3. Let function In: (0, ∞) → R be defined by ln(x) = A. The function In(x) is continuous at every x in its domain. Explain. (Hint: what property does the first fundamental theorem imply about function In(x).) B. The value of In (2) is positive and the value of In(1/2) is negative. Explain. C. Assume the identity In(ab) = ln(a) + In(b) holds for all positive a, b. Using this identity, we have the following results for any positive integer n: In(2¹) = n. ln(2) and ln() = n. ln(). Explain. - dt.
3. Let function In: (0, ∞) → R be defined by ln(x) = A. The function In(x) is continuous at every x in its domain. Explain. (Hint: what property does the first fundamental theorem imply about function In(x).) B. The value of In (2) is positive and the value of In(1/2) is negative. Explain. C. Assume the identity In(ab) = ln(a) + In(b) holds for all positive a, b. Using this identity, we have the following results for any positive integer n: In(2¹) = n. ln(2) and ln() = n. ln(). Explain. - dt.
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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Transcribed Image Text:3.
Let function In: (0, ∞0) → R be defined by ln(2) = [dt
A. The function In(x) is continuous at every x in its domain. Explain. (Hint: what property does the first fundamental theorem imply about function ln(x).)
B. The value of ln(2) is positive and the value of ln(1/2) is negative. Explain.
C. Assume the identity In(ab) = ln(a) + ln(b) holds for all positive a, b. Using this identity, we have the following results for any positive integer n:
In(2¹) = n. ln(2) and In() = n. ln(). Explain.
D. The results of Parts A, B, and C combine to yield that function In(x) has a range of (-∞, ∞). (Hint: Part A implies that the IVT applies, and the IVT
works well with the result from Part C.)
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