Study Aid for Exam # 2, Math 210, Fall 2011
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Actuarial Science Program
DEPARTMENT OF MATHEMATICS
Prof. Rick Gorvett
Fall, 2008
Math 210
Theory of Interest
Exam # 2 (17 Problems Max possible points = 40)
Thursday,
S E C T I O N 13.7
SOLUTION
Cylindrical and Spherical Coordinates
(ET Section 12.7)
411
2 A point P on the parabola C has the form P = x0 , ax 0 , c , hence the parametric equations of the line through the origin and P are
x = t x0 ,
2 y = tax
802
C H A P T E R 15
D I F F E R E N T I AT I O N I N S E V E R A L VA R I A B L E S
(ET CHAPTER 14)
1 + ln x n = + E n We subtract the last equation from the other equations to obtain ln xi - ln x n = (E i - E n ) , or ln xi = (E i - E n ) x
352
C H A P T E R 13
V EC T OR G E OME TRY
(ET CHAPTER 12)
13.4 The Cross Product
Preliminary Questions
(ET Section 12.4)
3 -5 4 4 -1 0 2 1 ? 3
1. What is the (1, 3) minor of the matrix
SOLUTION
The (1, 3) minor is obtained by crossing out the
378
C H A P T E R 13
V EC T OR G E OME TRY
(ET CHAPTER 12)
SOLUTION The two planes are parallel if vectors that are normal to the planes are parallel. The vector n = 1, 1, 1 is normal to the plane x + y + z = 1. We identify the following normals:
S E C T I O N 15.1
Functions of Two or More Variables
(ET Section 14.1)
625
Therefore, (1) gives
x 2 + y 2 2|x y|
x 2 + y2 |x y| 2
xy 1 1 x 2 + y2 1 |x y| |c| = 2 2 22 = < 1= = 2 2 x 2 + y2 + 1 2 2 x + y2 + 1 x + y2 + 1 x + y +1 That is,
S E C T I O N 14.5
Motion in Three-Space
(ET Section 13.5)
543
We now compute T(0): T(0) = Finally we find N = B T: 1 1 1 1 1 N(0) = 2, 0, 1 0, 1, 0 = (2i + k) j = (2i j + k j) = (2k - i) = -1, 0, 2 5 5 5 5 5 (b) Differentiating r(t) =
764
C H A P T E R 15
D I F F E R E N T I AT I O N I N S E V E R A L VA R I A B L E S
(ET CHAPTER 14)
We thus showed that if the Fermat point exists, then A < 120 . Similarly, one shows also that B and C must be smaller than 120 . We conclude that
S E C T I O N 13.6
A Survey of Quadric Surfaces
(ET Section 12.6)
397
or 1 4, -2, 4 x, y, z = 4 6 or 2 1 2 ,- , x, y, z = 4, in normal form. 3 3 3 - Let n = O P, where P = (x 0 , y0 , z 0 ) is a point on the sphere x 2 + y 2 + z 2 = r 2 , and
702
C H A P T E R 15
D I F F E R E N T I AT I O N I N S E V E R A L VA R I A B L E S
(ET CHAPTER 14)
72. Find the curve y = g(x) passing through (0, 1) that crosses each level curve of f (x, y) = y sin x at a right angle. If you have a computer a
MULT IPLE 16 INTEGRATION
16.1 Integration in Several Variables
Preliminary Questions
1. In the Riemann sum S8,4 for a double integral over R = [1, 5] [2, 10], what is the area of each subrectangle and how many subrectangles are there?
SOLUTION
(ET
Math 210 Fall 2011
Basic Interest Theory Material
Old FM Exam Problems
1) Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits 100 into his
bank account, and Robbie deposits 50 into his. Each account earns the same annual effective
in
Study Aid for Exam # 1, Math 210, Fall 2011
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Actuarial Science Program
DEPARTMENT OF MATHEMATICS
Prof. Rick Gorvett
Fall, 2008
Math 210
Theory of Interest
Exam # 1 (17 Problems Max possible points = 40)
Thursday,
University of Illinois at Urbana-Champaign Department of MathematicsActuarial Science Program Math 210 Theory of Interest Instructor Jared Thompson Spring 2006 Homework Assignment #6 (10 points) Due at the beginning of class on Friday, March 31, 2006
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