Here is a linear demand function:Q = 12 - .5PFind its price function by inverting the demand function.Then find its total revenue function by multiplying through by Q.EXAMPLE:The linear demand function Q = 400 -250P inverts into the price function P = 1.6 -0.004Q.Multiplying this by Q gives its total revenue function TR = 1.6Q -0.004Q².This skill will be useful in assignment 4.Show the algebra involved.

a.Derive the price function from the demand function Q = 12 - .5P:

P =

b.Derive the total revenue function from your price function in (a.):

TR =

2.Evaluate the following TR function: TR = 24Q - 2Q².EXAMPLE: When Q = 6, TR = $72.

a.When Q = 4, TR = $_____b.When Q = 12, TR = $_____

3.Evaluate the following expression.Y = 2X² + 5X(X + 6)²EXAMPLE:When X = 1, Y = 247.

a. When X = 0, Y = ___b.When X = 2, Y = ___

c. When X = 3, Y = ___d.When X = 5, Y = ___

4.Evaluate the following exponentials.You may need to use a calculator with a Y^x key.EXAMPLE:X ^-1/4.If X = 16, this gives _0.5__. Compute to two decimal places or more.

a.X^-1/4, When X = 2, this gives ____.

b.X^-1, When X = 5, this gives ____.

c.X⁰, When X = 5, this gives ____.

d.X^2/3, When X = 7, this gives ____.

5.Find the two roots of each of the following quadratic functions (that is find the two X values that make Y = 0).This skill may be useful in assignment 11.EXAMPLE:Y = 3X² -11X +6 = (3X -2)(X -3).If you let Y = 3X-2 then X = 2/3 will make Y = 0.If you let Y = X-3 then X = 3 will make Y = 0.Thus both X = 2/3 and X = 3 are roots. Show the algebra involved.

a. Y = 4X² +5X -6.The two roots are X = ______ and X = ______.

b.Y = 300 +5X -X² . The two roots are X = ______ and X = ______.

6.Exponential functions are useful in business and economics.Lesson 7 discusses them.Show how the values are entered into your functions and also calculate the amounts of each of the following:

a1. You learn on the business channel that inflation was about 0.5% last month.Assume this rate is maintained each month for a year.What will be the annualized rate be?EXAMPLE:A rate of 0.1% per month represents (1 + 0.001)¹² -1 = 0.0121 or 1.21% annually.

a2.In order for same store sales to contribute satisfactorily to earnings, same store sales need to grow about 5 1/2% per annum. What monthly growth rate is implied?EXAMPLE: A 2% growth rate for the year would require 1.02 = (1 + r)¹².Solve this for r:; r = .00165 or .165% per month on average.

b1. F = Pe^rt , which assumes continuous compounding, says that the Future value (F) of an amount (P) invested today at an annual rate (r), expressed as a decimal for the time (t), in years is given by the function.Thus if you invested $100 at the annual rate of 5 1/2% for 6 years and 3 months you would get back (at the end of the time), F = $100e^{(0.055)(6.25)} = $100e^(0.3438) = $100(1.4102) = $141.02.If you invest $15000 today, what amount does the formula say you will get back if you leave it for 5 years and 3 months in a savings account paying 4 1/2% annually?

b2.Alternatively, if a borrower tells you that heneeds a loan for 6 years and 3 months and will pay you an annual rate of 5 1/2% for the loan, but will giveyou only $141.02 back at the end of the loan term , you should only loan him $100 today.The sports coup auto of your dreams will cost you about $45500.What amount do you need to invest today, P = F/e^rt , such that you willhave $45500 in 12 years if you can earn 4 3/4% per annum on your money?