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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![## Problem 1: Using the Fundamental Theorem of Calculus to Find Derivatives
### (a)
\[
\frac{d}{dx} \int_{x}^{13} \sqrt{9 - y^2} \, dy
\]
### (b)
\[
\frac{d}{dx} \int_{0}^{3^{\tan x}} \frac{9 + y^2}{2y + 1} \, dy
\]
### (c)
\[
\frac{d}{dx} \int_{2\sqrt{x}}^{3^{x^3}} (1 + t) \, dy
\]
**Explanation:**
- **Part (a):** The integral's limits of integration are from \(x\) to 13. The function inside the integral is \(\sqrt{9 - y^2}\).
- **Part (b):** The limits are from 0 to \(3^{\tan x}\), with the integrand \(\frac{9 + y^2}{2y + 1}\).
- **Part (c):** The limits are from \(2\sqrt{x}\) to \(3^{x^3}\) and the integrand is \(1 + t\).
Each problem involves differentiating a definite integral with respect to \(x\), applying the Fundamental Theorem of Calculus, which relates differentiation and integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7866e7eb-0860-4fe3-a693-6d7d638dcf8c%2Fca158515-7c14-44ba-8776-25e7c2139102%2F99zqk2x4_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem 1: Using the Fundamental Theorem of Calculus to Find Derivatives
### (a)
\[
\frac{d}{dx} \int_{x}^{13} \sqrt{9 - y^2} \, dy
\]
### (b)
\[
\frac{d}{dx} \int_{0}^{3^{\tan x}} \frac{9 + y^2}{2y + 1} \, dy
\]
### (c)
\[
\frac{d}{dx} \int_{2\sqrt{x}}^{3^{x^3}} (1 + t) \, dy
\]
**Explanation:**
- **Part (a):** The integral's limits of integration are from \(x\) to 13. The function inside the integral is \(\sqrt{9 - y^2}\).
- **Part (b):** The limits are from 0 to \(3^{\tan x}\), with the integrand \(\frac{9 + y^2}{2y + 1}\).
- **Part (c):** The limits are from \(2\sqrt{x}\) to \(3^{x^3}\) and the integrand is \(1 + t\).
Each problem involves differentiating a definite integral with respect to \(x\), applying the Fundamental Theorem of Calculus, which relates differentiation and integration.
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