of Newton's method with
MATH 16B - SPRING 2009 - FINAL EXAM REVIEW PROBLEM GUIDE
Problem 1 Let f (x, y) = 3x2 + 6xy + 4y 2 + 4y. Find all maxima, minima, and saddle points of f (x, y). Solution. At a maximum, minimum, or saddle point, the partial derivatives have f (1) = 6x + 6y