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Chapter 2: Problem 1

Rewrite the sentences to express how rapidly, on average, the quantity changed over the given interval. Apple Stock Prices During a media event at which CEO Steve Jobs spoke, Apple shares opened at \(\$ 156.86\) and dropped to \(\$ 151.80\) fifty minutes into Jobs's keynote address.

Short Answer

Expert verified
The stock fell by $0.1012 per minute on average.

Step by step solution

01

Identify Initial and Final Values

First, note the initial stock price, which is $156.86, and the final stock price, which is $151.80. These represent the price at the start and end of the interval, respectively.
02

Calculate the Change in Stock Price

Subtract the final stock price from the initial stock price to determine how much the price changed: $156.86 - $151.80 = $5.06.
03

Determine the Time Interval

Recognize the time interval over which this change occurred. In this case, it is 50 minutes.
04

Calculate the Rate of Change

Divide the change in stock price by the time interval to find the average rate of change: \( \frac{5.06}{50} = 0.1012 \text{ dollars per minute} \).
05

Rewrite the Sentence

Express the rate of change in words: "During Jobs's keynote address, Apple shares decreased on average by $0.1012 per minute over the 50 minutes."

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Rate
Understanding the **average rate** of change can provide insightful information on how a quantity changes over time. It is a simple yet powerful concept where you divide the change in the variable of interest by the period over which it changed.
When looking at stocks or prices, calculating the average rate offers insight into how rapidly a stock's price rises or falls. This is done by subtracting the initial value from the final value to find the total change in price. In our exercise, the stock price fell from \(156.86\) to \(151.80\), so the change is \(156.86 - 151.80 = 5.06\). The time interval in minutes or seconds determines how quickly this occurs when calculating the average rate.
An average rate of change is then found by dividing \(5.06\) by \(50\) minutes, resulting in approximately \(0.1012\) dollars per minute, indicating that the stock price decreased by around 10 cents each minute on average.
Stock Price Change
**Stock price changes** often reflect investor reactions to news or events, and can fluctuate due to many factors like earnings reports, market trends, or news announcements. Accurate measurement of these changes is crucial for investors and analysts trying to understand market dynamics.
  • An increase or decrease in stock price conveys how the market is valuing a company at a given moment.
  • Short-term drops or rises can be indicators of investor sentiment during major announcements or events, such as a CEO's speech.
In this exercise, Apple's stock price dropped as the speech occurred, showing how rapidly investor perceptions can impact stock valuations. A change of \(5.06\) dollars represents a significant moment-to-moment shift for investors tracking this stock's performance.
Calculus Application
Exploring the **calculus application** in stock prices involves applying mathematical techniques to understand changes in continuous variables over time. Calculus helps in analyzing rates of change more comprehensively, going beyond average rates to assess instantaneous rates and analyze deeper trends in data.
In the context of stock prices:
  • The average rate of change is like finding a secant line on a graph, connecting two points over a specified interval.
  • Calculus extends this to differential calculus, which considers tangent lines or instantaneous rate changes, providing more nuanced insights.
This becomes particularly important in financial mathematics where understanding not just the average but the instantaneous rate or how stock prices could potentially behave next is key to making informed predictions and decisions.

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Most popular questions from this chapter

Drivers The table gives the percentage of licensed drivers in 2006 who are females of at a specific age. Percentage of Licensed Drivers Who Are Female $$ \begin{array}{c|c} \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Drivers } \\ \text { (percent) } \end{array} \\ \hline 16 & 0.6 \\ \hline 17 & 1.1 \\ \hline 18 & 1.4 \\ \hline 19 & 1.5 \\ \hline 20 & 1.6 \\ \hline 21 & 1.6 \end{array} $$ a. Find a quadratic model for the data. Round the coefficients in the equation to three decimal places. b. Use the algebraic method to develop the derivative formula for the rounded equation. c. Evalute the rate of change of the equation in part \(a\) when the input is 18 years of age. Interpret the result. d. Calculate the percentage rate of change in the number of female licensed drivers 18 years old. Interpret the result.

Use the limit definition of the derivative (algebraic method) to confirm the statements. The derivative of \(f(x)=2 x^{0.5}\) is \(f^{\prime}(x)=x^{-0.5}\).

Doubling Time The function \(D\) gives the time, in years, that it takes for an investment to double if interest is continuously compounded at \(r \% .\) a. What are the units on \(D^{\prime}(9)\) ? b. Why does it make sense that \(\left.\frac{d D}{d r}\right|_{r=a}\) is negative for every positive \(a\) ? c. Write a sentence of interpretation for each of the following statements: i. \(D(9)=7.7\) ii. \(D^{\prime}(5)=-2.77\) iii. \(\left.\frac{d D}{d r}\right|_{r=12}=-0.48\) iv. \(D(16)=5.79\)

Flu Shots The percentage of adults who said they got a flu shot before the winter of year \(t\) is given by $$ S(t)=-0.18 t^{2}+5.24 t+9 \text { percent } $$ where \(t\) is the number of years since 2000 , data from \(2004 \leq t \leq 2009 .\) (Source: Based on data in USA Today, p. \(1 \mathrm{~A}, 5 / 18 / 2009)\) a. Find the derivative formula using the algebraic method. b. Evaluate the derivative of \(s\) in \(2007 .\) Interpret the result.

Why is it important to understand horizontal-axis intercepts to sketch a rate- of-change graph?
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