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I have 12 questions that...

I have 12 questions that I need answers for by Monday. These are not that difficult. They are for MTH/221, pre calculus. Can you do this?

**12 Questions **

** **

**MTH/221**

**Exercise 4.1:**

** **

**4. **A wheel of fortune has the integers from 1 to 25 placed on it

in a random manner. Show that regardless of how the numbers

are positioned on the wheel, there are three adjacent numbers

whose sum is at least 39.

**7. **A lumberjack has 4*n *+ 110 logs in a pile consisting of *n *layers.

Each layer has two more logs than the layer directly above

it. If the top layer has six logs, how many layers are there?

**Exercise 4.2:**

**16. **Give a recursive definition for the set of all

**a) **positive even integers

**b) **nonnegative even integers

**Exercise 4.3:**

**10. **If *n *∈ **Z**+, and *n *is odd, prove that 8|*(n*2 − 1*)*.

**15. **Write each of the following (base-10) integers in base 2 and

base 16.

**a) **22 **b) **527 **c) **1234 **d) **6923

**Exercise 4.4:**

**14. **An executive buys $2490 worth of presents for the children

of her employees. For each girl she gets an art kit costing $33;

each boy receives a set of tools costing $29. How many presents

of each type did she buy?

**Exercise 5.1:**

**8. **Logic chips are taken from a container, tested individually,

and labeled defective or good. The testing process is continued

until either two defective chips are found or five chips are tested

in total. Using a tree diagram, exhibit a sample space for this

process.

**Exercise 5.2:**

**2. **Does the formula *f (x) *_ 1*/(x*2 − 2*) *define a function

*f *: **R**→**R**? A function *f *: **Z**→**R**?

**12.**For *n, k *∈ **Z**+, prove that _*n/k*_ _ *(n *− 1*)/k*_ + 1.

** **

** **

** **

**Exercise 5.7:**

**1. **Use the results of Table 5.11 to determine the best “big-Oh”

form for each of the following functions *f *: **Z**+ →**R**.

**a) ***f (n) *_ 3*n *+ 7 **b) ***f (n) *_ 3 + sin*(*1*/n)*

**c) ***f (n) *_ *n*3 − 5*n*2 + 25*n *− 165

**d) ***f (n) *_ 5*n*2 + 3*n *log2 *n*

**e) ***f (n) *_ *n*2 + *(n *− 1*)*3

**f ) ***f (n) *_

*n(n *+ 1*)(n *+ 2*)*

*(n *+ 3*)*

**g) ***f (n) *_ 2 + 4 + 6 + ・ ・ ・ + 2*n*

* *

* *

**5. **The following pseudocode procedure can be used to evaluate

the polynomial 8 − 10*x *+ 7*x*2 − 2*x*3 + 3*x*4 + 12*x*5*,*

when *x *is replaced by an arbitrary (but fixed) real number *r*.

For this particular instance, *n *_ 5 and *a*0 _ 8*, a*1 _ −10*,*

*a*2 _ 7*, a*3 _ −2*, a*4 _ 3, and *a*5 _ 12.

**procedure ***PolynomialEvaluation1*

(*n*: nonnegative integer;

*r*,*a*0,*a*1,*a*2,*. . .*,*a**n*: real)

**begin**

*product *:= 1.0

*value *:= *a*0

**for ***i *:= 1 **to ***n ***do**

**begin**

*product *:= *product ** *r*

*value *:= *value *+ *a**i ** *product*

**end**

**end**

**a) **How many additions take place in the evaluation of

the given polynomial? (Do not include the *n *− 1 additions

needed to increment the loop variable *i*.) How many multiplications?

**b) **Answer the questions in part (a) for the general polynomial

*a*0 + *a*1*x *+ *a*2*x*2 + *a*3*x*3 + ・ ・ ・ + *a**n*−1*x**n*−1 + *a**n**x**n**,*

where *a*0*, a*1*, a*2*, a*3*, . . . , a**n*−1*, a**n *are real numbers and *n*

is a positive integer.

**6. **We first note how the polynomial in the previous exercise

can be written in the *nested multiplication method*:

8 + *x(*−10 + *x(*7 + *x(*−2 + *x(*3 + 12*x)))).*

Using this representation, the following pseudocode procedure

(implementing *Horner’s method*) can be used to evaluate the

given polynomial.

**procedure ***PolynomialEvaluation2*

(*n*: nonnegative integer;

*r*,*a*0,*a*1,*a*2,*. . .*,*a**n*: real)

**begin**

*value *:= *a**n*

**for ***j *:= *n *- 1 **down to **0 **do**

*value *:= *a**j *+ *r ** *value*

**end**

Answer the questions in parts (a) and (b) of Exercise 5 for the

new procedure given here.** 6. **We first note how the polynomial in the previous exercise

can be written in the *nested multiplication method*:

8 + *x(*−10 + *x(*7 + *x(*−2 + *x(*3 + 12*x)))).*

Using this representation, the following pseudocode procedure

(implementing *Horner’s method*) can be used to evaluate the

given polynomial.

**procedure ***PolynomialEvaluation2*

(*n*: nonnegative integer;

*r*,*a*0,*a*1,*a*2,*. . .*,*a**n*: real)

**begin**

*value *:= *a**n*

**for ***j *:= *n *- 1 **down to **0 **do**

*value *:= *a**j *+ *r ** *value*

**end**

Answer the questions in parts (a) and (b) of Exercise 5 for the

new procedure given here.

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