Mathematics AI HL

Sample Internal Assessment

Table of content

Rationale

Aim

Background information

Data collection and analysis

Process of calculation

Conclusion

Reflection

Bibliography

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One subject which can be used to prove any practical application is mathematics. According to me, knowledge is acquired when the gap between practicality and theory is met. The IB has always maintained the balance between both practical and theories and hence, has been able to provide the proper understanding of education which can be used in our daily life. Since childhood, I have always been interested in business and subjects related to the stock market. Moreover, hailing from a family which owns a business, this knowledge has helped me to understand the concept of investment better and to aid my company. Furthermore, I am also interested in knowing about the health sector and the recent pandemic has motivated me to think about other widespread infections that have affected the health and the economic sector alike. The IB has groomed me to be an inquirer which has further led to me knowing about the outbreak of the Ebola virus in south-eastern Guinea and how it has affected the economy of the country. However, one aspect of the economy that has gained profit from this pandemic is the pharmaceutical companies. Since last month, I have spent most of my time in reading articles and researches and in watching videos that have helped me to comprehend the relation between stock prices of pharmaceutical companies and Ebola cases. One company that has been involved in helping the case of Ebola virus is the BioCryst pharmaceutical. Hence, this fascinated me to discover the relation between the increase in Ebola virus cases in south-eastern Guinea and the stock prices of BioCryst pharmaceutical by the help of calculus which is a part of our syllabus

Hence, I have selected the topic of this exploration:

*Investigation on the increase of stock price of BioCryst Pharmaceuticals Inc corresponding to the exponential increase in Ebola infection in South-eastern Guinea and hence determining a correlation between them.*

The main goal of this investigation is to discover the relationship between the increase in Ebola cases in south-eastern Guinea and the price of stocks of BioCryst Pharmaceuticals.

The outbreak of the Ebola virus in Guinea was reported on June 19, 2021. Making a vaccine for this virus has been difficult as the virus has mutated several times and hence, has negatively affected the health and economic sectors of the economy.

Stock prices refer to the current cost of shares of a particular company. These shares are traded in the market. Numerous varied factors such as variations in the industries, wars, political events, economy, and environmental changes affect the stock prices.

This exploration has been shown in two parts. Primarily, in order to find the mean and the standard deviation included in the relations of trigonometry and statistics the data had been collected which further helped in the determination in the increase of cases of Ebola over a period of time.

Next, trigonometric functions were used to analyse the increase in the stock price of Biocryst pharmaceutical in relation to the Ebola cases.

A graphical representation of variation of stock price of BioCryst Pharmaceuticals in 2014:

The graph clearly represents the change in the stock price of BioCryst pharmaceutical. It shows that the price has increased over the time period.

Sample Calculation:

Mean = \(\frac{9.10+9.11+9.17}{3}\) = 9.13

SD = \(\frac{(9.13-9.10)^2+(9.13-9.11)^2+(9.13-9.17)^2}{3}\) = 0.04

The Desmos Graphical Calculator was used to calculate the equation suitable for the best-fit sine wave:

*y* = 0.05*x*^{2 }- 0.43*x* + 9.24…(*equation *- 1)

Date in 2014

Number of fresh cases

**31 - 03**

49

**07 - 04**

159

**14 - 04**

485

**21 - 04**

1292

**28 - 04**

1670

**05 - 05**

2376

**12 - 05**

3185

**19 - 05**

4682

**26 - 05**

6983

**02 - 06**

8790

**09 - 06**

9830

**16 - 06**

10321

**23 - 06**

11049

**30 - 06**

11956

Using Desmos Graphical Calculator, the equation of the best-fit function was calculated:

*y* = 36.50*x*^{2 }+ 493.84*x* - 1147.89…(*equation *- 2)

**Rate of change of stock price of BioCryst during Ebola virus outbreak:**

Figure 2 shows the difference of stock price of BioCryst pharmaceuticals over three months which has been calculated over a time period of 14 days in 13 weeks each:

