(1) Find the most general antiderivative of the function given. What notation would be used for this antiderivative (i.e. f(x), f(x), F(x) ...)?. (Easy to check you answer by differentiation =) ) f(x) = 4x³ +2sin.x b) g'(x)=5√x-7x²/3 a)
(1) Find the most general antiderivative of the function given. What notation would be used for this antiderivative (i.e. f(x), f(x), F(x) ...)?. (Easy to check you answer by differentiation =) ) f(x) = 4x³ +2sin.x b) g'(x)=5√x-7x²/3 a)
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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Please help answer for a and b
![### Problem Statement:
Find the most general antiderivative of the function given. What notation would be used for this antiderivative (i.e., \( f(x) \), \( F(x) \), \( \mathcal{F}(x) \), etc.)? (Easy to check your answer by differentiation =)
**a)** \( f(x) = 4x^3 + 2 \sin x \)
**b)** \( g'(x) = 5\sqrt{x} - 7x^{2/3} \)
### Solution Explanation and Graphs:
1. For part **a**:
The given function is \( f(x) = 4x^3 + 2 \sin x \).
- The antiderivative or integral of \( 4x^3 \) is \( x^4 \) as the power of \( x \) increases by 1 and we divide by the new power.
- The antiderivative or integral of \( 2 \sin x \) is \( -2 \cos x \) because the integral of \( \sin x \) is \( -\cos x \).
- Therefore, the most general antiderivative of \( f(x) \) is:
\[
F(x) = x^4 - 2 \cos x + C
\]
Here, \( C \) is the constant of integration.
2. For part **b**:
The given function is \( g'(x) = 5\sqrt{x} - 7x^{2/3} \).
- The antiderivative or integral of \( 5\sqrt{x} \) can be rewritten as \( 5x^{1/2} \). The integral of \( x^{1/2} \) is \( \frac{2}{3} x^{3/2} \), so scaling it, \( \int 5x^{1/2} \, dx = 5 \cdot \frac{2}{3} x^{3/2} = \frac{10}{3} x^{3/2} \).
- The antiderivative or integral of \( -7x^{2/3} \) is \( -7 \cdot \frac{3}{5} x^{5/3} \) as the power of \( x \) increases by 1 and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99284a24-5b6d-4c49-8bfd-674d8761bb54%2Fc4dd1a47-d8fe-4235-88cf-dad6367e3b38%2Fkn0jcm8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Find the most general antiderivative of the function given. What notation would be used for this antiderivative (i.e., \( f(x) \), \( F(x) \), \( \mathcal{F}(x) \), etc.)? (Easy to check your answer by differentiation =)
**a)** \( f(x) = 4x^3 + 2 \sin x \)
**b)** \( g'(x) = 5\sqrt{x} - 7x^{2/3} \)
### Solution Explanation and Graphs:
1. For part **a**:
The given function is \( f(x) = 4x^3 + 2 \sin x \).
- The antiderivative or integral of \( 4x^3 \) is \( x^4 \) as the power of \( x \) increases by 1 and we divide by the new power.
- The antiderivative or integral of \( 2 \sin x \) is \( -2 \cos x \) because the integral of \( \sin x \) is \( -\cos x \).
- Therefore, the most general antiderivative of \( f(x) \) is:
\[
F(x) = x^4 - 2 \cos x + C
\]
Here, \( C \) is the constant of integration.
2. For part **b**:
The given function is \( g'(x) = 5\sqrt{x} - 7x^{2/3} \).
- The antiderivative or integral of \( 5\sqrt{x} \) can be rewritten as \( 5x^{1/2} \). The integral of \( x^{1/2} \) is \( \frac{2}{3} x^{3/2} \), so scaling it, \( \int 5x^{1/2} \, dx = 5 \cdot \frac{2}{3} x^{3/2} = \frac{10}{3} x^{3/2} \).
- The antiderivative or integral of \( -7x^{2/3} \) is \( -7 \cdot \frac{3}{5} x^{5/3} \) as the power of \( x \) increases by 1 and
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