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 5Q 50 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 004Q 2 This skill will be useful in assignment 4 Show the algebra involved a Derive the price function from the demand function 5Q 50 5P P b Derive the total revenue function (TR) from the price function in (a ) TR 2 Evaluate the following TR function TR 10Q Q 2 EXAMPLE When Q 5 TR $25 a WhenQ 3, TR $ b When Q 10, TR $ 3 Evaluate the following expression Y 2X 3 (2X 5) 2 EXAMPLE When X 1, Y 51 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 2 3 , When X 3, this gives b X 0 , When X 2, this gives c X 2 3 , When X 3, this gives d X 5 3 , When X 2, 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 3X 2 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 4X 2 2X 6 The two roots are X and X b Y X 2 4X 12 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 55 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 The price of beef increased by about 7 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) in years, is given by the function 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 What amount would you expect to get back at maturity in 5 years if you invested $40000 today in a note paying 2 annually 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 Here is a loan proposition more in line with current rates A borrower agrees to pay you 5 3 4 annually for 3 years and 6 months At the end of the term he will make a balloon payment of $25000 to repay the loan and interest What amount (P) does the formula P F e rt indicate you should loan this prospect

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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

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:5Q = 50 -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 5Q = 50 - 5P:

P =

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

TR =

2.Evaluate the following TR function: TR = 10Q - Q². EXAMPLE: When Q = 5 TR= $25.

a. WhenQ = 3, TR = $_____b. When Q = 10, TR = $_____

3.Evaluate the following expression.Y = 2X³+ (2X + 5)²EXAMPLE:When X = 1, Y = 51.

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^xkey.EXAMPLE:X^-1/4.If X = 16, this gives _0.5__. Compute to two decimal places or more.

a.X^-2/3, When X = 3, this gives ____.

b.X⁰, When X = 2, this gives ____.

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

d.X^5/3, When X = 2, 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 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 = -4X²-2X +6.The two roots are X = ______ and X = ______.

b. Y = X²-4X -12. 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 0.55% 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.The price of beef increased by about 7% 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) in years, is given by the function.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.What amount would you expect to get back at maturity in 5 years if you invested $40000 today in a note paying 2% annually?

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.Here is a loan proposition more in line with current rates.A borrower agrees to pay you 5 3/4% annually for 3 years and 6 months.At the end of the term he will make a balloon payment of $25000 to repay the loan and interest.What amount (P) does the formula P = F/e^rtindicate you should loan this prospect?