*y* = 0.05*x*^{2 }- 0.43*x* + 9.24

By differentiating both sides with respect to time (*x*), the rate of change of wind speed could be obtained:

\(\frac{dy}{dx}=\frac{d}{dx}\) [0.05*x*^{2 }- 0.43*x* + 9.24]

\(\frac{dy}{dx}=\frac{d}{dx}\) [0.05*x*^{2}] - \(\frac{d}{dx}\) [0.43*x*] + \(\frac{d}{dx}\) [9.24]

\(\frac{dy}{dx}\) = 0.10*x* - 0.43

Let, the price of stock be presented as *b*. Hence, the resulting equation would be:

\(\frac{db}{dx}\) = 0.10*x* - 0.43…(*equation *- 3)

**Rate of change of Ebola cases during the second wave:**

Figure 2 shows the difference of Ebola cases over three months, which has been calculated over a time period of 14 days in 13 weeks each, by the equation:

*y* = 36.50*x*^{2 }+ 493.84*x* - 1147.89

By differentiating both sides with respect to time (*x*), the rate of change of wind speed could be obtained:

\(\frac{dy}{dx}=\frac{d}{dx}\) [36.50*x*^{2} + 493.84*x* - 1147.89]

\(\frac{dy}{dx}=\frac{d}{dx}\) [36.50*x*^{2}] + \(\frac{d}{dx}\) [493.84*x*] + \(\frac{d}{dx}\) [1147.89]

\(\frac{dy}{dx}\) = 73*x* + 493.84…(*equation *- 4)

**Variation of Ebola cases in 2014 with respect to stock price:**

The difference in the stock price of BioCryst pharmaceuticals in regard to the difference in the cases of Ebola in south-eastern Guinea can be figured by dividing the rate of variation in the stock prices by the rate of variation in Ebola cases:

By equation (4) ÷ equation (3):

\(\frac{\frac{dc}{dx}}{\frac{db}{\frac{dx}{}}}=\frac{73x+493.84}{0.10x-0.43}\)

\(\frac{de}{db}=\frac{73x+493.84}{0.10x-0.43}\)…(equation - 5)

From *equation*(1):

*y* = 0.05*x*^{2 }- 0.43*x* + 9.24

Representing a dependent variable of a quadratic polynomial function in terms of the independent variable is not possible hence, to calculate this value, graphical methods have been used to discover the ordinate from the abscissa.

Stock Price ($)

No of week

9.13

1

8.92

2

8.33

3

8.04

4

8.46

5

7.71

6

8.34

7

9.03

8

9.94

9

10.92

10

11.34

11

11.86

12

12.54

13

12.66

14

*x* = 2.06*y* - 12.66

As stock price is expressed as *b*. Hence the modified equation would be:

*x* = 0.07*b* - 60.95

Plugging in the value of *x* in equation (5):

\(\frac{de}{db}=\frac{730.07b-60.95+493.84}{0.10 0.07b-60.95-0.43}\)

\(\frac{de}{db}=\frac{730(b-774.07)}{b-932.14}\)…(equation - 6)

To determine a relationship between number of Ebola cases and stock price of BioCryst Pharmaceuticals, the above equation (6) should be integrated with respect to b (stock price of Cipla Pharmaceuticals).

\(\frac{de}{db}=\frac{730b-774.07}{b-932.14} \)

*de = *\(\frac{730b-774.07}{b-932.14}\)db…(equation - 7)

Integrating both sides:

\(\displaystyle\int de=\displaystyle\int\frac{730b-774.07}{b-932.14}db\)

*e* = 115391(ln *ln *|*b* - 932.143|) + 0.006326(*x* + 3894.11)

*e* ≈ 115391(ln *ln *|*b* - 932.143|)

- Variation of the cases of the Ebola virus is found to be exponential in nature.
- A LogRhythmic relationship can be seen between the prices of stocks of BioCryst pharmaceuticals and the cases of Ebola.
- A negative relation can be seen in the analysis of the stock prices of Cipla pharmaceuticals and COVID cases during the second wave.
- In order to represent the variation in the stock price of BioCryst, the best fit sine-wave’s equation is
*y*= 0.05*x*^{2 }- 0.43*x*+ 9.24 - In order to represent the variation in the cases of Ebola, the best fit function is
*y*= 36.50*x*^{2 }+ 493.84*x*- 1147.89 - The equation e ≈ 115391(ln|b - 932.143| ) can be used to represent the relation between the prices of stocks of BioCryst pharmaceuticals and the cases of Ebola.

