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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![The given mathematical expression on the image is as follows:
"If \( f(x) = \int_{x}^{8} t^3 \, dt \), then
\[ f'(x) = \quad \]"
This format is often used to ask students to find the derivative of a function defined as an integral with variable limits. In this specific example, \( f(x) \) is defined as the definite integral of \( t^3 \) with the upper limit of 8 and the lower limit of \( x \).
To solve this problem using the Fundamental Theorem of Calculus Part 1, note that if \( F(t) \) is an antiderivative of \( t^3 \), then:
\[ f'(x) = - t^3 \Bigg|_{t=x} \]
\[ f'(x) = - x^3 \]
Thus, the derivative \( f'(x) \) is \( -x^3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dc9f679-6754-47d7-a5b7-cef477361a1e%2F87f85b5a-da42-4caa-923c-3e9475cb6e8b%2F4q8qr3v_processed.png&w=3840&q=75)
Transcribed Image Text:The given mathematical expression on the image is as follows:
"If \( f(x) = \int_{x}^{8} t^3 \, dt \), then
\[ f'(x) = \quad \]"
This format is often used to ask students to find the derivative of a function defined as an integral with variable limits. In this specific example, \( f(x) \) is defined as the definite integral of \( t^3 \) with the upper limit of 8 and the lower limit of \( x \).
To solve this problem using the Fundamental Theorem of Calculus Part 1, note that if \( F(t) \) is an antiderivative of \( t^3 \), then:
\[ f'(x) = - t^3 \Bigg|_{t=x} \]
\[ f'(x) = - x^3 \]
Thus, the derivative \( f'(x) \) is \( -x^3 \).
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