fundamental algebra

This assignment exercises your knowledge of fundamental algebra. Rusty algebra gives more problems than economics. You are hereby warned that some problems have nice, clean answers and some do not (even if calculations are carried to several decimal places). Please carry out calculations to two decimals before rounding off, as a general rule. Also, please show your work where possible.

1. Here is a linear demand function: Q = 10 – 0.5P. 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 = 10 – 0.5P:
P =

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

2. Evaluate the following TR function: TR = 20Q – 2Q2. EXAMPLE: When Q = 5, TR = $50.
a. When Q = 2, TR = $_____ b. When Q = 10, TR = $_____

3. Evaluate the following expression. Y = 3X3 -X2 +3X -6 EXAMPLE: When X = 3, Y = 75.

a. When X = 2, Y = ___ b. When X = 4, Y = ___
c. When X = 0, Y = ___ d. When X = 1, Y = ___

4. Evaluate the following exponentials. You may need to use a calculator with a Yx key. EXAMPLE: X -2. If X = 4, this gives _0.0625__. Compute to two decimal places or more.

⦁ X2, When X = 2, this gives ____.
⦁ X2, When X = -2, this gives ____.
⦁ X1/3, When X = 8, this gives ____.
⦁ X0, When X = 0, 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 = X2 +2X -8. The two roots are X = ______ and X = ______.

b. Y = 3X2 +11X -4. The two roots are X = ______ and X = ______.

6. Exponential functions are useful in business and economics. Power Point 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 0.6% 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. Your sales grew 6% last year. What average monthly growth rate is needed to achieve this 6% growth? 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 = Pert , 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. Suppose you put $6000 in a savings account paying 3% per annum for 36 years and 6 months for your retirement. What should you have in your retirement account at the end of the time?

b2. You decide you need a small nest egg of $400,000 by the time you hang it all up (retire). You have 36 years and 6 months to accumulate this amount. If you can earn 3% annually on it, what amount does the formula P = F/ert say you need to place in the savings account today in order to have the $400000 by the time you need it? EXAMPLE: 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 give you only $141.02 back at the end of the loan term , you should only loan him $100 today.