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Math 421 - Practice Problems 2
The exercises below are practice problems for the final exam.
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LECTURE SLIDES
1. Lecture slides 3: Exercises on pages 16, 17,18
2. Lecture slides 4: Exercises on pages 14, 16, 17, 18, 20
3. Lecture slides 7: Exercises on pages 6, 8, 9, 14, 16 (Prove the Theorem)
4. Lecture slides 8: Exercises on pages 14, 15, 16 (Prove the Theorems)
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HOMEWORK
1. Find all numbers x ∈ R for which
a. |x + 2| > 3.
b. |x + 5| > 9.
c. x2 + 4x + 4 > 1.
2. Prove that for x, y ∈ R
a.
x2 + 2xy + y 2 ≥ 0.
b.
x4 + 3x2 y 2 + y 4 ≥ 0.
3. Prove by mathematical induction that
1 + 3 + 5 + · · · + (2n + 1) = (n + 1)2 .
4. Prove by induction: If h > −1 then (1 + h)n ≥ 1 + nh.
5. Prove the following modification of the Principle of Mathematical Induction: If A ⊂ N contains a number n0 ∈ N and for all k ∈ Z, k ∈ A implies
k + 1 ∈ A, then A contains all natural numbers ≥ n0 .
6. In Mathland the currency consists of 3 and 4 penny coins. Suppose you
want to buy an item that costs 9 pennies, then you would be paying with
three 3 penny coins.
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Show that if you have an unlimited number of 3 and 4 penny coins you
can pay any price greater than or equal to six pennies, without receiving
change.
Hint: use complete induction.
√
7. Show that 15 is irrational.
8. Call a function f : R → R even if for all x ∈ R, f (x) = f (−x); and odd if
for all x ∈ R, f (x) = −f (−x). Show:
a. The sum of two even functions is even; and the sum of two odd functions
is odd.
b. The product of two even functions is even; and the product of two odd
functions is even.
c. The product of an even function and an odd function is odd.
d. Must the sum of an even function and an odd function be even? Odd?
9. Call a function f : R → R periodic (with period A) if for all x ∈ R,
f (x) = f (x + A).
a. Show that the sum of periodic functions with period A has period A.
b. Show that the sum of periodic functions, one with period 2A and the
other with period 3A, has period 6A.
10. Suppose f (x) > 0 for all x, and also that limx→a f (x) exists.
a) Show that limx→a f (x) ≥ 0.
b) Give an example where limx→a f (x) = 0.
11. Prove that limx→2
3
x+1
= 1 directly by using the definition of the limit.
12. Prove that limx→2
a theorem.
1
x+1
=
1
3
directly from the definition, and not by using
13. Give an example to show that the following definition of limx→a f (x) = L
is *NOT* correct:
For all > 0 there is a δ > 0 such that if |f (x)−L| < , then 0 < |x−a| < δ.
14. Prove or give a counter example: if |f | is continuous at a, then f is continuous at a
15. Prove or give a counter example: if f is continuous at a, then |f | is continuous at a
16. Give an example of a function f : R → R, that is discontinuous at all
x ∈ R, such that |f (x)| is continuous for all x.
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17. Suppose that f and g are two functions continuous on R such that |f | = |g|
and such that f (x) 6= 0 and g(x) 6= 0 for all x ∈ R. Show that f = g or
f = −g.
18. Let limx→x0 f (x) = l and limx→x0 g(x) = m. Show that
(a) lim max{f (x), g(x)} = max{l, m};
x→x0
(b) lim min{f (x), g(x)} = min{l, m}.
x→x0
19. Show that x2 + 2 = ex has at least one solution in R.
20. Let f : (−1, 1) → R be continuous at x = 0 and suppose that f (x) = f (x2 )
for all x ∈ (−1, 1). Show that f (x) = f (0) for all x ∈ (−1, 1).
21. Let f : [a, b] → R be continuous and suppose that there is some c ∈ (a, b)
such that f (c) 6= 0. Show that there exists some δ > 0 such that f (x) 6= 0
for every x ∈ (c − δ, c + δ).
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SPIVAK BOOK
1. Exercises 1, 2, 5, 6, 9, 12 Chapter 2
2. Exercises 1, 2, 6, 8, 9, 10, 12, 13, 17 Chapter 5
3. Exercises 3, 4, 5, 6, 7, 8 Chapter 6
4. Exercises 6, 8, 10 Chapter 7
5. Exercise 3 Chapter 8
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