1. Here is a linear demand function: Q = 25 - 5P. Find 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 = 25 - 5P:

P =

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

TR =

2. Evaluate the following TR function TR = 5Q - 0.2 Q². EXAMPLE: When Q =12.5,TR = $31.25.

a. When Q = 6, TR = $_____ b. When Q = 25, TR = $_____

3. Evaluate the following expression. Y = 3X³ + 2(X + 2)² EXAMPLE:When X = 1, Y = 21.

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.

1. X^-1/4, When X = 3, this gives ____.
2. X⁰, When X = 2, this gives ____.
3. X^1/2, When X = 3, this gives ____.
4. X^5/3, When X = 2, this gives ____.

5. Find the two roots of each of the following 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 is the product of (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 -14X -6. The two roots are X = ______ and X = ______.

b. Y = 1200 +10X -X². The two roots are X = ______ and X = ______.

6. Exponential functions are useful in business and economics.PowerPoint 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 0.65% last month. Assume this rate is maintained each month for a year. What will 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. Population in Durant increased 10% last year. What average 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, expressed in years) is given by the function. e is usually a key on a scientific calculator or it may be considered a constant approximately equal to 2.7183. EXAMPLE: invest $100 at the annual rate of 5 1/2% for 6 years and 3 months and you should get back (at the end of the time), F = $100e^{(0.055)(6.25)} = $100e^(0.3438)= $100(1.4102) = $141.02. YOUR PROBLEM: Some auto dealerships are advertising, "78 months to pay, no interest, no payments." However, on the maturity date you have to pay the whole amount owed. Moreover, if you are late, you must also pay interest computed from the beginning of the loan. You want a $50000 auto for which, after down payment, you qualify for a $40000 loan as advertised. What total amount will you owe when the loan becomes due if you are late (interest will be computed at 7% per annum)?

b2. Alternatively, if a borrower tells you that he needs a loan for 6 years and 3 months and will pay you an annual rate of 5 1/2% for the loan, but will only give you $141.02 back at the end of the loan term, you should only loan him $100 today YOUR PROBLEM: A borrower agrees to pay you 5% annually for 4 years. At the end of the term, he agrees to make a balloon payment of $10000 to repay the loan and interest. What amount (P) does the formula P = F/e^rt indicate you should loan this prospect?