Find An Equation Of The Line Tangent To $y = X + \frac{4}{x}$ At The Point (4, 5).

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Introduction

In calculus, finding the equation of a tangent line to a curve at a given point is a fundamental concept. It involves using the concept of derivatives to determine the slope of the tangent line and then using the point-slope form of a line to find the equation of the tangent line. In this article, we will find the equation of the line tangent to the curve $y = x + \frac{4}{x}$ at the point (4, 5).

Understanding the Curve

The given curve is $y = x + \frac{4}{x}$. To find the equation of the tangent line, we need to find the derivative of the curve, which represents the slope of the tangent line at any point on the curve.

Finding the Derivative

To find the derivative of the curve $y = x + \frac{4}{x}$, we will use the power rule and the quotient rule of differentiation.

The derivative of $x$ is 1, and the derivative of $\frac{4}{x}$ is $\frac{-4}{x^2}$.

Using the sum rule of differentiation, we can write the derivative of the curve as:

$$\frac{dy}{dx} = 1 - \frac{4}{x^2}$$

Finding the Slope of the Tangent Line

To find the slope of the tangent line at the point (4, 5), we need to substitute $x = 4$ into the derivative of the curve.

$$\frac{dy}{dx} = 1 - \frac{4}{4^2} = 1 - \frac{4}{16} = 1 - \frac{1}{4} = \frac{3}{4}$$

Finding the Equation of the Tangent Line

Now that we have the slope of the tangent line, 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 point on the line, and $m$ is the slope of the line.

Substituting the values of the point (4, 5) and the slope $\frac{3}{4}$ into the point-slope form, we get:

$$y - 5 = \frac{3}{4}(x - 4)$$

Simplifying the Equation

To simplify the equation, we can multiply both sides by 4 to eliminate the fraction.

$$4(y - 5) = 3(x - 4)$$

Expanding the left-hand side, we get:

$$4y - 20 = 3x - 12$$

Adding 20 to both sides, we get:

$$4y = 3x + 8$$

Dividing both sides by 4, we get:

$$y = \frac{3}{4}x + 2$$

Conclusion

In this article, we found the equation of the line tangent to the curve $y = x + \frac{4}{x}$ at the point (4, 5). We used the concept of derivatives to determine the slope of the tangent line and then used the point-slope form of a line to find the equation of the tangent line. The final equation of the tangent line is $y = \frac{3}{4}x + 2$.

Final Answer

The final answer is $y = \frac{3}{4}x + 2$.

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Introduction

In our previous article, we found the equation of the line tangent to the curve $y = x + \frac{4}{x}$ at the point (4, 5). In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q1: What is the derivative of the curve $y = x + \frac{4}{x}$?

A1: The derivative of the curve $y = x + \frac{4}{x}$ is $\frac{dy}{dx} = 1 - \frac{4}{x^2}$.

Q2: How do you find the slope of the tangent line at a given point on the curve?

A2: To find the slope of the tangent line at a given point on the curve, you need to substitute the x-coordinate of the point into the derivative of the curve.

Q3: What is the point-slope form of a line?

A3: The point-slope form of a line is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point on the line, and $m$ is the slope of the line.

Q4: How do you find the equation of the tangent line using the point-slope form?

A4: To find the equation of the tangent line using the point-slope form, you need to substitute the values of the point and the slope into the point-slope form and simplify the equation.

Q5: What is the final equation of the tangent line to the curve $y = x + \frac{4}{x}$ at the point (4, 5)?

A5: The final equation of the tangent line to the curve $y = x + \frac{4}{x}$ at the point (4, 5) is $y = \frac{3}{4}x + 2$.

Q6: How do you use the concept of derivatives to determine the slope of the tangent line?

A6: You use the concept of derivatives to determine the slope of the tangent line by finding the derivative of the curve and substituting the x-coordinate of the point into the derivative.

Q7: What is the significance of finding the equation of the tangent line to a curve?

A7: Finding the equation of the tangent line to a curve is significant because it helps us understand the behavior of the curve at a given point and can be used to make predictions about the curve's behavior.

Q8: How do you simplify the equation of the tangent line?

A8: You simplify the equation of the tangent line by multiplying both sides by a common factor, adding or subtracting the same value to both sides, or dividing both sides by a common factor.

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of the line tangent to the curve $y = x + \frac{4}{x}$ at the point (4, 5). We hope that this article has been helpful in clarifying any doubts you may have had about the topic.

Final Answer

The final answer is $y = \frac{3}{4}x + 2$.