Problem 9 - Calculus Concepts

Question

October Madness The scatter plot shows the number of shares traded each day during October of \(1987 .\) The behavior of the graph on October 19 and 20 has been referred to as "October Madness." a. Calculate the percentage change and average rate of change in the number of shares traded per trading day between October 1 (when 193.2 million shares were traded) and October 30,1987 (when 303.4 shares were traded). b. Draw a secant line whose slope is the average rate of change between October 1 and October \(30,1987 .\) c. Write a sentence describing how the number of shares traded changed throughout the month. How well does the average rate of change calculated in part \(a\) reflect what occurred throughout the month?

Step-by-Step Solution

Answer

The percentage change in shares traded was 57.05%, with an average increase of 3.8 million shares per day.

Step 1: Calculate the Percentage Change

To calculate the percentage change from October 1 to October 30, we use the formula: \[ \text{Percentage Change} = \left( \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \right) \times 100 \% \] Substitute the given values: \[ \text{Percentage Change} = \left( \frac{303.4 - 193.2}{193.2} \right) \times 100 = \frac{110.2}{193.2} \times 100 \approx 57.05 \% \] Thus, the percentage increase in the number of shares traded from October 1 to 30 is approximately 57.05%.

Step 2: Calculate the Average Rate of Change

The average rate of change is calculated as follows: \[ \text{Average Rate of Change} = \frac{\text{Change in Value}}{\text{Change in Time}} \]. Here, the change in the number of shares is 303.4 - 193.2 = 110.2 million shares, and the change in time is from October 1 to October 30, which is 29 days.Thus: \[ \text{Average Rate of Change} = \frac{110.2\text{ million shares}}{29\text{ days}} \approx 3.8\text{ million shares per day} \].

Step 3: Draw the Secant Line

In a graph plotting the number of shares traded against the days in October, the secant line between October 1 and October 30 represents the average rate of change. You would draw a straight line going through the point on October 1 (193.2 million) and the point on October 30 (303.4 million). The slope of this line is approximately 3.8 million shares per day.

Step 4: Describe the Share Trading Trend

Throughout October, the number of shares traded increased dramatically, with substantial increases on key days referred to as 'October Madness'. The average rate of change calculated reflects a generalized trend of increase but may not capture the spikes and drops on specific days, notably on October 19 and 20.

Key Concepts

Percentage Change Calculation

Calculating percentage change is a fundamental concept used to measure how much a quantity has increased or decreased over time. It's particularly useful in stock trading, where we often want to compare changes in trading values over different periods. The formula for percentage change is:

• \( \text{Percentage Change} = \left( \frac{\text{Final Value} - \text{Initial Value}}{\text{Initial Value}} \right) \times 100 \% \)

In the October Madness scenario, we see an increase from 193.2 million shares to 303.4 million shares. Plugging these values into our formula:

• \( \frac{303.4 - 193.2}{193.2} \times 100 \approx 57.05 \% \)

This tells us there was a 57.05% increase in the number of shares traded. This metric helps traders understand growth rates relative to the initial amounts at the start of the period.

Average Rate of Change

The average rate of change provides an overview of how a quantity changes over a specific interval. It's a useful metric for understanding trends without getting lost in the day-to-day fluctuations. For stock trading, it helps provide a clearer picture of trading activity over time. The formula is:

• \( \text{Average Rate of Change} = \frac{\text{Change in Value}}{\text{Change in Time}} \)

In our example involving October shares, this quantity is calculated by dividing the total change in shares by the number of days:

• \( \frac{110.2\text{ million shares}}{29\text{ days}} \approx 3.8\text{ million shares per day} \)

This tells us that, on average, 3.8 million more shares were traded each day from October 1 to October 30. It's a smoothed-out measure that helps in understanding the overall trend.

Secant Line

In mathematics, a secant line is a straight line that intersects a curve at two or more points. In the context of stock trading graphs, the secant line between two points provides a visual representation of the average rate of change.
To draw a secant line for October's trading activity, one would plot a graph of the number of shares traded over each day and draw a straight line connecting October 1 (193.2 million shares) and October 30 (303.4 million shares).

• The slope of this line, approximately 3.8 million shares per day, mirrors the average rate of change.

The secant line gives us a snapshot of the general movement in trading activity over the month, highlighting the changes despite the intermediate fluctuations.

Trading Volume Analysis

Trading volume analysis involves examining the quantity of shares traded in a given timeframe to deduce activity and interest in a stock. In the case of October 1987, a significant increase in trading volume illustrated a period of great market activity.
Key analysis notes include:

• A consistent increase in volume, such as the one from 193.2 to 303.4 million shares, often reflects growing interest and activity in the stock market.
• Specific days, like October 19 and 20, noted for 'October Madness', show spikes which can correlate with market events affecting investor sentiment.
• While the average rate of change suggests a general trend, it may smooth over daily spikes or drops, providing a broader perspective without losing sight of unusual events.

By looking at the volume trends, traders and analysts can make informed predictions about stock behavior and potential future movements.