Math 315-003
April 20, 2004
D. Wright
Final Exam
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Name_
1. Explain what if means for a function f : D R to be continuous.
a. Give the sequence definition.
b. Give the epsilon-delta definition.
2. Show a bounded increasing sequence conve
Math 315 Sections 1 and 2
1617 March 2007
D. Wright
Test 2
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Name _
1. Let f : A R be a real-valued function. Give the sequence definition of continuity and
the definition of continuity.
2. Prove that a closed interval has the property t
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Math 315, Section 2
Exam 1
Instructor: David G. Wright
29-31 January 2009
1. (20%) Give an example of each of the following or argue that such a request is impossible:
(a) an unbounded sequence with a convergent subsequence;
(b) a nested sequence of
Name:
Math 315, Section 2
Final Exam
Instructor: David G. Wright
17 April 2009, 11 AM 2 PM
1. (20%) Give an example of each of the following or argue that such a request is impossible:
(a) a nested sequence of open intervals whose intersection is empty;
(
Math 315 Sections 1 and 2
20 and 24 April 2007
D. Wright
Final
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Name _
1. Show that if lim an = a and lim bn = b , then lim an bn = ab .
2. Show that the Nested Interval Property implies the Axiom of Completeness.
3. Show that if 0 < r
Math 315-003
Test 3
Name_
April 2,3,5, 2004
Show relevant work!
D. Wright
1. State and prove the First Fundamental Theorem of Calculus.
2. Suppose the function f : R R is continuous. Define
x
G( x ) = 0 ( x t ) f ( t ) dt
Prove that G(x) = f (x) for all x
Math 315-003
2728 February 1 March 2004
D. Wright
Test 2
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Name_
1. State and prove the product rule for derivatives.
2. For a natural number n, let g(x) = x 1/n for x > 0. Prove that g' ( x ) =
1 1 / n 1
.
x
n
3. State and prove the (La
Math 315 Sections 1 and 2
810 February 2007
D. Wright
Test 1
Show relevant work!
Name _
1. State the Axiom of Completeness and use it to prove that a monotone increasing
sequence converges to its least upper bound.
2. Show that the rational numbers are co
Math 315-003
3031 January 2004
D. Wright
Test
Show relevant work!
Name _
1. Define what it means for a subset A R to be bounded.
2. State the Completeness Axiom.
3. State the Archimedian Property.
1
4. Prove lim = 0.
n n
5. Show a convergent sequence is b
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Math 315, Section 2
Exam 2
Instructor: David G. Wright
5-7 March 2009
1. (30%) Complete the following denitions:
(a) A set U in R is open if
(b) A point p is a limit point of a set A if
(c) Let f : A R be a function. Then lim f (x) = L means
xc
(d)