/*! 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 15 For Activities 7 through \(18,\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 15

For Activities 7 through \(18,\) write the first and second derivatives of the function. \(f(x)=3.2 \ln x+7.1\)

Short Answer

Expert verified
The first derivative is \( f'(x) = \frac{3.2}{x} \) and the second derivative is \( f''(x) = -\frac{3.2}{x^2} \).

Step by step solution

01

Identify the Function and Differentiate

The given function is the logarithmic function \[ f(x) = 3.2 \ln x + 7.1 \].To find the first derivative, we'll apply the derivative rule for the natural logarithm, \( (\ln x)' = \frac{1}{x} \).
02

First Derivative

Applying the derivative:\[f'(x) = 3.2 \cdot \frac{d}{dx}(\ln x) = 3.2 \cdot \frac{1}{x} \]So, the first derivative is:\[f'(x) = \frac{3.2}{x}\]
03

Differentiate Again for Second Derivative

Now we need to find the second derivative by differentiating the first derivative:\[f'(x) = \frac{3.2}{x}\]The derivative \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).
04

Second Derivative

Applying the derivative rule for \( \frac{1}{x} \):\[f''(x) = 3.2 \cdot -\frac{1}{x^2} = -\frac{3.2}{x^2}\]So, the second derivative is:\[f''(x) = -\frac{3.2}{x^2} \]

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.

Logarithmic Function
A logarithmic function is a mathematical expression where a constant base is raised to a power. The natural logarithm is represented by \( \ln x \), where the base is the irrational number \( e \), approximately equal to 2.71828. Logs are useful in various fields such as science, economics, and engineering.
The principles of logarithms rely on rules like:
  • \( \ln(a \cdot b) = \ln a + \ln b \)
  • \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \)
  • \( \ln(a^b) = b \ln a \)
When we differentiate a natural logarithmic function, often these rules help us simplify complex expressions. Differentiating \( \ln x \) directly results in \( \frac{1}{x} \), a foundation for other more complex derivatives. In the provided problem, the logarithm helps us identify the approach to differentiation.
First Derivative
Finding the first derivative refers to calculating the rate at which a function changes. In other words, it tells us how steeply the function is increasing or decreasing. For a function involving a natural logarithm like \( f(x) = 3.2 \ln x + 7.1 \), this involves simple rules of calculus.
To find the first derivative, apply the power rule and the derivative of \( \ln x \). The derivative for \( \ln x \) is \( \frac{1}{x} \). By multiplying with the constant 3.2 in the original function, we achieve \( f'(x) = \frac{3.2}{x} \).
  • This tells us that the rate of change depends on \( x \), being inversely proportional.
  • As \( x \) increases, \( f'(x) \) decreases.
  • This helps us identify the behavior of the logarithmic curve and make further conclusions on its intervals of increase or decrease.
The first derivative is crucial in understanding the dynamic nature of the original function.
Second Derivative
A second derivative represents the rate of change of the first derivative, or in simpler terms, the acceleration of the function. This can inform us how the "rate at which a function is changing" itself changes.
In the example, once we have \( f'(x) = \frac{3.2}{x} \), we differentiate again. The derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \), which results in \( f''(x) = -\frac{3.2}{x^2} \) after applying the constant multiple rule.
  • The negative sign indicates that the concavity of the graph of the original function is downwards.
  • It suggests that as \( x \) increases, the rate at which the slope of the function decreases.
  • This means the function flattens out more as \( x \) grows.
The second derivative is valuable for identifying points of inflection, concavity shifts, and more in the study of a function鈥檚 graph.

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

Discuss the options available for finding the relative extrema of a function.

Sketch the graph of a function \(f\) such that all of the following statements are true. \- \(\quad f^{\prime}(x)<0\) for \(x<-1\) \- \(\quad f^{\prime}(x)>0\) for \(x>-1\) \(f^{\prime}(-1)\) does not exist.

A software developer is planning the launch of a new program. The current version of the program could be sold for 100 . Delaying the release will allow the developers to package add-ons with the program that will increase the program's utility and, consequently, its selling price by 2 for each day of delay. On the other hand, if they delay the release, they will lose market share to their competitors. The company could sell 400,000 copies now but for each day they delay release, they will sell 2,300 fewer copies. a. If \(t\) is the number of days the company delays the release, write a model for \(P\), the price charged for the product. b. If \(t\) is the number of days the company will delay the release, write a model for \(Q,\) the number of copies they will sell. c. If \(t\) is the number of days the company will delay the release, write a model for \(R\), the revenue generated from the sale of the product. d. How many days should the company delay the release to maximize revenue? What is the maximum possible revenue?

For Activities 5 through \(10,\) given the units of measure for production and the units of measure for cost or revenue a. Write the units of measure for the indicated marginal. b. Write a sentence interpreting the marginal as an increase. Revenue is given by \(R(q)\) thousand dollars when \(q\) hundred units are sold; \(R^{\prime}(16)=0.15 .\)

Polo Ralph Lauren Revenue The revenue (in million dollars) of the Polo Ralph Lauren Corporation from 2002 through 2009 is given in the table. a. Use the data to estimate the year in which revenue was growing most rapidly. b. Find a model for the data. c. Give the first derivative for the model in part \(b\) with units. d. Determine the year in which revenue was growing most rapidly. Find rate of change of revenue in that year. $$ \begin{aligned} &\text { Polo Ralph Lauren Corporation Annual Revenue }\\\ &\begin{array}{|c|c|} \hline \text { Year } & \begin{array}{c} \text { Revenue } \\ \text { (billion dollars) } \end{array} \\ \hline 2002 & 2.36 \\ \hline 2003 & 2.44 \\ \hline 2004 & 2.65 \\ \hline 2005 & 3.31 \\ \hline 2006 & 3.75 \\ \hline 2007 & 4.30 \\ \hline 2008 & 4.88 \\ \hline 2009 & 5.02 \\ \hline \end{array} \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.