---### Calculus Tutorial: Using the Fundamental Theorem of Calculus#### Application of the Fundamental Theorem of Calculus - Part IIn this exercise, we will use Part I of the Fundamental Theorem of Calculus to find the derivative of the given function.\[ f(x) = \int_{-2}^{x} \sqrt{t^3 + 2^3} \, dt \]We aim to determine $ f'(x) $.---\[ f'(x) = \boxed{\text{Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary } ( \text{ , } ) \text{ , etc.}} \][NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e., remember to put all the necessary parentheses, commas, etc.]---In this problem, the notation provided involves an integral with variable limits of integration, specifically from -2 to x. The Fundamental Theorem of Calculus Part I tells us that if $ F(x) $ is an antiderivative of $ f(x) $, then:\[ \frac{d}{dx} \left( \int_{a}^{x} f(t) \, dt \right) = f(x) \]In this context, the integrand $ \sqrt{t^3 + 8} $ is a function of $ t $. Thus, applying the theorem, we find that:\[ f'(x) = \sqrt{x^3 + 8} \]This result leverages the property that taking the derivative of the integral with respect to its upper limit $ x $ effectively "evaluates" the integrand at its upper limit.### SummaryBy employing the Fundamental Theorem of Calculus Part I, we transformed our integral expression into a derivative, thereby simplifying our function to its derivative form.For further practice, ensure your syntax is correct when inputting mathematical expressions, always considering proper use of parentheses and other notation.

Name: ---### Calculus Tutorial: Using the Fundamental Theorem of Calculus##
Uploaded: 2024-03-20T06:46:42.000Z
Channel: Bartleby
Description: ---### Calculus Tutorial: Using the Fundamental Theorem of Calculus#### Application of the Fundamental Theorem of Calculus - Part IIn this exercise, we will use Part I of the Fundamental Theorem of Calculus to find the derivative of the given functi

Answered: Image /qna-images/answer/72407bbe-d41d-4600-923b-04782de1f6d6.jpg

Answered: Use part I of the Fundamental Theorem…

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...

See similar textbooks

Related questions

Topic Video

Question

100%

---
### Calculus Tutorial: Using the Fundamental Theorem of Calculus

#### Application of the Fundamental Theorem of Calculus - Part I

In this exercise, we will use Part I of the Fundamental Theorem of Calculus to find the derivative of the given function.

\[ f(x) = \int_{-2}^{x} \sqrt{t^3 + 2^3} \, dt \]

We aim to determine $ f'(x) $.

---

\[ f'(x) = \boxed{\text{Enter a function as your answer. Make sure that your syntax is correct, i.e. remember to put all the necessary } ( \text{ , } ) \text{ , etc.}} \]

[NOTE: Enter a function as your answer. Make sure that your syntax is correct, i.e., remember to put all the necessary parentheses, commas, etc.]

---

In this problem, the notation provided involves an integral with variable limits of integration, specifically from -2 to x. The Fundamental Theorem of Calculus Part I tells us that if $ F(x) $ is an antiderivative of $ f(x) $, then:
\[ \frac{d}{dx} \left( \int_{a}^{x} f(t) \, dt \right) = f(x) \]

In this context, the integrand $ \sqrt{t^3 + 8} $ is a function of $ t $. Thus, applying the theorem, we find that:

\[ f'(x) = \sqrt{x^3 + 8} \]

This result leverages the property that taking the derivative of the integral with respect to its upper limit $ x $ effectively "evaluates" the integrand at its upper limit.

### Summary

By employing the Fundamental Theorem of Calculus Part I, we transformed our integral expression into a derivative, thereby simplifying our function to its derivative form.

For further practice, ensure your syntax is correct when inputting mathematical expressions, always considering proper use of parentheses and other notation.

Expert Solution

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Chain Rule$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Calculus: Early Transcendentals

Calculus

ISBN:

9781285741550

Author:

James Stewart

Publisher:

Cengage Learning