/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Let \(P(t)\) represent the price... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 24

Let \(P(t)\) represent the price of a share of stock of a corporation at time \(t .\) What does each of the following statements tell us about the signs of the first and second derivatives of \(P(t) ?\) (a) "The price of the stock is rising faster and faster." (b) "The price of the stock is close to bottoming out."

Short Answer

Expert verified
(a) \( P'(t) > 0, P''(t) > 0 \); (b) \( P'(t) < 0, P''(t) > 0 \).

Step by step solution

01

Understanding Derivatives

The first derivative of a function, noted as \( P'(t) \), represents the rate of change of the function itself. It tells us whether the function is increasing or decreasing. The second derivative, \( P''(t) \), indicates the rate of change of the first derivative, describing whether the rate of change is increasing or decreasing.
02

Analyzing Statement (a)

The statement "The price of the stock is rising faster and faster" suggests that the stock price is increasing. This means \( P'(t) > 0 \). Since it's rising faster and faster, the rate at which the price is increasing is itself increasing, which means \( P''(t) > 0 \).
03

Analyzing Statement (b)

The statement "The price of the stock is close to bottoming out" suggests the stock is decreasing but nearing a point where it will stop decreasing and start to increase. This implies \( P'(t) \) is negative but approaching zero. Furthermore, since it is about to bottom out, \( P''(t) \) is positive, indicating an impending change from decreasing to increasing prices.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

First Derivative
In calculus, the first derivative of a function, represented as \( P'(t) \), is an essential concept. It acts as a tool to measure how a quantity like price changes over time. At its core, the first derivative tells us the rate of change at any given moment.
  • If \( P'(t) > 0 \), the function is increasing, which means the stock price is rising.
  • Conversely, if \( P'(t) < 0 \), the function is decreasing, indicating the stock price is falling.
Situations can also occur where \( P'(t) = 0 \), signaling a constant price, neither rising nor falling. Grasping this concept helps to predict and understand trends in various scenarios, like rising stock prices that do so at an accelerating pace.
Second Derivative
The second derivative, denoted as \( P''(t) \), serves a different but equally important purpose. Its role is to provide insight into the nature of the rate of change signified by the first derivative.
  • When \( P''(t) > 0 \), it means the rate at which the stock price is increasing or decreasing is itself increasing. This could mean the stock is rising faster over time.
  • \( P''(t) < 0 \) suggests the rate of increase or decrease is slowing down. For instance, if a stock is falling but at a decreasing rate, it might soon stop falling altogether.
The second derivative is crucial for understanding the acceleration or deceleration of changes. It's this powerful tool that helps predict future behavior of prices as it suggests turning points in the trend.
Rate of Change
The concept of rate of change in calculus predominantly comes into play through derivatives. It helps us understand the speed at which changes occur. These changes can relate to things like costs, speed, or as in this case, stock prices.
  • The first derivative \( P'(t) \) directly reflects the rate at which the stock price is changing at a specific time.
  • The second derivative \( P''(t) \) offers a deeper layer, indicating how the rate itself is changing.
Understanding the rate of change is invaluable in fields such as economics and sciences, facilitating insights into not just current trends, but future behavior, by examining how increases or decreases are accelerating or decelerating over time.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A company's revenue from car sales, \(C\) (in thousands of dollars), is a function of advertising expenditure, \(a,\) in thousands of dollars, so \(C=f(a).\) (a) What does the company hope is true about the sign of \(f^{\prime} ?\) (b) What does the statement \(f^{\prime}(100)=2\) mean in practical terms? How about \(f^{\prime}(100)=0.5 ?\) (c) Suppose the company plans to spend about 100,000 dollars on advertising. If \(f^{\prime}(100)=2,\) should the company spend more or less than 100,000 dollars on advertising? What if \(f^{\prime}(100)=0.5 ?\)

Chlorofluorocarbons (CFCs) were used as propellants in spray cans until their build up in the atmosphere started destroying the ozone, which protects us from ultraviolet rays. since the 1987 Montreal Protocol (an agreement to curb CFCs), the CFCs in the atmosphere above the US have been reduced from a high of 3200 parts per trillion (ppt) in 1994 to 2750 ppt in \(2010 .^{15}\) The reduction has been approximately linear. Let \(C(t)\) be the concentration of CFCs in ppt in year \(t\) (a) Find \(C(1994)\) and \(C(2010)\) (b) Estimate \(C^{\prime}(1994)\) and \(C^{\prime}(2010)\) (c) Assuming \(C(t)\) is linear, find a formula for \(C(t)\) (d) When is \(C(t)\) expected to reach 1850 ppt, the level before CFCs were introduced? (e) If you were told that in the future, \(C(t)\) would not be exactly linear, and that \(C^{\prime \prime}(t)>0,\) would your answer to part (d) be too early or too late?

Assume \(g(v)\) is the fuel efficiency, in miles per gallon, of a car going at a speed of \(v\) miles per hour. What are the units of \(g^{\prime}(v)=\frac{d g}{d v} ?\) There may be more than one option. (a) (miles) \(^{2} /(\text { gal })(\text { hour })\) (b) hour/gal (c) gal/hour (d) (gal)(hour)/(miles) \(^{2}\) (e) (miles/gallon)/(miles/hour)

Are the statements true or false? If a statement is true, give an example illustrating it. If a statement is false, give a counterexample. If a function is continuous, then it is differentiable.

Are the statements true or false? Give an explanation for your answer. If \(f(a) \neq g(a),\) then \(f^{\prime}(a) \neq g^{\prime}(a).\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.