Determine the instantaneous rate of change of the function lo sin (2³) f(x) = √V² sit dt 1+√t x3 at x = 1.
Determine the instantaneous rate of change of the function lo sin (2³) f(x) = √V² sit dt 1+√t x3 at x = 1.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### VI. Determine the Instantaneous Rate of Change of the Function
Given the function:
\[
f(x) = \int_{x^3}^{\sqrt{x}} \frac{\sin \left(\frac{\pi t^3}{2}\right)}{1 + \sqrt{t}} \, dt
\]
Determine the instantaneous rate of change at \( x = 1 \).
---
To solve this problem, you need to find the derivative of the function \( f(x) \) and evaluate it at \( x=1 \). This involves applying techniques from integral calculus and understanding the Fundamental Theorem of Calculus.
The integral's upper and lower limits are functions of \( x \), which requires the use of the Leibniz rule for differentiation under the integral sign. Detailed solution steps would include:
1. Differentiating the integral with respect to \( x \).
2. Applying the Fundamental Theorem of Calculus.
3. Evaluating the resulting expression at \( x=1 \).
---
Note: The large yellow "10" seen on the image is likely a watermark and is not related to the math problem.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F610d65bd-822d-435d-bd07-52365ff393da%2Fe0dd14e7-3937-49f1-9e65-ddf766fc4b48%2Fzwi6zm_processed.png&w=3840&q=75)
Transcribed Image Text:### VI. Determine the Instantaneous Rate of Change of the Function
Given the function:
\[
f(x) = \int_{x^3}^{\sqrt{x}} \frac{\sin \left(\frac{\pi t^3}{2}\right)}{1 + \sqrt{t}} \, dt
\]
Determine the instantaneous rate of change at \( x = 1 \).
---
To solve this problem, you need to find the derivative of the function \( f(x) \) and evaluate it at \( x=1 \). This involves applying techniques from integral calculus and understanding the Fundamental Theorem of Calculus.
The integral's upper and lower limits are functions of \( x \), which requires the use of the Leibniz rule for differentiation under the integral sign. Detailed solution steps would include:
1. Differentiating the integral with respect to \( x \).
2. Applying the Fundamental Theorem of Calculus.
3. Evaluating the resulting expression at \( x=1 \).
---
Note: The large yellow "10" seen on the image is likely a watermark and is not related to the math problem.
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