Use the Second Fundamental Theorem of Calculus to find F'(x). • L₁ VE 4 + 4 dt Vt Step 1 F(x) = Note that F(x) = Step 2 - Love [M Use the Second Fundamental Theorem of Calculus, which states that, if f is continuous on an open interval I containing a, then for every x in the interval, d X dx Therefore, In this problem, F(x) = F'(x) t4+ 4 dt. So assume f(t) f(t) dt = f(x). = f(x) X L √t4 + 4 dt. 6 d - + [[^r(1) d²] = dx -6 = +4 4 that is continuous on the entire real line.
Use the Second Fundamental Theorem of Calculus to find F'(x). • L₁ VE 4 + 4 dt Vt Step 1 F(x) = Note that F(x) = Step 2 - Love [M Use the Second Fundamental Theorem of Calculus, which states that, if f is continuous on an open interval I containing a, then for every x in the interval, d X dx Therefore, In this problem, F(x) = F'(x) t4+ 4 dt. So assume f(t) f(t) dt = f(x). = f(x) X L √t4 + 4 dt. 6 d - + [[^r(1) d²] = dx -6 = +4 4 that is continuous on the entire real line.
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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![Use the Second Fundamental Theorem of Calculus to find \( F'(x) \).
\[ F(x) = \int_{-6}^{x} \sqrt{t^4 + 4} \, dt \]
**Step 1**
Note that \( F(x) = \int_{-6}^{x} \sqrt{t^4 + 4} \, dt \). So assume \( f(t) = \sqrt{t^4 + 4} \) that is continuous on the entire real line.
Use the Second Fundamental Theorem of Calculus, which states that, if \( f \) is continuous on an open interval \( I \) containing \( a \), then for every \( x \) in the interval,
\[
\frac{d}{dx} \left[ \int_{a}^{x} f(t) \, dt \right] = f(x).
\]
**Step 2**
In this problem, \( F(x) = \int_{-6}^{x} \sqrt{t^4 + 4} \, dt \).
Therefore,
\[
F'(x) = \frac{d}{dx} \left[ \int_{-6}^{x} f(t) \, dt \right]
\]
\[
= f(x)
\]
\[
= \sqrt{x^4 + 4}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff9b6a03f-2298-4441-ab19-fb50f28ec42f%2F6bec99b3-5005-4eb8-a648-44394eaa2282%2Fxu9isir_processed.png&w=3840&q=75)
Transcribed Image Text:Use the Second Fundamental Theorem of Calculus to find \( F'(x) \).
\[ F(x) = \int_{-6}^{x} \sqrt{t^4 + 4} \, dt \]
**Step 1**
Note that \( F(x) = \int_{-6}^{x} \sqrt{t^4 + 4} \, dt \). So assume \( f(t) = \sqrt{t^4 + 4} \) that is continuous on the entire real line.
Use the Second Fundamental Theorem of Calculus, which states that, if \( f \) is continuous on an open interval \( I \) containing \( a \), then for every \( x \) in the interval,
\[
\frac{d}{dx} \left[ \int_{a}^{x} f(t) \, dt \right] = f(x).
\]
**Step 2**
In this problem, \( F(x) = \int_{-6}^{x} \sqrt{t^4 + 4} \, dt \).
Therefore,
\[
F'(x) = \frac{d}{dx} \left[ \int_{-6}^{x} f(t) \, dt \right]
\]
\[
= f(x)
\]
\[
= \sqrt{x^4 + 4}.
\]
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