Answer

MATH 107 QUIZ 4                                                                        Fall II, 2018                  Instructor: Bob Kudva               

NAME: _______________________________

I have completed this assignment myself, working independently and not consulting anyone except the instructor.

 

INSTRUCTIONS

• The quiz is worth 100 points. There are 13 problems. This quiz is open book and open notes. This means that you may refer to your textbook, notes, and online classroom materials, but you must work independently and may not consult anyone (and confirm this with your submission). You may take as much time as you wish, provided you turn in your quiz no later than Sunday, Dec 2.
• Show work/explanation where indicated. Answers without any work may earn little, if any, credit. You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also. In your document, be sure to include your name and the assertion of independence of work.
• General quiz tips and instructions for submitting work are posted in the Quizzes module.
• If you have any questions, please contact me by e-mail.

1. (4 pts) Solve the inequality x² ³ 7x and write the solution set in interval notation.

                                                                                                     (no explanation required)                      

1. [7, ¥)
2. (–¥, 0] È [7, ¥)
3. (–¥, 7] È [0, ¥)
4. [0, 7]
5. Answer: B: (–¥, 0]  È [7, ¥)

 

 

2. (4 pts) Solve £ 0 and write the solution set in interval notation.

                                                                                                      (no explanation required)                            

1. (1, 6)
2. (–¥, –3]
3. (–¥, –3] È (1, 6)
4. [–3, 1) È (6, ¥)

 

 

2. Answer: B. (–¥, –3]

 

 

3. (4 pts) For f (x) = x³ – 2x² – 7, use the Intermediate Value Theorem to determine which interval must contain a zero of f.              (no explanation required)

 

1. Between 0 and 1
2. Between 1 and 2
3. Between 2 and 3
4. Between 3 and 4

 

Answer: C: Between 2 and 3

 

 

4. (4 pts) Translate this sentence about area into a mathematical equation.

Answer

The area A of a regular pentagon is directly proportional to the square of the length s of its sides.

Direct proportionality is represented as y=kx where k is a constant. Therefore the sentence is translated as A=Ks²

But for a polygon, A=s²n/4tan(180⁰/n) where n=number of sides. Hence, A=s²*5/4tan(180⁰/5)

A=5s²/4tan(36⁰)=[5/(4tan36⁰)]s²

Therefore, for a regular pentagon, k=5/(4tan36⁰)

5. (8 pts) Look at the graph of the quadratic function and complete the table. [No explanations required.]

Graph	Fill in the blanks	Equation
	State the vertex:   _____(-1,-2)_______   State the range:   _______y≥-2______   State the interval on which the function is decreasing:   ___(-∞,-1)____	The graph represents which of the following equations?      Choice: Equation D___     A.    y  =   –x2 + 2x – 1   B.    y  =  –2x2 – 4x + 1   C.    y  =  2x2 + 4x – 1   D.    y  =   x2 + 2x – 1

 

 

6. (6 pts) Each graph below represents a polynomial function. Complete the following table.

(no explanation required)

Graph	Graph A	Graph B
Is the degree of the polynomial odd or even? (choose one)	Even Degree Polynomial	Odd Degree Polynomial
Is the leading coefficient of the polynomial positive or negative? (choose one)	Positive	Positive
How many real number zeros are there?	4	3

 

7. (12 pts) Let When factored,  

(a) State the domain.

p(x)→∞ as (x)→-∞  and p(x)→-∞ as (x)→∞

 

(b) Which sketch illustrates the end behavior of the polynomial function?

 

 

 

Answer:  __Sketch B______

 

 

 

 

 (c) State the y-intercept:

(0, -6)

 

(d) State the real zeros:

x=-1, x=3/2, and x=4

 

(e) State which graph below is the graph of P(x).

Answer: Graph B is the graph of P(x)

 

GRAPH A.  (below)                                                                      GRAPH  B.  (below)

 

GRAPH  C.  (below)                                                                     GRAPH  D.  (below)

 

8. (8 pts) Let . (no explanations required)

 

        (a)  State the y-intercept.

(0,4/9)

 

        (b) State the x-intercept(s).

