# MATHEMATICS FOR BUSINESS

This assignment comprises 10% of the assessment for ETF2700 and 10% for ETF5970.

You must submit a **printed** “hard copy” of your written work – with an Assignment Cover Sheet (from the “Assignments” tab on Moodle). Submit it in your tutorial before the due time, or submit it to your tutor’s mailbox, 5^{th} floor H Block, by the due time.

**ENSURE that you**

- submit a printed “hard copy” of your assignment to your tutor.
- Name your assignment:
*Surname-Initials_A3.docx*, eg,*Rowling-JK_A3.docx*. - Upload this file to Moodle (to prove that it was submitted, and that it was on time).

Go to the “ASSIGNMENTS” section. Click on the “ASSIGNMENT 3” link to upload.

The following message will appear **momentarily**, “*File uploaded successfully*.”

[To later confirm your upload was successful, go to the “ASSIGNMENTS” section and click

on the “*Assignment 3*” uploading link. The uploaded file’s name willbe shown.]

**NB**, DO NOT submit any *Excel* files. You may upload only **ONE** file.

* Retain your marked assignment until after the publication of final results for this unit.

- A maximum penalty of 20% of the total mark allocated to this assessment will be deducted for each working day that it is late.
- Extensions beyond the due date will only be allowed in special circumstances. Contact your tutor.
**Do not submit your assignment in a folder – stapled pages are easier for the marker.**- Save trees
*!*Double-sided printing is encouraged. - Retain your marked assignment work until after the publication of final results for this unit.
- Where you are asked to use Excel, you may use equivalent software – IF it can perform the task.

**Plagiarism**

*Intentional plagiarism amounts to cheating in terms of Monash University **Statute 4.1 – Discipline**.*

**Plagiarism**: Plagiarism means to take and use another person’s ideas and or manner of expressing them and to pass these off as one’s own by failing to give appropriate acknowledgement. This includes material from any source, staff, students or the Internet – published and unpublished works.

**Collusion**: Collusion is unauthorised collaboration with another person or persons.

Where there are reasonable grounds for believing that intentional plagiarism or collusion has occurred, this

will be reported to the Chief Examiner, who may disallow the work concerned by prohibiting assessment or

refer the matter to the Faculty Manager.

** **

**Question 1****SHOW YOUR WORKING**.

For the demand function P = 20 – 1/Q,

(a) obtain the first and second derivatives.

(b) describe the curvature for *Q*> 0.

(c) determine the turning point(s) and/or point(s) of inflection.

Explain whether any turning point(s) are maximum or minimum.

(d) wheredoes its graphintercept the horizontal axis?

** **

**Question2****SHOW YOUR WORKING**.

The output from a pottery is related to the number of labour units employed by the production function

Q = 9L^{2} – 0.1L^{3}.

- How many labour units must be employed to maximise Q?
- (i) Is there a point of inflection? If there is, then find it. If there is not, then explain why.

(ii) What information about the current problem is provided by this value?

- Derive the equation for

(i) the marginal product of labour (ii) the average product of labour.

- Determine the number of labour units which maximises
- MP
_{L}(ii) AP_{L}. - Show (WITHOUT graphing) that the MP
_{L}and AP_{L}curves intersect at the maximum point on the AP_{L} - (i) Graph the production function.

(ii) Graph MP_{L} and AP_{L} together on another set of axes.

(iii) Interpret MP_{L} and AP_{L}.

(iv) Discuss the relationship (if any) between the turning point(s) of the MP_{L} and AP_{L} graphs and the

point of inflection of the production function.

**Question 3****SHOW YOUR WORKING**.

The demand function for good A is Q_{A} = 200 – 8P_{A} + 5P_{B} + 0.5Y. At present, the price of good A (P_{A}) is $40, the price of good B (P_{B}) is $10, and the consumer’s income (Y) is $500.

- Find the partial price elasticity of demand for good A.
- Interpret your result from (a).
- Find the partial income elasticity of demand of good A if the competitor has now doubled the price of good B.
- Interpret your result from (c).
- Find the partial cross-price elasticity of demand for good A if the consumer’s income has now increased by 4% and the price of good B is from part (c).
- Interpret your result from (e).

