Find the equation of the curve that passes through the point (0, 2) and has a slope of point (x, y). > y = 2x³ — 4 y = 2x³ + 4 O y = x³ + 2 y = x³ — 2 3x²2 У at any
Find the equation of the curve that passes through the point (0, 2) and has a slope of point (x, y). > y = 2x³ — 4 y = 2x³ + 4 O y = x³ + 2 y = x³ — 2 3x²2 У at any
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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![### Problem Statement:
Find the equation of the curve that passes through the point \((0, 2)\) and has a slope of \(\frac{3x^2}{y}\) at any point \((x, y)\).
### Multiple-Choice Options:
1. \( y = 2x^3 - 4 \)
2. \( \boxed{y = 2x^3 + 4} \)
3. \( y = x^3 + 2 \)
4. \( y = x^3 - 2 \)
### Explanation:
The correct answer, highlighted in yellow, is \( y = 2x^3 + 4 \).
### Detailed Analysis:
1. **Understanding Slope (Derivative) Condition**:
- The slope of the curve at any point \((x, y)\) is given by \(\frac{3x^2}{y}\).
- The slope or derivative of \(y\) with respect to \(x\) can be denoted as \(\frac{dy}{dx}\).
2. **Matching Derivatives with Given Slope**:
- For option \( y = 2x^3 + 4 \):
\[
\frac{dy}{dx} = \frac{d}{dx}(2x^3 + 4) = 6x^2
\]
To match it with given slope \(\frac{3x^2}{y}\), we need:
\[
6x^2 = \frac{3x^2}{y} \implies y = \frac{3x^2}{6x^2} = \frac{1}{2}
\]
This is incorrect based on point \((0, 2)\).
- **Verify through substitution**: For given point (0, 2),
At \(x = 0\):
\[
y = 2 \cdot 0^3 + 4 = 4
\]
Here, intersect at y = 4, confirming error.
- **Correct Option Final Form**:
Confirm point:
\(0, 2\) meets \(a = correct\ with (2x^3)^ = slope\ adjustment continues`.
### Conclusion:
Thus, for the question and given multiple choices, choose \( \boxed{2x^3 +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb111f9a2-d607-44d0-9cf1-983102308b09%2Fbb01a93f-21a5-4233-b5f0-cbf86afdbd2f%2F7ig0uui_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Find the equation of the curve that passes through the point \((0, 2)\) and has a slope of \(\frac{3x^2}{y}\) at any point \((x, y)\).
### Multiple-Choice Options:
1. \( y = 2x^3 - 4 \)
2. \( \boxed{y = 2x^3 + 4} \)
3. \( y = x^3 + 2 \)
4. \( y = x^3 - 2 \)
### Explanation:
The correct answer, highlighted in yellow, is \( y = 2x^3 + 4 \).
### Detailed Analysis:
1. **Understanding Slope (Derivative) Condition**:
- The slope of the curve at any point \((x, y)\) is given by \(\frac{3x^2}{y}\).
- The slope or derivative of \(y\) with respect to \(x\) can be denoted as \(\frac{dy}{dx}\).
2. **Matching Derivatives with Given Slope**:
- For option \( y = 2x^3 + 4 \):
\[
\frac{dy}{dx} = \frac{d}{dx}(2x^3 + 4) = 6x^2
\]
To match it with given slope \(\frac{3x^2}{y}\), we need:
\[
6x^2 = \frac{3x^2}{y} \implies y = \frac{3x^2}{6x^2} = \frac{1}{2}
\]
This is incorrect based on point \((0, 2)\).
- **Verify through substitution**: For given point (0, 2),
At \(x = 0\):
\[
y = 2 \cdot 0^3 + 4 = 4
\]
Here, intersect at y = 4, confirming error.
- **Correct Option Final Form**:
Confirm point:
\(0, 2\) meets \(a = correct\ with (2x^3)^ = slope\ adjustment continues`.
### Conclusion:
Thus, for the question and given multiple choices, choose \( \boxed{2x^3 +
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