Find The Equation Of The Tangent Line To The Graph Of The Given Equation At The Indicated Point.Equation: $x^2 Y^2 + 3xy = 10$Point: $(2, 1$\]

Introduction

In mathematics, finding the equation of the tangent line to a given equation at a specific point is a crucial concept in calculus and differential equations. The tangent line is a line that just touches the curve at a given point, and it is used to approximate the behavior of the curve near that point. In this article, we will discuss how to find the equation of the tangent line to the graph of the given equation $x^2 y^2 + 3xy = 10$ at the point $(2, 1)$.

Understanding the Problem

To find the equation of the tangent line, we need to follow these steps:

1. Find the derivative of the given equation: We need to find the derivative of the given equation with respect to $x$.
2. Evaluate the derivative at the given point: We need to evaluate the derivative at the given point $(2, 1)$.
3. Use the point-slope form of a line: We need to use the point-slope form of a line to find the equation of the tangent line.

Finding the Derivative of the Given Equation

To find the derivative of the given equation, we can use implicit differentiation. Implicit differentiation is a technique used to find the derivative of an equation that is not in the form $y = f(x)$. We will differentiate both sides of the equation with respect to $x$.

Let's start by differentiating the given equation:

$$\frac{d}{dx} (x^2 y^2 + 3xy) = \frac{d}{dx} (10)$$

Using the product rule and the chain rule, we get:

$$2xy^2 + 2x^2 y \frac{dy}{dx} + 3y + 3x \frac{dy}{dx} = 0$$

Now, we can solve for $\frac{dy}{dx}$:

$$2x^2 y \frac{dy}{dx} + 3x \frac{dy}{dx} = -2xy^2 - 3y$$

Factoring out $\frac{dy}{dx}$, we get:

$$(2x^2 y + 3x) \frac{dy}{dx} = -2xy^2 - 3y$$

Now, we can solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx} = \frac{-2xy^2 - 3y}{2x^2 y + 3x}$$

Evaluating the Derivative at the Given Point

Now that we have the derivative, we can evaluate it at the given point $(2, 1)$.

Substituting $x = 2$ and $y = 1$ into the derivative, we get:

$$\frac{dy}{dx} = \frac{-2(2)(1)^2 - 3(1)}{2(2)^2 (1) + 3(2)}$$

Simplifying, we get:

$$\frac{dy}{dx} = \frac{-4 - 3}{8 + 6}$$

$$\frac{dy}{dx} = \frac{-7}{14}$$

$$\frac{dy}{dx} = -\frac{1}{2}$$

Using the Point-Slope Form of a Line

Now that we have the derivative, we can use the point-slope form of a line to find the equation of the tangent line.

The point-slope form of a line is given by:

$$y - y_1 = m(x - x_1)$$

where $(x_1, y_1)$ is the given point and $m$ is the slope of the line.

In this case, the given point is $(2, 1)$ and the slope is $-\frac{1}{2}$.

Substituting these values into the point-slope form, we get:

$$y - 1 = -\frac{1}{2}(x - 2)$$

Simplifying, we get:

$$y - 1 = -\frac{1}{2}x + 1$$

$$y = -\frac{1}{2}x + 2$$

Conclusion

In this article, we discussed how to find the equation of the tangent line to the graph of the given equation $x^2 y^2 + 3xy = 10$ at the point $(2, 1)$. We used implicit differentiation to find the derivative of the given equation, evaluated the derivative at the given point, and used the point-slope form of a line to find the equation of the tangent line. The equation of the tangent line is $y = -\frac{1}{2}x + 2$.

Example Problems

Here are some example problems that you can try to practice finding the equation of the tangent line:

• Find the equation of the tangent line to the graph of the equation $x^2 y^2 + 2xy = 5$ at the point $(1, 2)$.
• Find the equation of the tangent line to the graph of the equation $y^2 + 2xy = 3$ at the point $(2, 1)$.
• Find the equation of the tangent line to the graph of the equation $x^2 y^2 + 3xy = 10$ at the point $(3, 2)$.

