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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do number 4 pls and pls show all the work
![Below is the transcription of the given image, suitable for an Educational website. Each question is listed with detailed descriptions of any graphs or diagrams:
---
### Calculus Problems and Exercises
1. **Antiderivative Problem**
- **Task**: Find the antiderivative of \( f(x) \).
- **Condition**: Initial condition is \( f(0) = 3 \).
- **Given**: \( f'(x) = 4x^3 + x^2 - 3x \).
2. **Function \( g(x) \)**
- **Task**: Find the function \( g(x) \) where \( g'(x) = 4 \sin x + 7x - \frac{5}{x} \).
3. **Differentiation and Evaluation**
- **Task**: Find \( y' \) and then evaluate it for \((-2, 3)\).
- **Given**: \( 3x^3 + y^2 = -xy \).
4. **Limit Calculation**
- **Task**: Find the limit:
\[
\lim_{x \to 0} \frac{e^{2x} - 1}{\cos x - 1}
\]
5. **Maximum and Minimum of \( f(x) \)**
- **Task**: Find the absolute maximum and absolute minimum of
\[
f(x) = x^3 - 3x^2 + 1
\]
- **Domain**: \( [-2, 4] \).
6. **Compute \(\nabla y\) and \( \nabla x \)**
- **Task**: Compute \( \nabla y \) and \( \nabla x \) for \( f(x) = x - x^3 \).
- **Condition**: \(x = 1\) and \( \nabla x = 0.1 \).
7. **Rate of Change in Volume of a Sphere**
- **Scenario**: Air is being pumped into a spherical balloon so that the volume increases at a rate of 50 cubic feet per second.
- **Question**: How fast is the radius of the balloon increasing when the diameter is 26 feet?
- **Formulas Provided**:
- Diameter \( = 2 \times \text](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92b3472e-ab59-40f2-b296-ac3727a4d9f3%2F387287ea-75c7-44f8-bedd-e5adc26bbbc2%2Flgja08_processed.png&w=3840&q=75)
Transcribed Image Text:Below is the transcription of the given image, suitable for an Educational website. Each question is listed with detailed descriptions of any graphs or diagrams:
---
### Calculus Problems and Exercises
1. **Antiderivative Problem**
- **Task**: Find the antiderivative of \( f(x) \).
- **Condition**: Initial condition is \( f(0) = 3 \).
- **Given**: \( f'(x) = 4x^3 + x^2 - 3x \).
2. **Function \( g(x) \)**
- **Task**: Find the function \( g(x) \) where \( g'(x) = 4 \sin x + 7x - \frac{5}{x} \).
3. **Differentiation and Evaluation**
- **Task**: Find \( y' \) and then evaluate it for \((-2, 3)\).
- **Given**: \( 3x^3 + y^2 = -xy \).
4. **Limit Calculation**
- **Task**: Find the limit:
\[
\lim_{x \to 0} \frac{e^{2x} - 1}{\cos x - 1}
\]
5. **Maximum and Minimum of \( f(x) \)**
- **Task**: Find the absolute maximum and absolute minimum of
\[
f(x) = x^3 - 3x^2 + 1
\]
- **Domain**: \( [-2, 4] \).
6. **Compute \(\nabla y\) and \( \nabla x \)**
- **Task**: Compute \( \nabla y \) and \( \nabla x \) for \( f(x) = x - x^3 \).
- **Condition**: \(x = 1\) and \( \nabla x = 0.1 \).
7. **Rate of Change in Volume of a Sphere**
- **Scenario**: Air is being pumped into a spherical balloon so that the volume increases at a rate of 50 cubic feet per second.
- **Question**: How fast is the radius of the balloon increasing when the diameter is 26 feet?
- **Formulas Provided**:
- Diameter \( = 2 \times \text
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