The document contains a series of math word problems and questions. It asks the reader to: 1) Solve various math equations and systems of equations, showing the work. 2) Write mathematical statements and prove they are true for different values of n. 3) Graph functions and find limits. 4) Solve optimization problems to maximize profits or minimize costs given certain constraints.

1. Name: _________________________
Score: ______ / ______
1.
Find the indicated sum. Show your work.
k = 1, (-1)^k (k + 11) = (-1)^(1) (1 + 11)= -1*(12) = -12
k = 2, (-1)^k (k + 11) = (-1)^(2) (2 + 11)= 1*(13) = 13
k = 3, (-1)^k (k + 11) = (-1)^(3) (3 + 11)= -1*(14) = -14
k = 4, (-1)^k (k + 11) = (-1)^(4) (4 + 11)= 1*(15) = 15
(-12)+(13)+(-14)+(15)=2
2.
Locate the foci of the ellipse. Show your work.
X^2=(x-h)^2, then h=0
Y^2=(x-k)^2, then k=0
The centre is (0,0)
X^2/36+y^2/11=1
When x=0 y^2/11=1; y=0
When y=0,x=0

2. X^2/36=1;x=0
11+c^2=36
C=5
Foci (5,0) and (-5,0)
3.
Solve the system by the substitution method. Show your work.
2y - x = 5
x2 + y2 - 25 = 0
x:
2y - x = 5
2y - 5 = x
so x = 2y - 5
-Plug this into 2nd equation:
(2y - 5)² + y² - 25 = 0

3. -Use FOIL to solve the (2y - 5)² part:
(2y - 5)(2y - 5)
4y² - 10y - 10y + 25
4y² - 20y + 25
So :
4y² - 20y + 25 + y² - 25 = 0
Which can be simplified to:
4y² + y² - 20y = 0
4y² + y² - 20y = 0
y(4y + y - 20) = 0
So, because of the 0 multiplication rule,
y=0
x= -5 (plug in y=0 to original equations:
2y - x = 5
2(0) - x = 5, so x= -5)
(-5,0)
Y(4y+y-20)=0
So, y=0 or 4y+y-20=0
5y-20=0
Y=4
X=2y-5 when y=4
X=8-5=3
(-5,0) (3,4)

4. 4.
Graph the function. Then use your graph to find the indicated
limit. You do not have to provide the graph
f(x) = 5x - 3,
f(x)
22
5.
Use Gaussian elimination to find the complete solution to the
system of equations, or state that none exists. Show your work.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
6.

5. Solve the system of equations using matrices. Use Gaussian
elimination with back-substitution.
x + y + z = -5
x - y + 3z = -1
4x + y + z = -2
X=1/3, y=-(11/3),z=-(5/3)
7. A woman works out by running and swimming. When she
runs, she burns 7 calories per minute. When she swims, she
burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Write an inequality that describes the
situation. Let x represent the number of minutes running and y
the number of minutes swimming. Because x and y must be
positive, limit the boarders to quadrant I only.
7x+8y>=336
Short Answer Questions:
Type your answer below each question. Show your work.
8.
A statement S
n

6. about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
Show your work.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
S1=1(6*1^2-3(1)-1)/2=1
S2=1^2+4^2=17
S31^2+4^2+7^2=66

7. 9.
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely. Show your work.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3

8. 10.
Joely's Tea Shop, a store that specializes in tea blends, has
available 45 pounds of A grade tea and 70 pounds of B grade
tea. These will be blended into 1 pound packages as follows: A
breakfast blend that contains one third of a pound of A grade
tea and two thirds of a pound of B grade tea and an afternoon
tea that contains one half pound of A grade tea and one half
pound of B grade tea. If Joely makes a profit of $1.50 on each
pound of the breakfast blend and $2.00 profit on each pound of
the afternoon blend, how many pounds of each blend should she
make to maximize profits? What is the maximum profit?
Grade A = 45 lbs
Grade B = 70 lbs
Total weight = A+B = 45+70 = 115
Blend Br = 1/3A + 2/3B
Blend Af = 1/2A + 1/2B

9. Profit = 1.50*lbs Blend Br + 2.00*lbs Blend Af
Profit = 1.50*(1/3A + 2/3B)+2,00(1/2A + 1/2B) =
45 pounds of A and 70 pounds of B yields max 24A + 72B = 98
Pounds Max Br
" " " " max 38A + 76B = 114 Pounds Max Af
The rate of profit for A is .50 per pound in BR and 1.00 per
Pound in AF
For B, rate of profit for B is 1.00 for Br and 1.00 for Af.
Let X = # pounds in A
Let Y = # pounds in B
115 = A + B
A = 115-B

10. Let q = percent of Br
Let r = percent of Af
Let s = pounds of A in Br then 45-s = # pounds A in Af
Let t = pounds of B in Br and 70-t= # pounds in Af
.5s + 1.00*(45-s) + 1.00(t) + 1.00(70-t) = p
1.5s + 2.00t = p
If s = 45 pounds then 70 = 70/45 = 14/9s pounds
.5s + 1(45-s) + 1(14/9s) + 1(70-14/9s) = p
.5*45s + (45(1 - s)) + 14/9*45s + (45(1 - 14/9s) =p

