Advanced Calculus I
October 1, 2004
Test 1
Name:
D. P. Story
Instructions (100 points) Solve each problem, be neat and use good notation.
Denitions
(10pts )
ea.
1. Give a precise denition of each of the terms below.
(a) Let cfw_ xn be a sequence of real
Advanced Calculus I
Fall 2004
Assignment #4
Due 9/24/04
Dr. D. P. Story
1.3, pages 2729.
Problem 27. Let A be a nonempty bounded set. The maximum of A is a number x A
such that a x, x A. Prove that a nonempty bounded set has a maximum if and
only if it co
Advanced Calculus I
Fall 2004
Assignment #3
Due 9/17/04
Dr. D. P. Story
1.3, pages 2729.
Problem 2. Prove that the set S = cfw_ x R | 10 x x > 0 is bounded.
Solution : First we solve the inequality using standard algebraic techniques:
10 x x > 0 x(10 x)
Assignment #2
Advanced Calculus I
Fall 2004
Due 9/10/04
Dr. D. P. Story
1.1, pages 89.
Problem 12. This is the one I meant to assign on Assignment #1. Prove that 2 + 3
is irrational.
Solution : Suppose 2 + 3 is rational, then its square is rational as wel
Advanced Calculus I
November 5, 2004
Test 2
Name:
D. P. Story
Instructions (100 points) Solve each problem, be neat and use good notation.
Denitions
pts
(10ea. )
1. Give a precise denition of each of the terms below.
(a) Let cfw_ xn be a sequence of real
Assignment #1
Advanced Calculus I
Fall 2004
Due 9/3/04
Dr. D. P. Story
1.1, pages 89.
Problem 1. Show that Q is closed under addition and multiplication.
Proof : Let x, y Q, then there exists integers p, q , r, s Z such that
p
r
x = , y = , q = 0, s = 0
q
Assignment #5
Advanced Calculus I
Fall 2004
Due 10/8/04
Dr. D. P. Story
2.2, pages 6667.
n
Problem 2. Dene xn =
k=1
1
. Prove that cfw_xn converges.
n+k
Proof : After making some sample calculations,
x1 = 0.5
x6 = 0.65321
x2 = 0.58333
x6 = 0.65870
x3 = 0
Assignment #6
Advanced Calculus I
Fall 2004
Due 10/15/04
Dr. D. P. Story
2.3, pages 7476.
Problem 6. Let cfw_an be an sequence and suppose cfw_a2n and cfw_a2n1 both converge to
the same number, L. Prove that cfw_an converges to L.
Proof : Let
> 0. The
Assignment #11
Advanced Calculus I
Fall 2004
Due 11/29/04
Dr. D. P. Story
4.1, pages 137138.
Problem 7. Prove cfw_n f (c + 1/n) f (c) converges to f (c).
Proof : Let xn = c + 1/n, then cfw_xn converges to c, xn = c. Now apply Theorem 4.2
to obtain the d
Assignment #12
Advanced Calculus I
Fall 2004
Due 12/06/04
Dr. D. P. Story
4.2, pages 145147.
Problem 10. Prove x7 + x5 + x3 + 1 = 0 has exactly one solution.
Proof : Let f (x) = x7 + x5 + x3 + 1, and note that f (1) = 2 and f (0) = 1. By the
Intermediate
Assignment #10
Advanced Calculus I
Fall 2004
Due 11/19/04
Dr. D. P. Story
Here is my take on these problem, of course, each can be solved in a variety of ways.
3.4, pages 114115.
Problem 2. Those enrolled at the 400 level of this course, do parts (b) and
Assignment #32
Advanced Calculus I
Fall 2004
Due 11/12/04
Dr. D. P. Story
3.3, pages 106110.
Problem 2(a)(d). Develop the Bisection Method.
( a ) Explain why it is sucient to assume f (a) and f (b) have opposite signs and v = 0.
Solution : Dene g (x) = f
Assignment #7
Advanced Calculus I
Fall 2004
Due 10/22/04
Dr. D. P. Story
3.1, pages 9092.
Problem 4(b)(f ).
( b ) lim (x2 + x 1) = 1.
x1
Solution : Write the dierence of the polynomial and its limit as a polynomial around
x = 1:
(x2 + x 1) 1 = x2 + x 2 =
Assignment #8
Advanced Calculus I
Fall 2004
Due 10/29/04
Dr. D. P. Story
3.2, pages 9799.
Problem 4. Let f : [ a, b ] R be continuous at c [ a, b ] and suppose f (c) > 0. Prove
that there is a number m > 0 and an interval [ u, v ] [ a, b ] such that c [ u