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Mathematics - Algebra - Functions, Equations & Matrices

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17) The demand equation for a certain product is modeled by y = 50-√(0.01x+1) where x is the number of units demanded per day and y is the price per unit.
a) Present a graph of the function.
b) Approximate the demand if the price is $37.55.

18) Consider the function
a) Present a graph of f(x)
b) Solve f(x) = 0
c) Determine where the function is increasing.

19) The average cost per unit for a product for a certain business is given by
C(x) = (0.75x + 5000)/x
a) Present a graph of C
b) Find the average cost per unit when 2000 units are produced.
c) What is the horizontal asymptote?
d) What does the horizontal represent in the context of this problem?

20) Consider the function f(x) = (2x+9)/(4x2-3x).
a) What is the domain of this function?
b) What are the equations for all horizontal and vertical asymptotes?
c) Present a graph of the function.
d) What are the intercepts (both x and y)?

21) Consider the function f(x) = -3x3+20x2-36x+16.
a) Present a graph of the function.
b) Approximate all real roots.

22) Solve the linear system
2x+y = 6
-x+y = 0

23) A small business invests $250000 to produce an item that will sell for $9950. Each unit can be produced for $8650. How many units must be sold to break even?

24) Solve the system of linear equations
2x+ 4y+ z = 1
x- 2y -3z = 2
x+ y - z = -1

25) Given the matrix

A = find

a) A2
b) the inverse of A; that is, (A-1)

26) Given the matrices
A = B =

Find:

a) A+B
b) 4A-3B

27) Solve the system using matrices
x-2y = 1
2x-3y = -2

28) Consider the system of inequalities:

2x+3y≥6
3x-y<15
-x+y≤4
2x+5y≤27

a) Identify all corners of the solution.
b) Minimize the function g = 5x+7y

29) Determine whether each of the following sequences are arithmetic or geometric
a) 7,11,15,19,....
b) 1,5/4,3/2,7/4,...

30) Find the twelfth term of the geometric sequence

5,15,45,...

31) Find the sum of the first fifty terms of the arithmetic sequence

25,35,45,55,65,....

32) Find the sum of the infinite geometric series

∑(0.4)i , where 0≤i<∞

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The demand equation for a certain product is modeled by y = 50-√(0.01x+1) where x is the number of units demanded per day and y is the price per unit.
a) Present a graph of the function.
b) Approximate the demand if the price is $37.55.

(a) Graph:

(b) 37.55 = 50 - (0.01x + 1)
(0.01x + 1) = 50 - 37.55 = 12.45
0.01x + 1 = 12.45^2 = 155
0.01x = 154
x = 15400 units

18) Consider the function
a) Present a graph of f(x)
b) Solve f(x) = 0
c) Determine where the function is increasing.

(a) Graph:

(b) The graph intersects the x- axis at x = {-3, 1, 2}. These are the solutions of the equation f(x) = 0
(c) f(x) is increasing for {x | x < -3 or x > 2}

19) The average cost per unit for a product for a certain business is given by
C(x) = (0.75x + 5000)/x
a) Present a graph of C
b) Find the average cost per unit when 2000 units are produced.
c) What is the horizontal asymptote?
d) What does the horizontal represent in the context of this problem?

(a) Graph:

(b) C(2000) = (0.75 * 2000 + 5000)/2000 = $3.25
(c) C(x) = (0.75x + 5000)/x = 0.75 + (5000/x)
As x  ¥, C(x)  0.75, which means C(x) = 0.75 is the horizontal asymptote
(d) It means when a very very large number of units are produced, the average cost per unit approaches $0.75 and remains constant at that value

20) Consider the function ...

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Required Materials
Click here for a PowerPoint presentation on solving systems of linear equations.
After thoroughly reviewing the examples in the PowerPoint presentation, go through these two introductory tutorials below:
The Hofstra webpage has a good tutorial on systems of linear equations. You can skip portions that discuss matrices and matrix algebra, as you won't be required to know this. Their applications of linear equations section is also worth looking at.
Warner, S. & Costenoble, S.R. (2000). 2.1 Systems of Two Linear Equations in Two Unknowns. Finite Mathematics and Applied Calculus. Retrieved May 20, 2009, from: http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/tutorialsf1/fr ames2_1.html
The ThinkQuest Library has a Page on Systems of Equations that is worth going through.
Here is a good site that discusses solving systems of linear equations.
Stapel, E. (2004). Systems of linear equations. Purple Math Practical Algebra Lessons. Retrieved May 20, 2009, from: http://www.purplemath.com/modules/systlin1.htm
These following two links below are good for reviewing some of the basic formulas:
This page from S.O.S. Math on Systems of Equations in Two Variables is a good overview.
Marcus, N. (n.d.) Systems of equations. S.O.S. Mathematics Algebra. Retrieved May 20, 2009, from: http://www.sosmath.com/soe/SE/SE.html
Marcus, N. (n.d.) Systems of equations in Two Variables. S.O.S. Mathematics Algebra. Retrieved May 20, 2009, from: http://www.sosmath.com/soe/SE2001/SE2001.html
Visit the below web site review the tutorial on solving systems of linear equations in three variables. Pay particular attention to how consistent and inconsistent systems are defined.
Seward, K. (2002). College Algebra Tutorial 50: Solving Systems of Linear Equations in Three Variables. Accessed May 20, 2009, at: http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col _alg_tut50_systhree.htm.
For three equations with three variable see here
Optional Materials
Mathpower.com has a good collection of tutorials prepared by students for students for a variety of topics. So if your professor's explanations are not clear to you, maybe these tutorials will be clear to you. There are a number of tutorials on solving systems of equations at the bottom of this page.
Math.com has a lot of useful background material for those of you who wish to refresh your basic math skills such as fractions, percents, etc.
Warner, S. & Costenoble, S.R. (1998). Regression: Fitting functions to data. Calculus Applied to the Real World. Retrieved May 20, 2009, from: http://people.hofstra.edu/faculty/Stefan_Waner/calctopic1/regression.ht ml

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