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Using the derivatives of In x, tell in a complete sentence why Inr is increasing, and why its graph is concave downward.

Using the derivatives of In x, tell in a complete sentence why Inr is increasing, and why its graph is concave downward.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Problem 2 (a) **Explanation Using Derivatives of \( \ln x \):** The function \( \ln x \) is increasing because its derivative, \( \frac{d}{dx}(\ln x) = \frac{1}{x} \), is positive for all \( x > 0 \). The graph of \( \ln x \) is concave downward because the second derivative, \( \frac{d^2}{dx^2}(\ln x) = -\frac{1}{x^2} \), is negative for all \( x > 0 \). (b) **Approximating \( \ln y \):** Let \( y = 10^{100} \). Using \( \ln 10 \approx 2.3026 \), find an approximate numerical value of \( \ln y \). (c) **Finding Integer \( z \):** Again using \( \ln 10 \approx 2.3026 \), find an integer \( z \) such that \( \ln z > 1,000,000 \). (Together with part (a), this should convince one that \(\lim_{x \to \infty} \ln x = \infty\).) (d) **Finding Number \( w \):** Again using \( \ln 10 \approx 2.3026 \), find a number \( w \) such that \( \ln w < -1,000 \).
Transcribed Image Text:### Problem 2 (a) **Explanation Using Derivatives of \( \ln x \):** The function \( \ln x \) is increasing because its derivative, \( \frac{d}{dx}(\ln x) = \frac{1}{x} \), is positive for all \( x > 0 \). The graph of \( \ln x \) is concave downward because the second derivative, \( \frac{d^2}{dx^2}(\ln x) = -\frac{1}{x^2} \), is negative for all \( x > 0 \). (b) **Approximating \( \ln y \):** Let \( y = 10^{100} \). Using \( \ln 10 \approx 2.3026 \), find an approximate numerical value of \( \ln y \). (c) **Finding Integer \( z \):** Again using \( \ln 10 \approx 2.3026 \), find an integer \( z \) such that \( \ln z > 1,000,000 \). (Together with part (a), this should convince one that \(\lim_{x \to \infty} \ln x = \infty\).) (d) **Finding Number \( w \):** Again using \( \ln 10 \approx 2.3026 \), find a number \( w \) such that \( \ln w < -1,000 \).
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