Question 1

Draw a careful graph of the function f deﬁned by the formula $f ( n ) = \left\lfloor \frac { n } { 3 } \right\rfloor$ 
for all integers n.

Accepted Answer

The answer of Draw a careful graph of the function...

Question 2

(a) Consider the following algorithm segment:
for $ i:=1 $ to $ n $
for $ j:=1 $ to $ i $
$ x:=5 \cdot i+8 \cdot j $
next $ j $
next $ i $
How many additions and multiplications are performed when the inner loop of this algorithm segment is executed? How many additions and multiplications are performed when the entire algorithm segment is executed?
(b) Find an order for this algorithm segment from among the following: $\log _ { 2 } n , n , n \cdot \log _ { 2 } n$, $n ^ { 2 } , n ^ { 3 }$, and $n ^ { 4 }$. Give a reason for your answer.

Accepted Answer

a. Two multiplications and one addition are performed when the inner loop of the given
algorithm segment is executed for a total of three elementary operations for each iteration of
the inner loop. The number of iterations of the inner loop can be deduced from the following
table, which shows the values of i and j for which the inner loop is executed.
  
Therefore, by Theorem $5.2 .2$, the number of iterations of the inner loop is
$$n + ( n - 1 ) + \cdots + 2 + 1 = \frac { n ( n + 1 ) } { 2 }$$
The total number of elementary operations that must be performed when the algorithm is executed is the number performed during each iteration of the inner loop times the number of iterations of the inner loop:
$$3 \cdot \left( \frac { n ( n + 1 ) } { 2 } \right) = \frac { 3 } { 2 } n ^ { 2 } + \frac { 3 } { 2 } n .$$
b. By the theorem on polynomial orders, $\frac { 3 } { 2 } n ^ { 2 } + \frac { 3 } { 2 } n$ is $\Theta \left( n ^ { 2 } \right)$, and so the algorithm segment has order $n ^ { 2 }$.

Question 3

Express the following statement using Ω-notation.: $$\left| x ^ { 5 } \right| \leq \left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \quad \text { for all real numbers } x > 2$$

Accepted Answer

$\frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \text { is } \Omega \left( x ^ { 5 } \right) .$

Question 4

Explain why the following statement is true. (You may use the theorem on polynomial orders.) $$3 + 6 + 9 + \cdots + 3 n \text { is } O \left( n ^ { 2 } \right)$$

Accepted Answer

Note that
\[\begin{aligned}
3 + 6 + 9 +

Question 5

If $n$ and $k$ are positive integers and $2 ^ { k } \leq n < 2 ^ { k + 1 }$, what is $\left\lfloor \log _ { 2 } ( n ) \right\rfloor$ ? Be sure to justify each step of your answer.

Accepted Answer

If @#LAT-DLM& and @#LAT-DLM& are positive integers and @#LAT-DLM&, the

Question 6

If x is a real number and $x > 1 , \text { is } x ^ { 2 } > x ? \text { Why? Is } 5 x ^ { 3 } > 5$  Why?

Accepted Answer

If @#LAT-DLM& is a real number and @#LAT-DLM&, we can multip

Question 7

Use the definition of $O$-notation to prove that $2 x ^ { 2 } + 3 x + 4$ is $O \left( x ^ { 2 } \right)$. (Do not use the theorem on polynomial orders.)

Accepted Answer

For all real numbers @#LAT-DLM&, 
@#LAT-DLM& because @#LAT-DLM&, and 4

Question 8

Use the definition of $O$-notation to prove that $15 x ^ { 3 } + 8 x + 4$ is $O \left( x ^ { 3 } \right)$.(Do not use the theorem on polynomial orders.)

Accepted Answer

For all real numbers @#LAT-DLM&,
@#LAT-DLM& because @#LAT-DLM&, and 4

Question 9

Express the following statement using O-notation: $$\left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \leq 36 \left| x ^ { 5 } \right| \quad \text { for all real numbers } x > 2$$

Accepted Answer

The answer of Express the following statement using O-notation: \[\left|...

Question 10

Use O-notation to express the following statement: $$\left| 5 x + x \log _ { 2 } x \right| \leq 6 \left| x \log _ { 2 } x \right| \text { for all } x > 2$$

Accepted Answer

The answer of Use O-notation to express the following statement:...

Question 11

Describe the operation of the insertion sort algorithm.

Accepted Answer

The object of the insertion sort algorit

Question 12

Let h be the function whose graph is shown below. Carefully sketch the graph of 2h.

Accepted Answer

The answer of Let h be the function whose graph...

Question 13

Describe the operation of the binary search algorithm.

Accepted Answer

Binary search is a recursive algorithm.

