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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pls find the derivitve
![The image presents two mathematical expressions that could be applied in different contexts such as calculus or algebra studies.
(c) \( y = \ln |x^3 - x^2| \)
This equation involves the natural logarithm of the absolute value of a polynomial expression. It's important to note the requirement of the absolute value to ensure the argument of the logarithm is non-negative, as logarithms are only defined for positive values.
(d) \( G(x) = \ln \left( \frac{a-x}{a+x} \right) \)
This function uses the natural logarithm of a rational expression. The presence of the logarithm indicates that the composition of \( (a-x) \) and \( (a+x) \) must remain positive for the function to be well-defined. This kind of expression might appear in calculus, particularly in integration or differentiation contexts.
(e) \( y = x^2 \ln[\ln x] \)
This equation combines both powers and nested logarithms. It implies that \( x \) must be greater than 1 for the inner natural logarithm to be defined (since \( \ln x > 0 \) is needed), highlighting a constraint on the domain. Such expressions are common in complex function analysis.
Each of these equations requires careful consideration of their domains and potential applications in mathematical problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44d0f866-3879-4a95-bacc-717e14c65c8b%2Fd9d196d1-f714-49ea-9d59-26029a4069d5%2Fa840ogb_processed.png&w=3840&q=75)
Transcribed Image Text:The image presents two mathematical expressions that could be applied in different contexts such as calculus or algebra studies.
(c) \( y = \ln |x^3 - x^2| \)
This equation involves the natural logarithm of the absolute value of a polynomial expression. It's important to note the requirement of the absolute value to ensure the argument of the logarithm is non-negative, as logarithms are only defined for positive values.
(d) \( G(x) = \ln \left( \frac{a-x}{a+x} \right) \)
This function uses the natural logarithm of a rational expression. The presence of the logarithm indicates that the composition of \( (a-x) \) and \( (a+x) \) must remain positive for the function to be well-defined. This kind of expression might appear in calculus, particularly in integration or differentiation contexts.
(e) \( y = x^2 \ln[\ln x] \)
This equation combines both powers and nested logarithms. It implies that \( x \) must be greater than 1 for the inner natural logarithm to be defined (since \( \ln x > 0 \) is needed), highlighting a constraint on the domain. Such expressions are common in complex function analysis.
Each of these equations requires careful consideration of their domains and potential applications in mathematical problems.
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