Advanced Calculus II
Spring 2005
Due 02/04/05
Dr. D. P. Story
Assignment #2
Solutions (21 points) Below, please nd a set of solutions to this assignment.
5.1, page 169170.
pts
(2
)
1. Problem 3. Let P1 and P2 be partitions of [ a, b ]. Find a renement, P0
Advanced Calculus II
Spring 2005
Due 01/28/05
Dr. D. P. Story
Assignment #1
Solutions (16 points) Below, please nd a set of solutions to this assignment.
4.3, page 155.
(4pts )
1. Problem 15. Use the denition to prove f (x) = x2 is convex.
Solution : Let
Advanced Calculus II
Spring 2005
Due 02/11/05
Dr. D. P. Story
Assignment #3
Solutions (17 points) Below, please nd a set of solutions to this assignment.
Before beginning the solutions, I thought Id prove a fact that I have used numerous times in lecture,
Advanced Calculus II
Spring 2005
Due 03/11/05
Dr. D. P. Story
Assignment #6
Solutions
Solutions (28 points) Below, please nd a set of solutions to this assignment.
6.1, pages 214215.
(2pts )
1. Problem 2(b).
k=3
1
k2 4
Solution : Looks like a telescoping
Advanced Calculus II
Spring 2005
Due 04/22/05
Dr. D. P. Story
Assignment #10
Solutions
Solutions (27 points) Below, please nd a set of solutions to this assignment.
7.2, pages 251.
(3pts )
ea.
1. Problem 7(b),(c).
(b) gn (x) = x(1 x)n .
Solution: Note tha
1
Advanced Calculus II
Spring 2005
As I was walking back to my oce, I constructed a better argument than the one given in class, here it
is.
Example. Suppose f : [ 1, ) R is positive and decreasing. Dene
n
n
f (i)
xn =
f
1
i=1
Prove that cfw_xn is a con
1
Advanced Calculus II
Spring 2005
Here is the example I did in class, in precise detail.
Example. Dene a function
f (x) =
1 xQ
0 xQ
Then f is not Riemann integrable over [ 0, 1 ].
Solution : Assume f is Riemann integrable over [ 0, 1 ] and let L =
1/2, t