Domanda del libro di testo
1. Express the following logarithms in terms of ln 2 and ln 3.
d. ln ∛9
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 7, Problema 7.3.155d
155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
d. Conclude that
xᵉ < eˣfor all positivex ≠ e.
Guida verificata passo dopo passo1
Rewrite the expressions \( \pi^{e} \) and \( e^{\pi} \) in a form that allows comparison using logarithms. Consider taking the natural logarithm of both expressions to compare their sizes without directly calculating their values.
Express the comparison as \( \pi^{e} < e^{\pi} \) if and only if \( e \ln(\pi) < \pi \ln(e) \). Since \( \ln(e) = 1 \), this simplifies to comparing \( e \ln(\pi) \) and \( \pi \).
Define a function \( f(x) = \frac{\ln(x)}{x} \) for \( x > 0 \) to analyze the inequality \( x^{e} < e^{x} \) for \( x \neq e \). The inequality \( x^{e} < e^{x} \) is equivalent to \( e \ln(x) < x \), or \( \frac{\ln(x)}{x} < \frac{1}{e} \).
Find the critical points of \( f(x) = \frac{\ln(x)}{x} \) by differentiating: \( f'(x) = \frac{1 - \ln(x)}{x^{2}} \). Set \( f'(x) = 0 \) to find that the maximum occurs at \( x = e \).
Conclude that since \( f(x) \) attains its maximum at \( x = e \), for all positive \( x \neq e \), \( f(x) < f(e) = \frac{1}{e} \). Therefore, \( x^{e} < e^{x} \) for all positive \( x \neq e \), which includes the comparison between \( \pi^{e} \) and \( e^{\pi} \).
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Exponential and Power Functions
Understanding the difference between expressions like πᵉ and e^π requires familiarity with power functions (x raised to a constant) and exponential functions (constant raised to variable). Recognizing how these functions behave for positive real numbers is essential to compare their values without direct computation.
Video consigliato:
6:13
Exponential Functions
Function Comparison Using Logarithms
Comparing expressions like πᵉ and e^π can be simplified by taking natural logarithms, converting powers into products. This technique transforms the inequality into a comparison of products involving logarithms, making it easier to analyze without a calculator.
Video consigliato:
5:26Graphs of Logarithmic Functions
Monotonicity and Critical Points of the Function f(x) = x^{1/x}
The function f(x) = x^{1/x} reaches its maximum at x = e, which helps prove inequalities like xᵉ < eˣ for x ≠ e. Understanding how to find and interpret critical points and monotonicity of such functions is key to concluding the given inequality.
Video consigliato:
04:56Derivative of the Natural Exponential Function (e^x)
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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
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c. lim(x→∞) sinh x
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Domanda del libro di testo
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
c. 3ln ∛(t² - 1) - ln(t+1)
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In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
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72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
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