11. 22.5s + 45 - 45s + 70s + 35 - 70s = p
22.5s +70 - 115s = p
p = 137.5s + 70
p' = 137.5
22.5 A and 45 B for Br and 23.75 A and 23.75 B for Af.
Max profit is $137.5
11
Your computer supply store sells two types of laser printers.
The first type, A, has a cost of $86 and you make a $45 profit
on each one. The second type, B, has a cost of $130 and you
make a $35 profit on each one. You expect to sell at least 100
laser printers this month and you need to make at least $3850
profit on them. How many of what type of printer should you

12. order if you want to minimize your cost?
Let x represent the number of A printers
Let y represent the number of B printers
Minimize cost = 86x + 130y
subject to
Total printers equn: x + y ≥ 100
Total profit equn: 45x + 35y ≥ 3850
x ≥ 0, y ≥ 0
x and y must be whole numbers.
The vertices of the feasible region are: (0, 100), (100, 0) and
(35, 65)
If x = 35 and y = 65 the cost is 11460 and profit is 3850
if x = 100 and y = 0 the cost is 8600 and profit is 4500

13. If x = 0 and y = 100 the cost is 13000 and profit is 3500
The best result is x = 100 and y = 0
12
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
Show your work.
S
n
: 2 + 5 + 8 + . . . + ( 3
n
- 1) =
n
(1 + 3
n
)/2

14. 13
Use mathematical induction to prove that the statement is true
for every positive integer n. Show your work.
2 is a factor of n
2
- n + 2
14
A statement S
n
about the positive integers is given. Write statements S
1
, S

15. 2
, and S
3
, and show that each of these statements is true.
Show your work.
S
n
: 2 is a factor of n
2
+ 7n
15
(i.)
f(x)
(ii.)
f(x)
(iii.) What can you conclude about
f(x)? How is this shown by the graph?
(iv.) What aspect of costs of renting a car causes the graph to
jump vertically by the same amount at its discontinuities?
16
Use mathematical induction to prove that the statement is true
for every positive integer n.

16. 8 + 16 + 24 + . . . + 8n = 4n(n + 1)
8(1)=4(1)(1+1)
8=8
1
st
condition is true
8+16+24+…+8k+8(k+1)=4k(k+1)+8(k+1)…1
8+16+24+…8k=4k(k+1)
=4k(k+1)+8(k+1)
=4k^2+4k+8k+8
=4k^2+12k+8
=4(k^2+3k+2
=4(k+1)((k+1)+1)
17
The following piecewise function gives the tax owed, T(x), by a
single taxpayer on a taxable income of x dollars.
T(x) =
(i)
Determine whether T is continuous at 6061. Yes the function is
continuous at 6061
(ii) Determine whether T is continuous at 32,473. T is
continuous at 32473
(iii) If T had discontinuities, use one of these discontinuities to
describe a situation where it might be advantageous to earn less
money in taxable income. T is continuous at every point it has
not discontinuities, so earning less money for saving tax isn’t
good idea

17. 18
A statement S
n
about the positive integers is given. Write statements S
k
and S
k+1
, simplifying S
k+1
completely.
S
n
: 1 + 4 + 7 + . . . + (3
n
- 2) =
n
(3

18. n
- 1)/2
19
An artist is creating a mosaic that cannot be larger than the
space allotted which is 4 feet tall and 6 feet wide. The mosaic
must be at least 3 feet tall and 5 feet wide. The tiles in the
mosaic have words written on them and the artist wants the
words to all be horizontal in the final mosaic. The word tiles
come in two sizes: The smaller tiles are 4 inches tall and 4
inches wide, while the large tiles are 6 inches tall and 12 inches
wide. If the small tiles cost $3.50 each and the larger tiles cost
$4.50 each, how many of each should be used to minimize the
cost? What is the minimum cost?
0 small tile, 6 large tiles, and a minimum cost of 27$
20
The Fiedler family has up to $130,000 to invest. They decide
that they want to have at least $40,000 invested in stable bonds
yielding 5.5% and that no more than $60,000 should be invested
in more volatile bonds yielding 11%. How much should they
invest in each type of bond to maximize income if the amount in
the stable bond should not exceed the amount in the more
volatile bond? What is the maximum income?

19. X= amount of money invested in the stable bonds at 5.5%
Y= amount of money invested in the volatile bonds at 11%
Up to 130,000 invested which means the investment can not be
larger than 130000
X+y<=130000
Stable bonds= 130,000-60,000=70,000
At least 40000 invested in stable bonds yielding 5.5%
x>=40000
no more than 60,000 should be invested in more volatile bonds
yielding 11%
y<=60000
60000 in the stable bonds and 60000 in volatile bonds
Maximum income $9900