At Each Point { (x, Y)$}$ On A Certain Curve, The Slope Of The Curve Is $ 3x^2y\$} . If The Curve Contains The Point { (0, 8)$}$, Then Its Equation Is A. { Y = 8e {x 3 $}$B. { Y = X^3 + 8$} C . \[ C. \[ C . \[ Y =

Introduction

In calculus, the slope of a curve at a given point is a measure of how steep the curve is at that point. It is an important concept in understanding the behavior of functions and their graphs. In this article, we will explore a specific problem where the slope of a curve is given by a function of the form $3x^2y$. We will use this information to find the equation of the curve.

The Problem

The problem states that at each point $(x, y)$ on a certain curve, the slope of the curve is given by the function $3x^2y$. This means that if we take any point on the curve and find the slope at that point, it will be equal to $3x^2y$. We are also given that the curve contains the point $(0, 8)$.

The Slope of a Curve

The slope of a curve at a point $(x, y)$ is given by the derivative of the function that represents the curve. In other words, if we have a function $y = f(x)$, then the slope of the curve at the point $(x, y)$ is given by the derivative $f'(x)$. In this case, the slope of the curve is given by the function $3x^2y$, which means that the derivative of the function that represents the curve is equal to $3x^2y$.

Finding the Equation of the Curve

To find the equation of the curve, we need to find a function $y = f(x)$ whose derivative is equal to $3x^2y$. This means that we need to find a function that satisfies the equation $f'(x) = 3x^2f(x)$. To solve this equation, we can use the method of separation of variables.

Separation of Variables

The method of separation of variables is a technique used to solve differential equations of the form $f'(x) = g(x)f(x)$. To apply this method, we can divide both sides of the equation by $f(x)$, which gives us:

$$\frac{f'(x)}{f(x)} = 3x^2$$

This equation can be rewritten as:

$$\frac{d}{dx}(\ln|f(x)|) = 3x^2$$

Integrating Both Sides

To solve for $f(x)$, we can integrate both sides of the equation:

$$\int \frac{d}{dx}(\ln|f(x)|) dx = \int 3x^2 dx$$

This gives us:

$$\ln|f(x)| = x^3 + C$$

where $C$ is a constant.

Solving for $f(x)$

To solve for $f(x)$, we can exponentiate both sides of the equation:

$$|f(x)| = e^{x^3 + C}$$

Since $f(x)$ is a function, we can drop the absolute value sign and write:

$$f(x) = e^{x^3 + C}$$

Finding the Constant $C$

We are given that the curve contains the point $(0, 8)$. This means that when $x = 0$, $y = 8$. We can substitute these values into the equation $f(x) = e^{x^3 + C}$ to find the value of $C$:

$$8 = e^{0^3 + C}$$

This gives us:

$$C = \ln 8$$

The Equation of the Curve

Now that we have found the value of $C$, we can write the equation of the curve:

$$y = e^{x^3 + \ln 8}$$

This can be simplified to:

$$y = 8e^{x^3}$$

Conclusion

In this article, we used the method of separation of variables to find the equation of a curve whose slope is given by the function $3x^2y$. We found that the equation of the curve is $y = 8e^{x^3}$. This is an example of how the method of separation of variables can be used to solve differential equations and find the equation of a curve.

Discussion

The equation of the curve $y = 8e^{x^3}$ is a transcendental equation, which means that it cannot be written in the form of a polynomial equation. This is because the function $e^{x^3}$ is an exponential function, which is not a polynomial function.

The curve $y = 8e^{x^3}$ is a type of curve known as a transcendental curve. Transcendental curves are curves that cannot be written in the form of a polynomial equation. They are often used in mathematics and physics to model real-world phenomena.

References

• [1] "Calculus" by Michael Spivak
• [2] "Differential Equations" by Morris Tenenbaum and Harry Pollard

Additional Resources

• [1] "Calculus Online" by MIT OpenCourseWare
• [2] "Differential Equations Online" by University of Michigan

Final Answer

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Introduction

In our previous article, we explored a problem where the slope of a curve is given by a function of the form $3x^2y$. We used the method of separation of variables to find the equation of the curve. In this article, we will answer some common questions related to this problem.

Q: What is the slope of a curve?

A: The slope of a curve at a point $(x, y)$ is a measure of how steep the curve is at that point. It is given by the derivative of the function that represents the curve.

Q: How do you find the equation of a curve given its slope?

A: To find the equation of a curve given its slope, you can use the method of separation of variables. This involves dividing both sides of the equation by the function that represents the curve and then integrating both sides.

Q: What is the method of separation of variables?

A: The method of separation of variables is a technique used to solve differential equations of the form $f'(x) = g(x)f(x)$. It involves dividing both sides of the equation by the function that represents the curve and then integrating both sides.

Q: How do you apply the method of separation of variables?

A: To apply the method of separation of variables, you can follow these steps:

1. Divide both sides of the equation by the function that represents the curve.
2. Integrate both sides of the equation.
3. Exponentiate both sides of the equation to find the function that represents the curve.

Q: What is a transcendental curve?

A: A transcendental curve is a curve that cannot be written in the form of a polynomial equation. It is often used in mathematics and physics to model real-world phenomena.

Q: What is the equation of the curve in this problem?

A: The equation of the curve in this problem is $y = 8e^{x^3}$.

Q: How do you find the constant $C$ in the equation of the curve?

A: To find the constant $C$ in the equation of the curve, you can substitute the values of $x$ and $y$ into the equation and solve for $C$.

Q: What is the significance of the constant $C$ in the equation of the curve?

A: The constant $C$ in the equation of the curve represents the vertical shift of the curve. It is a constant that is added to the function that represents the curve.

Q: How do you use the equation of the curve to model real-world phenomena?

A: The equation of the curve can be used to model real-world phenomena such as population growth, chemical reactions, and electrical circuits.

Q: What are some common applications of the equation of the curve?

A: Some common applications of the equation of the curve include:

• Modeling population growth
• Studying chemical reactions
• Analyzing electrical circuits
• Predicting stock prices

Conclusion

In this article, we answered some common questions related to the problem of finding the equation of a curve given its slope. We discussed the method of separation of variables and how it can be used to solve differential equations. We also explored the concept of transcendental curves and how they can be used to model real-world phenomena.

Discussion

The equation of the curve $y = 8e^{x^3}$ is a transcendental equation, which means that it cannot be written in the form of a polynomial equation. This is because the function $e^{x^3}$ is an exponential function, which is not a polynomial function.

The curve $y = 8e^{x^3}$ is a type of curve known as a transcendental curve. Transcendental curves are curves that cannot be written in the form of a polynomial equation. They are often used in mathematics and physics to model real-world phenomena.

References

• [1] "Calculus" by Michael Spivak
• [2] "Differential Equations" by Morris Tenenbaum and Harry Pollard

Additional Resources

• [1] "Calculus Online" by MIT OpenCourseWare
• [2] "Differential Equations Online" by University of Michigan

Final Answer

The final answer is: $\boxed{y = 8e^{x^3}}$