Question 14

(a) Find the total number of additions and multiplications that must be performed when the following algorithm is executed. Show your work carefully. 
$\text { for } i:=1 \text { to } n$
$\text { for } j=i \text { to } n$
$a:=2 \cdot(5 \cdot i+j+1)$
$\text { next } j$
$\text { next } i$ 
(b) Find an order for the algorithm segment of part (a) from among the following: $\log _ { 2 } n , n$, $n \cdot \log _ { 2 } n , n ^ { 2 } , n ^ { 3 }$, and $n ^ { 4 }$. Give a reason for your answer.

Accepted Answer

a. For each iteration of the inner loop

Question 15

Describe the operation of the sequential search algorithm.

Accepted Answer

The object of the sequential search algo

Question 16

Consider the statement: $\left| 3 x ^ { 2 } + 17 x + 5 \right| \leq 25 \left| x ^ { 2 } \right| \quad \text { for all } x > 1$ 
Express this statement using O-notation.

Accepted Answer

The answer of Consider the statement: \(\left| 3 x ^...

Question 17

Describe the operation of the merge sort algorithm.

Accepted Answer

The object of the merge sort algorithm i

Question 18

Consider the statement: $3 \left| x ^ { 2 } \right| \leq \left| 3 x ^ { 2 } + 17 x + 5 \right| \leq 25 \left| x ^ { 2 } \right| \quad \text { for all } x > 1 \text {. }$ 
Express this statement using $\Theta \text {-notation. }$

Accepted Answer

The answer of Consider the statement: \(3 \left| x ^...

Question 19

Express the following statement using Θ-notation: $$\left| x ^ { 5 } \right| \leq \left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \leq 36 \left| x ^ { 5 } \right| \quad \text { for all real numbers } x > 2$$

Accepted Answer

The answer of Express the following statement using Θ-notation: \[\left|...

Question 20

Define a function $F : \mathbf { R } ^ { + } \longrightarrow \mathbf { R }$ by the formula $F ( x ) = \log _ { 2 } ( x )$ for all positive real numbers $x$.
(a) Graph $F$, marking units carefully on your axes.
(b) What is $F \left( \frac { 1 } { 8 } \right)$ ? Why?
(c) Write the equation $2 ^ { 20 } = 1,048,576$ in logarithmic form.

Accepted Answer

a.
@#IMG-DLM& b. @#LAT-DLM& be

Draw a careful graph of the function f deﬁned by the formula $f ( n ) = \left\lfloor \frac { n } { 3 } \right\rfloor$ for all integers n.

Express the following statement using Ω-notation.: $\left| x ^ { 5 } \right| \leq \left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \quad \text { for all real numbers } x > 2$

Explain why the following statement is true. (You may use the theorem on polynomial orders.) $3 + 6 + 9 + \cdots + 3 n \text { is } O \left( n ^ { 2 } \right)$

If $n$ and $k$ are positive integers and $2 ^ { k } \leq n < 2 ^ { k + 1 }$ , what is $\left\lfloor \log _ { 2 } ( n ) \right\rfloor$ ? Be sure to justify each step of your answer.

If x is a real number and $x > 1 , \text { is } x ^ { 2 } > x ? \text { Why? Is } 5 x ^ { 3 } > 5$ Why?

Use the definition of $O$ -notation to prove that $2 x ^ { 2 } + 3 x + 4$ is $O \left( x ^ { 2 } \right)$ . (Do not use the theorem on polynomial orders.)

Use the definition of $O$ -notation to prove that $15 x ^ { 3 } + 8 x + 4$ is $O \left( x ^ { 3 } \right)$ .(Do not use the theorem on polynomial orders.)

Express the following statement using O-notation: $\left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \leq 36 \left| x ^ { 5 } \right| \quad \text { for all real numbers } x > 2$

Use O-notation to express the following statement: $\left| 5 x + x \log _ { 2 } x \right| \leq 6 \left| x \log _ { 2 } x \right| \text { for all } x > 2$

Describe the operation of the insertion sort algorithm.

Let h be the function whose graph is shown below. Carefully sketch the graph of 2h.

Describe the operation of the binary search algorithm.

Describe the operation of the sequential search algorithm.

Consider the statement: $\left| 3 x ^ { 2 } + 17 x + 5 \right| \leq 25 \left| x ^ { 2 } \right| \quad \text { for all } x > 1$ Express this statement using O-notation.

Describe the operation of the merge sort algorithm.

Consider the statement: $3 \left| x ^ { 2 } \right| \leq \left| 3 x ^ { 2 } + 17 x + 5 \right| \leq 25 \left| x ^ { 2 } \right| \quad \text { for all } x > 1 \text {. }$ Express this statement using $\Theta \text {-notation. }$

Express the following statement using Θ-notation: $\left| x ^ { 5 } \right| \leq \left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \leq 36 \left| x ^ { 5 } \right| \quad \text { for all real numbers } x > 2$

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Relations

Counting and Probability

Graphs and Trees

Regular Expressions and Finite State Automata

Filters

Exam 11: Analyzing Algorithm Efficiency

Draw a careful graph of the function f deﬁned by the formula f(n)=⌊n3⌋f ( n ) = \left\lfloor \frac { n } { 3 } \right\rfloorf(n)=⌊3n​⌋ for all integers n.