- Several varied trusted sources have been used to collect the data hence resulting in the exploration being more reliable and accurate.
- TI – Nspire CAS CX which is allowed in the curriculum of IB has been used to represent the calculation and the specific screenshot of the calculation has also been used.
- The representations of the graphs have been done through Desmos which is one of the most reliable sources and is well known today.

- Accurate number of the Ebola cases is not available on the internet thus leading to a discrepancy in the data.
- The methods taught in the IB curriculum were not sufficient to calculate the indefinite integral between Ebola cases and the stock prices of BioCryst pharmaceuticals.
- The relation between Ebola cases and stock price of BioCryst Pharmaceuticals involves several other factors hence this relation cannot be applied to a real-life scenario.

Several different factors are involved in the relation between Ebola cases and stock price of BioCryst Pharmaceuticals hence, all these factors have to be analyzed to provide a real-life significance to this exploration. A mathematical model can be curated to analyse these factors along with the variations in the Ebola cases and the change in the price of stocks and a research question can be formed as:

*Analysing different factors involved in the change in Ebola cases and stock price of a pharmaceutical company and hence creating a mathematical model to establish a relation between the variation in Ebola cases and price of stocks of a pharmaceutical company.*

Three parts would be involved in this investigation. First of all, the different factors involved in the change of the price of stocks would be curated into a mathematical model by the usage of sine curves. Then, a parallel model will be created for the Ebola cases. Finally, the resultant trendlines would be used to evaluate the relation between the two by the use of calculus.

- BioCryst Pharmaceuticals, Inc. (BCRX) Stock Price, News, Quote & History - Yahoo Finance. https://finance.yahoo.com/quote/BCRX/. Accessed 18 July 2021.
- Desmos | Beautiful, Free Math. https://www.desmos.com/. Accessed 28 Jun. 2021.
- ‘Journal of Environmental Studies and Sciences’. Springer, https://www.springer.com/journal/13412. Accessed 28 Jun. 2021.
- “News: US News, Top News in India, US Election News, Business News, Sports & International News | Times of India.” The Times of India, https://timesofindia.indiatimes.com/us. Accessed 28 June 2021.
- Boland, Michael James. “Why Big Pharma Had a Responsibility to Profit from the Pandemic.” The Conversation, http://theconversation.com/why-big-pharma-had-a-responsibility-to-profit-from-the-pandemic-160826. Accessed 8 July 2021
- Stock Price - Definition, Price Changes, How to Determine.” Corporate Finance Institute, https://corporatefinanceinstitute.com/resources/knowledge/trading-investing/stock-price/. Accessed 17 July 2021.
- 2014-2016 Ebola Outbreak in West Africa | History | Ebola (Ebola Virus Disease) | CDC. 17 Mar. 2020, https://www.cdc.gov/vhf/ebola/history/2014-2016-outbreak/index.html.
- BioCryst Pharmaceuticals, Inc. (BCRX) Stock Price, News, Quote & History - Yahoo Finance. https://finance.yahoo.com/quote/BCRX/. Accessed 18 July 2021.
- History of Ebola Virus Disease (EVD) Outbreaks <B>Error Processing SSI File</B><BR>. 17 June 2021, https://www.cdc.gov/vhf/ebola/history/chronology.html.
- ‘Stock Market LIVE - BSE, NSE & Nifty | Sensex Share Price, Indian Stock/Share Market & Stock Exchange’. Groww, https://groww.in/stocks. Accessed 18 July 2021.