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Logan's logo

Extracts from this document...

Introduction

Logan’s Logo

Abstract

This internal assessment focuses on functions and areas under curves. The task at hand was to develop models of functions to best fit the characteristics/behaviours of the curve of Logan’s logo. Using an appropriate set of axes data points of the curve were measured using a ruler and then identified and recorded. 11 data points were recorded for each curve. The main objective of this IA was to determine which model, cubic or sin, would be the best fit. It was observed that the cubic function was superior. Another task was to refine the model function to fit on a t-shirt and a business card. The area between the two curves had to be calculated. In doing this assignment many pieces of technology were incorporated when doing this assignment, ranging from a pencil and a ruler to calculators and even spreadsheets and other programs.

Introduction:

A diagram of a 10cm by 10cm square is divided into three regions by two curves. The logo is the shaded region between the two curves. This investigation is aimed at answering several questions but mainly to develop mathematical functions to model the two curves represented by the two curves. Some key terms that should be understood are: sin function, cubic function, MAE (Mean Absolute Error). These terms will be elaborated on later on this assignment.

A few data points were taken from the curves, with the base of the square representing the x-axis and each unit being one centimetre, and the y-axis represented as the left side of the square with each unit being a centimetre as well. The x-axis begins at 0 and goes on till 10 centimetres; the same is applicable for the y-axis. The lower curve is denoted at f(x) and the upper curve denoted as g(x).

...read more.

Middle

10

2.27

2.2241

0.0459

Total

0.2816

Table 3: Absolute Error values for lower curve using cubic function

The MAE is calculated by adding the absolute error values together and dividing it by the number of the points chosen:

image65.png

This implies that on average the generated y value from the cubic function model misses the observed points by 0.0256 units

Upper curve

The same process used to calculate the lower curve will be used to calculate the upper curve. Recall that the function for best fit curve for a cubic function could be calculated by matrices or the cubic regression. Seeing as how the cubic regression gave a more accurate function we shall use this method and disregard the matrices method.

Recall that in order to calculate a more accurate function we must use a method that incorporates all the points observed, therefore we will use the cubic regression tool on a Ti-84 Plus Graphical Display Calculator (GDC).  Recall that o calculate this we must upload the 11 points observed, this is done by first pressing the “STAT” button followed by “1:Edit”. A table is then brought to the screen, this will be filled with the x and y values observed. The L1 table corresponds to the x values and the L2 for the y values. Once completed the table is as follows:

image67.png

Fig 3: table of points on upper curve

Using the values of L1 and L2, we can now calculate the values for a,b,c and d using the cubic regression tool of the GDC. To do this we must return to the Statistics menu on the GDC by pressing the “STAT” button once again and then scrolling to the “CALC” tab, a menu is presented then select “6:CubicReg”. We are brought back to the home screen with “CubicReg” displayed on the screen. Now we select the L1 and L2 lists with a comma in between them. The cubic regression tool gives the following values:

image68.png

Fig 4: Cubic Regression of upper curve

The cubic regression tool gives us the function:

g(x)= -0.0562x3+0.8496x2-2.5047x+3.4257

The R2 value is a measure of accuracy. The closer the value it is to 1, the more accurate the values of the parameters are. In this case, an R2 value of 0.9999 means that the graph 99.99% meets the measured points. Considering the accuracy of this function, this function will be used as a cubic function model of the lower curve. A comparative graph is listed below, illustrating the function formulated using the cubic regression tool and matrices, of which one set has a wider spread of points chosen.

image69.png

Graph 4: Cubic model for upper curve

Another way of measuring accuracy is through the Mean Absolute error (MAE) of the functions, and how much the function matches the observed points. To calculate this, the absolute value of the difference between the y observed and y expected values, where y expected is the value that was calculated by the function. The values for y-expected were found using Graphmatica. Microsoft Excel was used to calculate the absolute error. This is summarized in the table below:

x

y

observed

y

expected

Absolute

Error

0

3.45

3.44

0.01

1

1.69

1.7144

0.0244

2

1.35

1.3651

0.0151

3

2.00

2.0406

0.0406

4

3.47

3.4037

0.0663

5

5.12

5.1172

0.0028

6

6.85

6.0000

0.8500

7

8.25

6.8439

1.4061

8

8.95

8.2466

0.7034

9

8.71

9.7312

1.0212

10

7.15

7.0700

0.0800

Total

4.2199

Table 4: Absolute Error values for upper curve using cubic function

The MAE is calculated by adding the absolute error values together and dividing it by the number of the points chosen:

image70.png

This implies that on average the generated y value from the cubic function model misses the observed points by 0.3836 units

Sine Model

Another possible family of function that can be used to describe the behaviour of the curve is the sine function. The generic function:        image71.png,

Where x and y are the horizontal distance  the vertical distance from the origin, respectively. The parameter a represents the amplitude of the curve, which is the distance between a maximum (or minimum) point and the main axis. The parameter b represents the period of the curve, which is defined as the length of 1 repetition of a cycle. It should be noted that the original sine curve has a period of 2π, thus the period of a transformed sine function is, image72.png. The parameter c represents the horizontal shift of the curve, and d represents for the vertical shift of the curve.

