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 Statement
9. Find \(\frac{d}{dx} \int_{3}^{3x} (t^2 - t) \, dt\).
### Explanation
This problem involves finding the derivative of an integral with a variable upper limit. The expression is an example of applying the Leibniz rule for differentiating under the integral sign, often used in calculus to evaluate how the value of a definite integral changes as a parameter changes.
### Approach
To solve this, you can use the Fundamental Theorem of Calculus, part 2, which can be expressed as:
\[
\frac{d}{dx} \int_{a}^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)
\]
In this problem:
- \( f(t) = t^2 - t \)
- The upper limit is \( g(x) = 3x \)
- The lower limit is a constant, 3
First, substitute \( g(x) \) into the function \( f(t) \):
\[ f(g(x)) = (3x)^2 - 3x = 9x^2 - 3x \]
Compute the derivative of \( g(x) \):
\[ g'(x) = \frac{d}{dx}(3x) = 3 \]
Finally, apply the chain rule (Fundamental Theorem of Calculus, part 2):
\[
\frac{d}{dx} \int_{3}^{3x} (t^2 - t) \, dt = (9x^2 - 3x) \cdot 3
\]
Simplify the expression:
\[
= 27x^2 - 9x
\]
This is the derivative of the given integral with respect to \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fce08c4ed-eb77-44e7-b778-51357cdb5486%2F5254caf6-c32c-4040-bcee-8d2bdd8d58d3%2Fel6qzkt_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
9. Find \(\frac{d}{dx} \int_{3}^{3x} (t^2 - t) \, dt\).
### Explanation
This problem involves finding the derivative of an integral with a variable upper limit. The expression is an example of applying the Leibniz rule for differentiating under the integral sign, often used in calculus to evaluate how the value of a definite integral changes as a parameter changes.
### Approach
To solve this, you can use the Fundamental Theorem of Calculus, part 2, which can be expressed as:
\[
\frac{d}{dx} \int_{a}^{g(x)} f(t) \, dt = f(g(x)) \cdot g'(x)
\]
In this problem:
- \( f(t) = t^2 - t \)
- The upper limit is \( g(x) = 3x \)
- The lower limit is a constant, 3
First, substitute \( g(x) \) into the function \( f(t) \):
\[ f(g(x)) = (3x)^2 - 3x = 9x^2 - 3x \]
Compute the derivative of \( g(x) \):
\[ g'(x) = \frac{d}{dx}(3x) = 3 \]
Finally, apply the chain rule (Fundamental Theorem of Calculus, part 2):
\[
\frac{d}{dx} \int_{3}^{3x} (t^2 - t) \, dt = (9x^2 - 3x) \cdot 3
\]
Simplify the expression:
\[
= 27x^2 - 9x
\]
This is the derivative of the given integral with respect to \( x \).
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