(-1,0), (1,0), and (4,0)

 

        (c) State the vertical asymptote(s).

x=-3 and x=3

 

        (d) State the horizontal asymptote.

y= 4

 

 

9. (8 pts) Solve the equation. Check all proposed solutions. Show work in solving and in checking, and state your final conclusion.

 

 

Answer

Simplifying the equation gives:[x(x+1)-6]/x(x-2) =0

x²+x-6=0

Using factorial method, x=3 and x=-2 gives the requisite factors.

Substituting in the equation gives: (x+3)(x-2)=0

Therefore: X=2 or x=-3

When x=2: (2)²+2-6=0

When x=-3: (-3)^2-3-6=0

Conclusion: The function cuts through the x-axis at points -3 and 2 respectively. The x intercepts are (-3,0) and (2.0).

10. (8 pts) Which of the following functions is represented by the graph shown below? Explain your answer choice. Be sure to take the asymptotes into account in your explanation.

 

 

10. _D____

 

 

 

Answer

The graph represents function D.

from the equation, f(x)=3/(x²+4x). Simplifying gives: f(x)=3/[x(x+4)]

Thus, x=0 and x=-4

For vertical asymptotes of a function, the value of x in the denominator=0 but the numerator≠0. Thus asymptotes for the graph lie at x=0 and x=-4.

Since the solution for the function corresponds to the graphical representation, the graph manifests function D.

11. (8 pts) For z = 4  3i and w = 7  i, find z/w. That is, determine and simplify as much as possible, writing the result in the form a + bi, where a and b are real numbers. Show work.

Answer

=[(4-3i)(7+7)] / [(7-i)(7+i)]

=(28+4i-21i+3)/(49+7i-7i+1)

=(31-17i)/50

=31/50 + 17/50i

=0.62+0.34i

 

 

 

12. (8 pts) Consider the equation 5x² + 20 = 16x. Find the complex solutions (real and non-real) of the equation, and simplify as much as possible. Show work.

Answer

Rearranging the equation gives: 5x²-16x+20=0

For quadratic equations, x= [-b ± √(b²-4ac)] /2a

x= [16 ± √((-16)²-4*5*20)] /2*5

x=[16 ± √(256-400)] /10

=[16 ± √-144] /10

=[16 ± √-1√144] /10

=[16±12i] /10

Therefore, the complex solutions are x=1.6+1.2i or x= 1.6-1.2i

13. (18 pts)

The cost, in dollars, for a company to produce x items is given by C(x) = 26 + 2x for x  0, and the price-demand function, in dollars per item, is p(x) = 30  2x for 0  x  15.

 

(a) Determine the profit function P(x);

Answer

P(x)=R(x)-C(x) where R(x) is the revenue function given by price-demand function times the number of units so;d

R(x)= p(x)*x = x(30-2x) = 30x-2x²

Profit function P(x)= 30x-2x²-( 26 + 2x) = -2x²+28x-26

 

Since it is a quadratic function, its graph is a parabola. Does the parabola open up or down?

 

The parabola opens down since the coefficient of x² is negative

 

 

(b) Find the vertex of the profit function P(x) using algebra. Show algebraic work.

Answer

P(x)=y=-2x²+28x-26

but h=-b/2a= -28/(2*(-2))=7

P(-7)=k=-2*(7)²+(28*(7)-26=-98+196-26=72

The vertex (h,k)=(7, 72)

 

 

(c) State the maximum profit and the number of items which yield that maximum profit:

 

 The maximum profit is _________$ 72______  when ____7____  items are produced and sold.

 

 

 

(d) Determine the price to charge per item in order to maximize profit.

Price p(x)= 30  2x=30-(2*7)=$16

 

 

 

(e) Find and interpret the break-even points. Show algebraic work.

Answer

At break-even point total cost=total Revenue

therefore, 26+2x=30x-2x²

2x²-30x+2x+26=0

2x²-28x+26=0

x²-14x+13=0

factors: x=-13 and x=-1

(x-13)(x-1)=0

The company breaks even when it produces either x=13 units or x=1 unit for which their production cost and the revenue collected are equal.