Exam 3 Solutions 1. Using the bilinearity of the inner product we have x+y
2
xy
2
= x + y, x + y x y, x y
= x, x + 2 x, y + y , y ( x, x + 2 x, y + y, y ) = x, x + 2 x, y + y , y ( x, x 2 x, y + y , y ) = 4 x, y .
2. For (x, y ) = (0, 0) we have
2 2 f =
Exam 2 Solutions
1. False. For example, the odd terms of the alternating harmonic series
n=1
(1)n1 form the n
series
k=1
1 which diverges by the Integral Test. 2k 1
23 , 24 119 . 120 (n+1)!1 (n+1)!
5 2. (i) s1 = 1 , s2 = 6 , s3 = 2
and s4 = =1
1 2
It see
Exam 1 Solutions 1. Claim(1). an an+1 for all n 1. Proof. By induction on n. Since a1 = 2 < 2 + 2 = a2 the base step is valid. Now assume that am am+1 for some m 1. Then am+1 = 2 + am 2 + am+1 = am+2 so the induction step is also valid. Proof. By inductio
Math 330 Statistics I
Test #1 Review
I. Descriptive statistics Graphical: Histogram Numerical: Mean, Variance, Standard deviation Range (Ex. 1.17) Empirical rule II. Probability Probability models - discrete case Experiments Sample space - outcomes Probab
Math 330 Statistics I
Problem Set 2
Fall 2009
This assignment will be graded, so you must do your own work. You may use your class notes and text, but no outside references. You may not receive assistance from anyone else. If you have questions about the
Math 330 Statistics I
Problem Set 3
Fall 2009
This assignment will be graded, so you must do your own work. You may use your class notes and text, but no outside references. You may not receive assistance from anyone else. If you have questions about the
Math 330 Statistics I
Problem Set 4
Fall 2009
This assignment will be graded, so you must do your own work. You may use your class notes and text, but no outside references. You may not receive assistance from anyone else. If you have questions about the
Math 330
Statistics I
Fall 2009
Instructor: Bob Snyder Office: College Center, 2nd Floor Office phone: Ext. 7211 (528-7211) E-mail: [email protected] Office Hours: M 2:30 - 3:30pm, T 2 - 4pm, W 2:30 - 3:30pm Statistics I & II provide an introduction to
Math 331 Statistics II
Problem Set 1
Spring 2010
This assignment will be graded, so you must do your own work. You may use your class notes and text, but no outside references. You may not receive assistance from anyone else. If you have questions about t
Math 331
Statistics II
Spring 2010
Instructor: Bob Snyder Office: College Center, 2nd Floor Office phone: Ext. 211 (528-7211) E-mail: [email protected] Office Hours: M 2:30pm - 3:30pm, T 2 - 4pm, W 2:30pm - 3:30pm Statistics II is a continuation of Stat
Math 364 Ordinary Differential Equations Instructor: Bob Snyder Office: College Center, 2nd Floor Office phone: Ext. 7211 (528-7211) E-mail: [email protected] Office Hours: MW11 am - 12 noon, T 2 - 4pm
Spring 2008
This is an introductory course o
Math 364 Ordinary Differential Equations Show your work. Be complete and precise.
Practice Final Exam
1. Find the solution to the initial value problem y + y = sin x, y(0) = 0, y (0) = 1. 2. Find the general solution to the differential equation y
CHAPTER 0
Preliminaries
1. Introduction
In this preliminary chapter we consider briey some important concepts
from calculus and algebra which we shall require for our study of differen-
tial equations. Many of these concepts may be familiar to the stud
Math 364 Ordinary Differential Equations
Practice Test #2
1. Find two linearly independent solutions to the equation y + 2xy + 4 y = 0 . Be sure to show convergence of any series and to show linear independence of solutions. 2. Find the general s
Danielle M. Henak Calculus II September 26, 2007 David Sharpe Manuscript 1 Considering that on the parabola y=x2, there are two points A and B which can be connected by a line. If the x value for A is c and the x value for B is d, the slope of the li
Danielle M. Henak Calculus II David Sharpe November 16, 2007 Manuscript #3 In this manuscript, we will be exploring hydrostatic forces, specifically, those hydrostatic forces that are exerted by a reservoir of water on a dam. The formula for the forc
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Danielle M. Henak Calculus II David Sharpe October 26, 2007 Maunscript #2 The goal is to make a hydrometer. Because the hydrometer is produced by rotating the ellipse around the x-axis, it could also be made by rotating half of that ellipse around th