** **

**Question 4SHOW YOUR WORKING**.

The total cost for a firm consists of the cost of capital inputs (P_{K}, K) and the cost of labour inputs (P_{L}, L). Here, P_{K }and P_{L }are the costs of a unit of capital and labour, respectively, and K and L are the number of units of inputs of capital and labour.

The total cost is given by C = P_{L}L+P_{K}K, where input costs are P_{L} = 3andP_{K} = 5.

The relation between units of output Q and inputs K and L, is the Cobb-Douglas production function

Q = AL^{α}K^{β}, where A = 10, α = 0.75 and β = 0.25.

The firm wishes to minimise the cost of producing 100 units.

- (i) Write down the function which is to be minimised.

(ii) Write down the constraint.

- Write down the Lagrangian function for this problem.
- Calculate the capital and labour inputs, and the Lagrange multiplier, at optimal values.
- What is the minimum cost of producing the 100 units?
- Interpret the value of the Lagrange multiplier for this problem.

**Question 5****SHOW YOUR WORKING**.

Bradley, p472: Test Exercises 8.

Question 1(b): y = 4 – x^{2} ONLY.

**Question 6****SHOW YOUR WORKING**.

Bradley, p473: Test Exercises 8, question 2.

**Question 7****SHOW YOUR WORKING**.

Bradley, p473: Test Exercises 8, question 5(b).

**Question 8****SHOW YOUR WORKING**.

The rate of change of revenue (R) from the showing of a newly released film is given by the equation:

, where *t* is the number of days since the film was released.

(a) Graph the rate of change of revenue.

(b) Calculate the total revenue collected

(i) from the first seven days of screenings.

(ii) from the second seven days of screenings.

*Excel***Tips**

**If you need help with Excel, **

- use Excel’s Help/Microsoft Excel Help menu item.
- see the “EXCEL” icon on the Moodle site for some Help files.

**To display the formulas which underlie your spreadsheet calculations**

While pressing the Ctrl key, press the key. (You may now need to widen some columns of the spreadsheet in order to fully display the revealed formulas.)

Repeat these key strokes to return to the usual display.

**To alter the page margins,in order to optimise the use of a page before printing it**

Click on the *Office Button *and select *Print/Print Preview/Preview/ShowMargins. * Then,click on and drag the margins.

**To print in “Landscape” view,rather than in “Portrait” ** **view**

Click on *Page Layout *(at the top of the screen)*/Orientation/Landscape*.

**To print the gridlines which border the spreadsheet cells**

- Select Click on
*Page Layout*(at the top of the screen)*/Sheet Options/Gridlines/Print*

**To print the Row (“1”, “2”, “3”, etc) and Column (“A”, “B”, “C”, etc) headings **

- Before printing, select Click on
*Page Layout*(at the top of the screen)*/Sheet Options/Headings/Print*

**To Copy part of your spreadsheet, eg to Paste it into a Word document**

- Select the desired cells in Excel.
- Click on the
*Copy* - Go to the Word document and select the triangle under the
*Paste*command and select

*Paste Special/Picture(Enhanced Metafile).*

An alternative to the procedures above is to press the “Print Scrn” key when your spreadsheet is displayed. This captures an image of your screen, which may then be *Paste*d into a Word document. Use Word’s “Crop” tool to reduce the *Paste*d image to your desired selection.

If you are unable to make Excel display the row and column headings clearly on your printout, then you **MUST** write them on by hand.

* *

*Word***Tips**

**To capture a screen image & Paste it into Word**

To capture the image (any image, eg, an Excel page, an internet page, a photo, etc) currently displayed on your screen,

- press the
*Print Scrn*button on your keyboard. - go to the required location in your Word document. Hold the
*Ctrl*key and press the letter “V”. - to trim (“Crop”) this image, select View/Toolbars/Picture and click on the “Crop” icon:

Then, apply the “Crop” icon to the small square “handles” of the image.

The “Snipping Tool” (available via START button/Programs/Accessories) is very useful too.

** **

**To arrange graphs and pictures within a “Word” document **

- Graphs and pictures can be more manageable if you insert a table into
*Word*and place your graphs and pictures (and text) into cells of the table.