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Mathematics - Algebra - Functions, Equations & Matrices

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 ...

Solution Summary

The expert solves algebraic functions, equations and matrices. A complete, neat and step-by-step solutions are provided in the attached file.

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