Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 47
Use the properties of integrals to verify that $$0\leqslant \int_{1}^{3} \ln x d x \leqslant 2 \ln 3$$ without evaluating the integral.
Short Answer
Expert verified
The inequality holds: \( 0 \leq \int_{1}^{3} \ln x \, dx \leq 2 \ln 3 \).
Step by step solution
01
Understand the Integral Properties
We need to verify the given inequality without solving the integral. The properties of integrals that can be useful are: (1) if a function \( f(x) \) is non-negative on the interval \([a, b]\), then \( \int_{a}^{b} f(x) \, dx \geq 0 \), and (2) if \( f(x) \) is less than or equal to some constant \( M \) on the interval \([a, b]\), then \( \int_{a}^{b} f(x) \, dx \leq M(b-a) \).
02
Analyze the Function \( \ln x \)
The natural logarithm function \( \ln x \) is increasing on its domain and non-negative for \( x \geq 1 \). Thus on the interval \([1, 3]\), \( \ln x \geq 0 \). This verifies the first part of the inequality: \( 0 \leq \int_{1}^{3} \ln x \, dx \).
03
Establish an Upper Bound for \( \ln x \)
On the interval \([1, 3]\), the maximum value of \( \ln x \) occurs at \( x = 3 \). Therefore, \( \ln x \leq \ln 3 \) for all \( x \in [1, 3] \).
04
Apply the Integral Properties for the Upper Bound
Using the upper bound \( \ln x \leq \ln 3 \), apply the integral property: \( \int_{1}^{3} \ln x \, dx \leq \int_{1}^{3} \ln 3 \, dx \). Since \( \ln 3 \) is a constant, this integral simplifies to \( \ln 3 \cdot (3-1) = 2 \ln 3 \).
05
Conclude the Inequality Verification
By combining the results from Steps 2 and 4, the integral inequality \( 0 \leq \int_{1}^{3} \ln x \, dx \leq 2 \ln 3 \) is verified without evaluating the integral.
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.
Understanding the Natural Logarithm
The natural logarithm, denoted as \( \ln x \), is a mathematical function with several notable properties. It is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.718. The natural logarithm has a domain for all positive real numbers, meaning you can only plug in positive numbers, and its range spans all real numbers.
Key characteristics of the natural logarithm \( \ln x \):
Key characteristics of the natural logarithm \( \ln x \):
- It is continuous for all \( x > 0 \), meaning there are no jumps or breaks in the function.
- On its domain, \( \ln x \) is an increasing function. This means as \( x \) increases, \( \ln x \) also increases.
- The function is non-negative for values of \( x \geq 1 \). This can help establish lower bounds when integrated.
- For integration purposes, the properties of \( \ln x \) allow us to evaluate or estimate integrals over certain intervals as seen in inequalities.
Grasping Integral Inequality
Integral inequalities involve comparing the integral of a function to a constant or another integral. They are useful in estimating or bounding the values of integrals without computing the exact integral. Here, we focus on bounding the integral of \( \ln x \) over the interval \([1, 3]\) by using properties of inequalities.
To prove \( 0 \leq \int_{1}^{3} \ln x \, dx \leq 2 \ln 3 \) without solving, follow these steps:
To prove \( 0 \leq \int_{1}^{3} \ln x \, dx \leq 2 \ln 3 \) without solving, follow these steps:
- If a function \( f(x) \) is non-negative on an interval \([a, b]\), then \( \int_{a}^{b} f(x) \, dx \geq 0 \). Given \( \ln x \) is non-negative for \( x \geq 1 \), establishing the lower bound is straightforward.
- Find an upper bound by ensuring \( f(x) \leq M \) on \([a, b]\). If \( \ln x \leq \ln 3 \) on \([1, 3]\), then \( \int_{1}^{3} \ln x \, dx \leq \int_{1}^{3} \ln 3 \, dx \).
- The integral of a constant \( \ln 3 \) over \([1, 3]\) is simply \( (3-1) \cdot \ln 3 = 2 \ln 3 \), justifying the upper bound of the original inequality.
Exploring Function Analysis
Function analysis involves dissecting the function's behavior over a given domain. For the function \( \ln x \), this means understanding how it acts over the specified interval \([1, 3]\). This analysis not only supports the inequality assessment but also broadens your mathematical perspective.
Key considerations in function analysis:
Key considerations in function analysis:
- Monotonicity: As an increasing function, \( \ln x \) ensures continuity and predictability over the interval \([1, 3]\). This trait aids in finding bounds for integrals.
- Critical Points: Although \( \ln x \) doesn't have critical points where \( x \in (1,3) \) (where its derivative changes sign), its continuous increase makes \( x=3 \) its maximum on the closed interval.
- Function Boundaries: Setting realistic upper limits, such as \( \ln 3 \), helps contextualize the function's behavior while strengthening understanding of how it integrates over the interval.