Express the following statement using Ω-notation.: ∣x5∣≤∣12x5(3x+4)x+2∣ for all real numbers x>2\left| x ^ { 5 } \right| \leq \left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \quad \text { for all real numbers } x > 2​x5​≤​x+212x5(3x+4)​​ for all real numbers x>2

Explain why the following statement is true. (You may use the theorem on polynomial orders.) 3+6+9+⋯+3n is O(n2)3 + 6 + 9 + \cdots + 3 n \text { is } O \left( n ^ { 2 } \right)3+6+9+⋯+3n is O(n2)

If nnn and kkk are positive integers and 2k≤n<2k+12 ^ { k } \leq n < 2 ^ { k + 1 }2k≤n<2k+1 , what is ⌊log⁡2(n)⌋\left\lfloor \log _ { 2 } ( n ) \right\rfloor⌊log2​(n)⌋ ? Be sure to justify each step of your answer.

If x is a real number and x>1, is x2>x? Why? Is 5x3>5x > 1 , \text { is } x ^ { 2 } > x ? \text { Why? Is } 5 x ^ { 3 } > 5x>1, is x2>x? Why? Is 5x3>5 Why?

Use the definition of OOO -notation to prove that 2x2+3x+42 x ^ { 2 } + 3 x + 42x2+3x+4 is O(x2)O \left( x ^ { 2 } \right)O(x2) . (Do not use the theorem on polynomial orders.)

Use the definition of OOO -notation to prove that 15x3+8x+415 x ^ { 3 } + 8 x + 415x3+8x+4 is O(x3)O \left( x ^ { 3 } \right)O(x3) .(Do not use the theorem on polynomial orders.)

Use O-notation to express the following statement: ∣5x+xlog⁡2x∣≤6∣xlog⁡2x∣ for all x>2\left| 5 x + x \log _ { 2 } x \right| \leq 6 \left| x \log _ { 2 } x \right| \text { for all } x > 2∣5x+xlog2​x∣≤6∣xlog2​x∣ for all x>2

Describe the operation of the insertion sort algorithm.

Let h be the function whose graph is shown below. Carefully sketch the graph of 2h.

Describe the operation of the binary search algorithm.

Describe the operation of the sequential search algorithm.

Consider the statement: ∣3x2+17x+5∣≤25∣x2∣ for all x>1\left| 3 x ^ { 2 } + 17 x + 5 \right| \leq 25 \left| x ^ { 2 } \right| \quad \text { for all } x > 1​3x2+17x+5​≤25​x2​ for all x>1 Express this statement using O-notation.

Describe the operation of the merge sort algorithm.

Speaking Mathematically

The Logic of Compound Statements

The Logic of Quantified Statements

Elementary Number Theory and Methods of Proof

Sequences, Mathematical Induction, and Recursion

Set Theory

Functions

Relations

Counting and Probability

Graphs and Trees

Regular Expressions and Finite State Automata

Filters

Draw a careful graph of the function f deﬁned by the formula $f ( n ) = \left\lfloor \frac { n } { 3 } \right\rfloor$ for all integers n.

Express the following statement using Ω-notation.: $\left| x ^ { 5 } \right| \leq \left| \frac { 12 x ^ { 5 } ( 3 x + 4 ) } { x + 2 } \right| \quad \text { for all real numbers } x > 2$

Explain why the following statement is true. (You may use the theorem on polynomial orders.) $3 + 6 + 9 + \cdots + 3 n \text { is } O \left( n ^ { 2 } \right)$

If $n$ and $k$ are positive integers and $2 ^ { k } \leq n < 2 ^ { k + 1 }$ , what is $\left\lfloor \log _ { 2 } ( n ) \right\rfloor$ ? Be sure to justify each step of your answer.

If x is a real number and $x > 1 , \text { is } x ^ { 2 } > x ? \text { Why? Is } 5 x ^ { 3 } > 5$ Why?

Use the definition of $O$ -notation to prove that $2 x ^ { 2 } + 3 x + 4$ is $O \left( x ^ { 2 } \right)$ . (Do not use the theorem on polynomial orders.)

Use the definition of $O$ -notation to prove that $15 x ^ { 3 } + 8 x + 4$ is $O \left( x ^ { 3 } \right)$ .(Do not use the theorem on polynomial orders.)

Use O-notation to express the following statement: $\left| 5 x + x \log _ { 2 } x \right| \leq 6 \left| x \log _ { 2 } x \right| \text { for all } x > 2$

Consider the statement: $\left| 3 x ^ { 2 } + 17 x + 5 \right| \leq 25 \left| x ^ { 2 } \right| \quad \text { for all } x > 1$ Express this statement using O-notation.