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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![**Topic: Application of the Fundamental Theorem of Calculus**
**Objective:** Use Part 1 of the Fundamental Theorem of Calculus (and properties of definite integrals and the Chain Rule) to evaluate the expression.
**Expression to Evaluate:**
\[ \frac{d}{dx} \left[ \int_{x^3}^{10} \sqrt{t^6 + 1} \, dt \right] \]
**Explanation:**
This problem requires the use of the Fundamental Theorem of Calculus, which connects differentiation and integration. Part 1 of this theorem states that if \( F(x) \) is an antiderivative of \( f(x) \), then:
\[ \frac{d}{dx} \left[ \int_{a}^{g(x)} f(t) \, dt \right] = f(g(x)) \cdot g'(x) \]
In this case, we will apply this theorem along with the Chain Rule to evaluate the integral, treating the variable limit of integration as a function of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3f88845-b007-4fd1-915c-0698883559f2%2F62f68ec2-94af-4972-a1da-52d965b6a295%2Ffw8f5d_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Application of the Fundamental Theorem of Calculus**
**Objective:** Use Part 1 of the Fundamental Theorem of Calculus (and properties of definite integrals and the Chain Rule) to evaluate the expression.
**Expression to Evaluate:**
\[ \frac{d}{dx} \left[ \int_{x^3}^{10} \sqrt{t^6 + 1} \, dt \right] \]
**Explanation:**
This problem requires the use of the Fundamental Theorem of Calculus, which connects differentiation and integration. Part 1 of this theorem states that if \( F(x) \) is an antiderivative of \( f(x) \), then:
\[ \frac{d}{dx} \left[ \int_{a}^{g(x)} f(t) \, dt \right] = f(g(x)) \cdot g'(x) \]
In this case, we will apply this theorem along with the Chain Rule to evaluate the integral, treating the variable limit of integration as a function of \( x \).
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