Expert: Sherman D. - 11/27/2005

Question
Hi,
could you please help me with these,
1.   Distinguish between a relation and a function. State a relation between the variables X and Y which is not a function.
   2.   Describe the constant function y = b0 by giving a business-oriented example.
   3.   A function relating annual family income X to annual purchases in dollars of new clothing by a family Y is the following. All the measurements are in dollars.

Y = - 60 + 0.06X

(a)   Give a realistic interpretation of the value of the Y intercept for this function.
(b)   Interpret the slope of the coefficient for this function.
(c)   What are expected new clothing purchases for a family with an annual income of $15,000?
(d)   What is the X intercept for this function? Interpret this point in the context of this problem.
   4.   Find the following demand function where price P(Q) is a function of the quantity demanded,

P(Q) = 600 – 0.02Q

a.   Find the total revenue function.
b.   Find the vertex of the total revenue function and state whether it is a maximum or a minimum point.
c.   Find the price at the vertex of the total revenue function.
d.   Solve the total revenue function for Q where total revenue equals zero.
e.   Briefly describe the points found in part d.
5.   Find the derivative f' (X) for each of the following functions:
(a)   f(X) = 1/(2X + 6)2
(b)   f(X) = (2X + 5)(2X – 5)
(c)   f(X) = (3X2 + 5X – 16) /(X – 2)
(d)   f(X) = - 3X3 + X2 – 10X + 18
(e)   f(X) = 27X1/3
                 
6.   For the total cost TC or total revenue TR function, find the corresponding marginal cost and marginal revenue function.

a.   TC = 0.04Q3 + 0.5Q1/2 + 1700
b.   TR = Q3 + 0.2Q2 + 6Q
                    
                 
7.   A furniture company knows the total revenue TR and total cost TC functions for dinning room tables.

  TR = - Q2 + 550Q
  TC = 0.03Q3 – Q2 – 26Q + 12000
  
       where Q designates number of dinning room tables.
  
The firm has decided to sell 75 dinning room tables. Is this a profit-maximizing quantity to offer for sale? If so, why? If not, what is the profit-maximizing quantity?
thanks in advance dear,
jesslyn

Answer
Here are the ones i can solve or understand.

1.)
A relation is a set of inputs and outputs, often written as ordered pairs (input, output).

A function is a relation in which each input has only one output.

Info found at www.sparknotes.com/math/algebra2/functions/section1.html

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2.)
by this do you mean y = b^0, if so, then no matter what "b" is, "y" will always be 1.

Sorry i can't show or give you any additional info.

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3.) A function relating annual family income X to annual purchases in dollars of new clothing by a family Y is the following. All the measurements are in dollars.

Y = - 60 + 0.06X

Assumining you do mean y = -60 + .06x, and not y = -60 + .06x^2

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y = .06x - 60

(a)
y = .06(0) - 60
y = 0 - 60
y = -60

y-intercept is -60

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(b)
.06 is the slope

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(c)
y = .06x - 60
y = .06(15000) - 60
y = 900 - 60
y = $840

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(d)
y = .06x - 60
0 = .06x - 60
60 = .06x
x = $1000

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4.) Find the following demand function where price P(Q) is a function of the quantity demanded,

P(Q) = 600 – 0.02Q

On this one do you mean 600 - .02Q^2, because if not, then you can't have a maximum, minimum, or vertex. since something in y = mx + b form will give you a straight line which has no vertex.

also would the total revenue function be -.02Q + 600

a. Find the total revenue function.
b. Find the vertex of the total revenue function and state whether it is a maximum or a minimum point.
c. Find the price at the vertex of the total revenue function.
d. Solve the total revenue function for Q where total revenue equals zero.
e. Briefly describe the points found in part d.

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If by these you mean

5.) Find the derivative f' (X) for each of the following functions:
(a)
f(X) = 1/(2X + 6)2
f(x) = (2x + 6)^-2
Using the chain rule
f'(x) = ((2x + 6)^-2)' * (2x + 6)'
f'(x) = ((2x + 6)^(-2 - 1)) * 2
f'(x) = -2(2x + 6)^(-3) * 2
f'(x) = -4(2x + 6)^(-3)
or
f'(x) = -4/((2x + 6)^3)

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(b)
f(X) = (2X + 5)(2X – 5)
f'(x) = (2x + 5)'(2x - 5) + (2x + 5)(2x - 5)'
f'(x) = 2(2x - 5) + 2(2x + 5)
f'(x) = 4x - 10 + 4x + 10
f'(x) = 8x

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(c)
f(X) = (3X^2 + 5X – 16) /(X – 2)
f'(x) = ((x-2)(3x^2 + 5x - 16)') - ((x-2)'(3x^2 + 5x - 16))/(x-2)^2
f'(x) = ((x - 2)(6x + 5)) - 1(3x^2 + 5x - 16))/(x - 2)^2
f'(x) = ((6x^2 + 5x - 12x - 10) - 3x^2 - 5x + 16)/(x - 2)^2
f'(x) = (6x^2 - 7x - 10 - 3x^2 - 5x + 16)/(x - 2)^2
f'(x) = (3x^2 - 12x + 6)/((x - 2)^2)

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(d)
f(X) = -3X3 + X2 – 10X + 18
f'(x) = (-3*3)x^(3-1) + (2*1)x^(2-1) - (10*1)x^(1-1)
f'(x) = -9x^2 + 2x - 10

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(e)
f(X) = 27X^1/3
f'(x) = (27 * (1/3))x^((1/3) - 1)
f'(x) = 9x^((1/3) - (3/3))
f'(x) = 9x^(-2/3)

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6. For the total cost TC or total revenue TR function, find the corresponding marginal cost and marginal revenue function.

a. TC = 0.04Q3 + 0.5Q1/2 + 1700
by this do you mean TC = .04Q^3 + .5Q^(1/2) + 1700

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b.)
TR = Q3 + 0.2Q2 + 6Q
TR = Q^3 + .2Q^2 + 6Q
TR = Q(Q^2 + .2Q + 6)   

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7.) A furniture company knows the total revenue TR and total cost TC functions for dinning room tables.

TR = -Q^2 + 550Q
TC = 0.03Q^3 – Q^2 – 26Q + 12000

TR = -(75)^2 + 550(75)
TR = -5625 + 41250
TR = 35625

TC = .03(75)^3 - (75)^2 - 26(75) + 12000
TC = .03(421875) - 5625 - 1950 + 12000
TC = 12656.25 - 5625 - 1950 + 12000
TC = 7031.25 - 1950 + 12000
TC = 5081.25 + 12000
TC = 17081.25

I hope you can figure out the rest.

That is as much as i can tell you, i will have to get back to you on the rest when i receive some help from someone else, and i may have to correct some of the work i have done.

P.S. Please work on your typing. instead of putting Q2, you can put it as (Q^2) so that i will know that it is a value by itself. Also when you have a negative in front of the problem, as in - 60 + .06x, since the negative is in front, you can write it as y = -60 + .06x