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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Question
![**Simplify the following expression.**
\[
\frac{d}{dx} \int_{9}^{x} (19t^2 + t + 6) \, dt
\]
The expression involves finding the derivative of an integral with respect to \(x\). This can be approached using the Fundamental Theorem of Calculus, which states that if \(F(x) = \int_{a}^{x} f(t) \, dt\), then \(\frac{d}{dx} F(x) = f(x)\).
In this case, the function inside the integral is \(19t^2 + t + 6\), and the limits of integration are from 9 to \(x\). Therefore, the derivative with respect to \(x\) is simply the integrand evaluated at \(x\):
\[
19x^2 + x + 6
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F26050964-be59-4b06-942d-ab1b8c265eb4%2Fd316b02f-c71e-4de7-b50d-88d62bba5223%2Fgzsmip_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Simplify the following expression.**
\[
\frac{d}{dx} \int_{9}^{x} (19t^2 + t + 6) \, dt
\]
The expression involves finding the derivative of an integral with respect to \(x\). This can be approached using the Fundamental Theorem of Calculus, which states that if \(F(x) = \int_{a}^{x} f(t) \, dt\), then \(\frac{d}{dx} F(x) = f(x)\).
In this case, the function inside the integral is \(19t^2 + t + 6\), and the limits of integration are from 9 to \(x\). Therefore, the derivative with respect to \(x\) is simply the integrand evaluated at \(x\):
\[
19x^2 + x + 6
\]
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