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Let f(x) = ex and g(x) = 2*. For all x > 0, f(x) > g(x). O True O False

Let f(x) = ex and g(x) = 2*. For all x > 0, f(x) > g(x). O True O False

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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### Topic: Comparison of Exponential Functions **Problem Statement:** Let \( f(x) = e^x \) and \( g(x) = 2^x \). For all \( x > 0 \), \( f(x) > g(x) \). - True ☐ - False ☐ **Analysis:** In this problem, you are asked to compare two exponential functions, \( f(x) = e^x \) and \( g(x) = 2^x \), to determine if \( f(x) \) is always greater than \( g(x) \) for all positive values of \( x \). **Steps for Analysis:** 1. **Understand the functions:** - \( f(x) = e^x \) represents an exponential function with base \( e \approx 2.71828 \). - \( g(x) = 2^x \) represents an exponential function with base 2. 2. **Evaluate functions at specific points:** - For \( x = 0 \): - \( f(0) = e^0 = 1 \) - \( g(0) = 2^0 = 1 \) - For \( x = 1 \): - \( f(1) = e^1 \approx 2.71828 \) - \( g(1) = 2^1 = 2 \) - For \( x = 2 \): - \( f(2) = e^2 \approx 7.38906 \) - \( g(2) = 2^2 = 4 \) 3. **General Comparison:** - The base \( e \) is greater than 2, therefore, \( e^x \) grows faster than \( 2^x \) for all \( x > 0 \). **Conclusion:** Based on the observations, it can be concluded that \( e^x \) is indeed greater than \( 2^x \) for all \( x > 0 \). **Answer:** - True ☐ - False ☐ The correct answer is "True." **Explanation of Graphs and Diagrams:** If there were a graph provided to visually represent this comparison, it would display two curves: - The curve for \( f(x) = e^x \) would grow faster as
Transcribed Image Text:### Topic: Comparison of Exponential Functions **Problem Statement:** Let \( f(x) = e^x \) and \( g(x) = 2^x \). For all \( x > 0 \), \( f(x) > g(x) \). - True ☐ - False ☐ **Analysis:** In this problem, you are asked to compare two exponential functions, \( f(x) = e^x \) and \( g(x) = 2^x \), to determine if \( f(x) \) is always greater than \( g(x) \) for all positive values of \( x \). **Steps for Analysis:** 1. **Understand the functions:** - \( f(x) = e^x \) represents an exponential function with base \( e \approx 2.71828 \). - \( g(x) = 2^x \) represents an exponential function with base 2. 2. **Evaluate functions at specific points:** - For \( x = 0 \): - \( f(0) = e^0 = 1 \) - \( g(0) = 2^0 = 1 \) - For \( x = 1 \): - \( f(1) = e^1 \approx 2.71828 \) - \( g(1) = 2^1 = 2 \) - For \( x = 2 \): - \( f(2) = e^2 \approx 7.38906 \) - \( g(2) = 2^2 = 4 \) 3. **General Comparison:** - The base \( e \) is greater than 2, therefore, \( e^x \) grows faster than \( 2^x \) for all \( x > 0 \). **Conclusion:** Based on the observations, it can be concluded that \( e^x \) is indeed greater than \( 2^x \) for all \( x > 0 \). **Answer:** - True ☐ - False ☐ The correct answer is "True." **Explanation of Graphs and Diagrams:** If there were a graph provided to visually represent this comparison, it would display two curves: - The curve for \( f(x) = e^x \) would grow faster as
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