Lower curve

In order to formulate a sine model for the lower curve, the parameters (a.b,c,d) must be calculated.

The minimum point of the lower curve was measured with a ruler, this came out to be (1.13, 0.45) and the maximum point was (7.41, 5.52). Since the amplitude is the distance between a maximum (or minimum) point and the main axis, a is calculated as follows:

image73.png

The period is the horizontal distance between two identical stationary points. However, in the diagram we are restricted by the domain and range image74.png.According to the diagram only a minimum and a maximum point can be seen. Thus, the difference between the two stationary points must be multiplied by 2 in order to obtain b. b is calculated as follows:

image75.png

Recall that c is the horizontal translation of the graph. Theimage76.png curve has its minimum point at image78.png, however the lower curve has its first minimum point at x=1.13. It should be noted that since the period has been altered, the place of the minimum points have been changed as well, therefore the difference of the original minimum and the recent minimum point has to be multiplied by the period, b.

image79.png

Recall that dis the vertical translation of the curve. An original image76.png function has its first maximum point at image80.png and the lower curve has its maximum point at y=5.52. The value of d is the difference of the lower curve’s maximum point and the original sine curve’s maximum point multiplied by the amplitude.

image81.png

Having found the values of the parameters, it is now possible to come up with a function that characterizes the lower curve using the sine model:

image82.png

Just as there is a cubic regression tool there is also a sin regression tool on the GDC. Using the sin regression tool gives us a more accurate sin function to fit the curve based on the points observed. To calculate this we must upload the 11 points observed, this is done by first pressing the “STAT” button followed by “1:Edit”. A table is then brought to the screen, this will be filled with the x and y values observed. The L1 table corresponds to the x values and the L2 for the y values. Once completed the table is as follows:

image62.png

Fig 5: table of points on lower curve

Using the values of L1 and L2, we can now calculate the values for a,b,c and d using the sin regression tool of the GDC. To do this we must return to the Statistics menu on the GDC by pressing the “STAT” button once again and then scrolling to the “CALC” tab, a menu is presented then select “C:SinReg”. We are brought back to the home screen with “SinReg” displayed on the screen. Now we select the L1 and L2 lists with a comma in between them. The sin regression tool gives the following values:

image83.png

Fig 6: Sin Regression of lower curve

The sin regression tool gives us the function:

image84.png

A comparison of the two functions is given below:

image85.pngimage08.pngimage09.pngimage06.pngimage07.png

Graph 5: Comparison of sin models for the lower curve

This graph shows that sin regression tool is not very accurate.

We shall calculate the MAE of both functions to see which is better.

Recall,  another way of measuring accuracy is through the Mean Absolute error (MAE) of the functions, and how much the function matches the observed points. To calculate this, the absolute value of the difference between the y observed and y expected values, where y expected is the value that was calculated by the function. The values for y-expected were found using Graphmatica. Microsoft Excel was used to calculate the absolute error. This is summarized in the table below:

x

y observed

y1 expected regression function

y2 expected measured function

Absolute Error Y1

Absolute Error Y2

0

0.9

1.73

0.9

0.83

0

1

0.41

0.7411

0.4554

0.3311

0.0454

2

0.73

0.3801

0.6864

0.3499

0.0436

3

1.6

0.7638

1.4809

0.8362

0.1191

4

2.7

1.7794

2.6441

0.9206

0.0559

5

3.93

3.1279

3.8908

0.8021

0.0392

6

4.91

4.4124

4.9155

0.4976

0.0055

7

5.45

5.2548

5.467

0.1952

0.017

8

5.32

5.4073

5.4101

0.0873

0.0901

9

4.31

4.8249

4.7588

0.5149

0.4488

10

2.27

3.6791

3.6726

1.4091

1.4026

Total

6.774

2.2672

...read more.

Conclusion

image37.pngcm2

To be more accurate the area between the two curves will be calculated using graphmatica.

image38.png

Fig 7: Area between 2 curves using graphmatica

According to graphmatica the area between the two curves is 9.2742 cm2. The total surface area of the business card is image39.pngcm2 . Using the area between the two curves on the business card (9.2742 cm2), thus the logo occupies image40.png part of the entire business card, which is 20.6093% of the total surface area of the business card.

It is important to know the area of the logo using in a business card because if the logo appears to small on the business card the message of the company will not get spread and resulting in lower rate of customers and a loss in profits due to the making of the business cards. On the other hand if the logo is too big on the card there is no room for text and therefore the message of the company is inhibited once again.

Conclusion

To find an appropriate model for the two  curves on Logan’s logo to showcase their behaviours, sine and cubic models were tested. A best fit curve for a cubic function was calculated using the cubic regression of the 11 points measured by ruler. It was through this calculation that the parameters of the function were discovered as well. In the end the cubic function seemed to fit better than the sin function. Logan wished to reshape her logo to fit either a t-shirt of a business card, in order to keep the nature of the curve the same, the function had to be altered so that it would be able to fit in the new dimension while still displaying the same characteristics as the original curve for the t-shirts the logo had to be doubled along with the function. When it came to printing the logo on business cards, the functions had to be altered to fit the new 9 x 5 cm dimensions

...read more.

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