Applications of Finding the Equation of the Tangent Line

Finding the equation of the tangent line has many applications in mathematics and science. Some of the applications include:

• Physics: Finding the equation of the tangent line is used to model the motion of objects in physics.
• Engineering: Finding the equation of the tangent line is used to design and optimize systems in engineering.
• Economics: Finding the equation of the tangent line is used to model the behavior of economic systems.

Conclusion

In conclusion, finding the equation of the tangent line is a crucial concept in mathematics and science. It has many applications in physics, engineering, and economics. By following the steps outlined in this article, you can find the equation of the tangent line to any given equation at any point.

Introduction

Finding the equation of the tangent line is a crucial concept in mathematics and science. However, it can be a challenging topic for many students and professionals. In this article, we will answer some of the most frequently asked questions (FAQs) about finding the equation of the tangent line.

Q: What is the tangent line?

A: The tangent line is a line that just touches a curve at a given point. It is used to approximate the behavior of the curve near that point.

Q: Why is finding the equation of the tangent line important?

A: Finding the equation of the tangent line is important because it has many applications in physics, engineering, and economics. It is used to model the motion of objects, design and optimize systems, and understand the behavior of economic systems.

Q: How do I find the equation of the tangent line?

A: To find the equation of the tangent line, you need to follow these steps:

1. Find the derivative of the given equation: Use implicit differentiation to find the derivative of the given equation with respect to x.
2. Evaluate the derivative at the given point: Evaluate the derivative at the given point (x, y).
3. Use the point-slope form of a line: Use the point-slope form of a line to find the equation of the tangent line.

Q: What is implicit differentiation?

A: Implicit differentiation is a technique used to find the derivative of an equation that is not in the form y = f(x). It involves differentiating both sides of the equation with respect to x.

Q: How do I use the point-slope form of a line?

A: To use the point-slope form of a line, you need to substitute the given point (x, y) and the slope (m) into the equation:

y - y1 = m(x - x1)

Q: What is the point-slope form of a line?

A: The point-slope form of a line is given by:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

Q: How do I find the slope of the tangent line?

A: To find the slope of the tangent line, you need to evaluate the derivative at the given point (x, y).

Q: What is the equation of the tangent line?

A: The equation of the tangent line is given by:

y - y1 = m(x - x1)

where (x1, y1) is the given point and m is the slope of the line.

Q: Can I use the equation of the tangent line to approximate the behavior of the curve?

A: Yes, you can use the equation of the tangent line to approximate the behavior of the curve near the given point.

Q: What are some common mistakes to avoid when finding the equation of the tangent line?

A: Some common mistakes to avoid when finding the equation of the tangent line include:

• Not using implicit differentiation: Make sure to use implicit differentiation to find the derivative of the given equation.
• Not evaluating the derivative at the given point: Make sure to evaluate the derivative at the given point (x, y).
• Not using the point-slope form of a line: Make sure to use the point-slope form of a line to find the equation of the tangent line.

Conclusion

In conclusion, finding the equation of the tangent line is a crucial concept in mathematics and science. By following the steps outlined in this article and avoiding common mistakes, you can find the equation of the tangent line to any given equation at any point.

Additional Resources

If you need additional help or resources, here are some suggestions:

• Textbooks: Check out textbooks on calculus and differential equations for more information on finding the equation of the tangent line.
• Online resources: Check out online resources such as Khan Academy, MIT OpenCourseWare, and Wolfram Alpha for more information on finding the equation of the tangent line.
• Practice problems: Practice finding the equation of the tangent line with practice problems and exercises.

Final Thoughts

Finding the equation of the tangent line is a challenging but rewarding topic. By following the steps outlined in this article and avoiding common mistakes, you can find the equation of the tangent line to any given equation at any point. Remember to practice regularly and seek help when needed. Good luck!