# Math 107 Test 2

**READ ALL DIRECTIONS CAREFULLY:
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-Write your name, test version, and row number on the outside of your blue book.
-Write ALL work and answers in your blue book - we will NOT look for, or grade,
work on the test sheet.
-You must SHOW ALL WORK for full credit.
-When you are finished, fold your test so that it fits inside your blue book and
turn
in the test and blue book.**

1. (20 Points) The price, p, in dollars, and the quantity, x, sold of a
certain product obey the

demand equation below :

(a) Express the revenue R as a function of x.

(b) What quantity maximizes revenue?

(c) What is the maximum revenue?

(d) What price should the company charge to maximize revenue?

2. (10 Points) A farmer with 6000 meters of fencing wants to enclose a
rectangular plot split in

two that borders on a river. The farmer does not fence the side along the river.
Express the area

A of the rectangle as a function of x.

**Things we know: A = xy and P = 3x + y = 6000
So we can solve for y in terms of x using the perimeter and get y = 6000 - 3x.
Then we plug this
in to our equation for A.
A = xy = x(6000 - 3x) = 6000x - 3x^2**

3. (20 Points) Given the following polynomial

(a) What is the degree of the function?

(b) List each real zero , its multiplicity, and whether the graph crosses or
touches the x-axis at

each zero. Write it in a form similar to the chart below. (Hint: The
multiplicities add up to the degree.)

zero | multiplicity | graph behavior |

(c) Find the power function that the graph of f(x) resembles for large values of |x|.

**(a) The degree is 11.**

zero |
multiplicity |
graph behavior |

0-7 3 -3 |
12 4 4 |
crosstouch touch touch |

**(c) f(x) resembles -5x ^{11} for large values of |x|.**

4. (12 Points) Given the following rational function:

(a) What is the domain of R(x)?

(b) Find any vertical asymptotes or holes. Show work to justify your answer.

(c) Find any horizontal asymptotes. Show work to justify your answer.

**(a) Domain: All reals except x = -1 and x = -3.
Know because Red Flag! Cannot have 0 in the denominator.
(b) VA or holes:**

**So, there is a hole at x = -1 and VA at x = -3.
(c) Degree of the numerator is 3 and degree of the denominator is 2, so we do
not have a HA.**

5. (12 Points) Solve this inequality algebraically :

** Step One : done**

**Step Two: Since the function can be factored to be
** **we see that we have VAs at
x = 3 and x = -3 and a zero at x = 4.**

**Steps Three and Four: We can see that our intervals are
Thus
we pick test points in each interval and check whether the value of f at these
points are positive
or negative .**

**The answer is that
when x is in [-3, 3] or x is in **

6. (10 Points) Given the following function:

(a) List all potential rational zeros.

(b) Use synthetic division to find the quotient and remainder when f(x) is
divided by (x - 1).

Write f(x) in the form f(x) = (x - 1)*quotient + remainder.

(c) What conclusions can you make about (x-1)? What conclusion can you make
about x = 1?

(b) Using synthetic division, we find

(c) (x - 1) is a factor of f(x) and x = 1 is a root /zero/x-intercept of f(x).

7. (5 Points) What is the remainder if is divided by (x - 1)?

8. (5 Points) Write the equation (make one up) of a rational function that has a horizontal asymptote at y = 0.

**Any function where the degree on the bottom is more
than the degree on top will work.**

9. (6 Points) Given f(x) = -4x^2 + 5x - 2:

(a) Does this f(x) have a maximum or a minimum? How do you
know?

(b) Where does the maximum or minimum occur and what is its value?

**(a) f(x) has a maximum because a is negative, so the
parabola opens downward.
(b) This maximum occurs at the vertex , so it occurs when x = 5/8 and the maximum
value is f(5/8) = -.4375.**

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