Math 4603 Advanced Calculus I
Homework 1: Sections 0.1, 0.2, 0.3
Due date: Friday, June 21
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to e
Math 4603 Midterm 1
Name:
Instructions:
This exam contains 6 questions. Each question is worth 20 points. The exam is worth 120
points in total.
1
1. Prove that Q is countable.
Solution. Dene f : Q Z N by f (0) = (0, 1) and f (n/m) = (n, m) where it is as
Math 4603 Midterm 3
Name:
Instructions:
This exam contains 6 questions. Each question is worth 20 points. The exam is worth 120
points in total.
1
1. Dene f : (0, ) R by f (x) = 1/x. Use the denition of derivative to show that f is
dierentiable and f (x)
4603 HW3
1. Prove that Cauchy sequences are bounded.
Solution. Let cfw_an be Cauchy. Choose N such that n, m N implies |an am | < 1.
n=1
Then in particular n N implies aN 1 < an < aN + 1. So cfw_an is bounded above by
n=1
maxcfw_a1 , . . . , aN 1 , aN +
4603 HW8
1. Let A and B be disjoint subsets of R, and f : A B R a continuous function. Assume
f is uniformly continuous on A and on B. Must it be true that f is uniformly continuous on
A B? Prove it or provide a counterexample.
Solution. No. Let A = (0, 1
4603 HW5
1. Dene f : R R by
f (x) =
x2 , x = 2
11, x = 2
Prove that limx2 f (x) = 4.
Solution. Let
> 0 and choose = mincfw_1, /5. Then 0 < |x 2| < implies
|f (x) 4| = |x2 4| = |x 2|x + 2| < |x + 2| < 5 .
2. Let f, g : D R, let a be an accumulation point o
4603 HW9
1. Suppose f : [a, b] R is continuous and injective. Prove that f is either strictly increasing
or strictly decreasing.
Solution. Suppose f is not strictly increasing or strictly decreasing. Then there exist x, y, z
[a, b] such that x < y < z an
4603 HW7
1. If f + g is continuous a, must it be true that both f and g are continuous at a? Prove it, or
provide a counterexample.
Solution. No. Let f (x) = 0 if x < 0 and f (x) = 1 if x 0, and let g = f . Then f + g 0 is
continuous at 0 but neither f no
4603 HW4
1. Let cfw_an be a sequence of real numbers. Prove that if x is an accumulation point of
n=1
cfw_an : n N, then cfw_an has a subsequence converging to x.
n=1
Solution. Let x be an accumulation point of cfw_an : n N. Then for each > 0, (x , x +
4603 HW6
1. If f + g has a limit at a, must it be true that both f and g have limits at a? Prove it, or
give a counterexample.
Solution. No, for example dene f : R R by f (x) = 0 for x < 0 and f (x) = 1 for x 0, and
let g = f . Then f + g 0, so f + g has
4603 HW11
1. Suppose that f : R R satises f (0) = f (0) = 0 and f (x) f (x) = 0 for all x R. Use
Taylors theorem to show that f (x) = 0 for all x R.
Solution. Note that f
= (f ) = f , f
= (f ) = f ,. and in general
f (n) =
f,
f,
n even
n odd
Since f (0) =
4603 HW12
1. Dene f : [0, 1] R by f (x) = x2 . Use FTC to show that
1
0 f
dx = 1/3.
Compare this to Problem 4 of HW11, where you computed this integral by hand.
Solution. Dene F (x) = x3 /3. Then F (x) = x2 . As F is continuous, it is (Riemann)
integrable
4603 HW2
1. Let S R be nonempty. Prove the following statements:
(i) S is countable if and only if there exists an injective function f : S N.
(ii) S is countable if and only if there exists a sequence cfw_an of real numbers such that
n=1
S = cfw_an : n
4603 HW1
1. Give examples of the following:
(i) A function which is bijective;
(ii) A function which is neither injective nor surjective;
(iii) A function which is injective but not surjective;
(iv) A function which is surjective but not injective.
Soluti
Math 4603 Midterm 2
Name:
Instructions:
This exam contains 6 questions. Each question is worth 20 points. The exam is worth 120
points in total.
1
1. Dene f : (0, ) R by f (x) = x. Prove that f is continuous.
Hint: | x a| = | x a| x+a .
x+ a
Solution. L
Math 4603 Advanced Calculus I
Homework 2: Sections 0.4, 0.5, 1.1, 1.2
Due date: Wednesday, June 26
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be a
Math 4603 Advanced Calculus I
Homework 3: Sections 1.3, 1.4, 2.1
Due date: Friday, June 28
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to e
Math 4603 Advanced Calculus I
Homework 4: Sections 2.2, 2.3
Due date: Wednesday, July 3
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to expl
Math 4603 Advanced Calculus I
Homework 5: Sections 2.4, 3.1
Due date: Monday, July 8
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to explain
Math 4603 Advanced Calculus I
Homework 6: Sections 3.2, 3.3
Due date: Wednesday, July 10
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to exp
Math 4603 Advanced Calculus I
Homework 7: Sections 3.3, 3.4
Due date: Friday, July 12
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to explai
Math 4603 Advanced Calculus I
Homework 8: Sections 4.1-4.3
Due date: Wednesday, July 17
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to expl
Math 4603 Advanced Calculus I
Homework 9: Sections 4.4, 5.1
Due date: Friday, July 19
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to explai
Math 4603 Advanced Calculus I
Homework 10: Sections 5.2-5.4
Due date: Wednesday, July 24
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to exp
Math 4603 Advanced Calculus I
Homework 11: Sections 5.5-5.7
Due date: Friday, July 26
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to explai
Math 4603 Advanced Calculus I
Homework 12: Sections 6.1, 6.2
Due date: Wednesday, July 31
Short Answer
State the answer to each of the following questions. It is not necessary to write down
any justication (though of course one would want to be able to ex
4603 HW10
1. Use IVT and MVT to show that x3 + x + 1 = 0 has exactly one solution.
Solution. Let f (x) = x3 + x + 1. Note that f (1) = 1 < 0 and f (1) = 3 > 0 so by IVT there
is a solution to x3 + x + 1 = 0 between 1 and 1. Suppose there are two such solu