Find An Equation Of The Tangent Line To The Graph Of $f(x)=\frac{(x-1)}{(x+1)}$ When $x=1$.A. Y = 2 ( X + 1 ) 2 ( X − 1 Y=\frac{2}{(x+1)^2}(x-1 Y = ( X + 1 ) 2 2 ​ ( X − 1 ] B. X − 2 Y = 1 X-2y=1 X − 2 Y = 1 C. Y = 2 ( X − 1 Y=2(x-1 Y = 2 ( X − 1 ] D. Y = − X + 1 Y=-x+1 Y = − X + 1

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Introduction

In calculus, the concept of a tangent line is crucial in understanding the behavior of functions. A tangent line is a line that touches a curve at a single point, and its slope is equal to the derivative of the function at that point. In this article, we will explore how to find the equation of a tangent line to the graph of a given function, specifically the function $f(x)=\frac{(x-1)}{(x+1)}$ when $x=1$.

Understanding the Function

The given function is $f(x)=\frac{(x-1)}{(x+1)}$. This is a rational function, which means it is the ratio of two polynomials. The function has a vertical asymptote at $x=-1$, where the denominator becomes zero. The function also has a hole at $x=1$, where both the numerator and denominator become zero.

Finding the Derivative

To find the equation of the tangent line, we need to find the derivative of the function. The derivative of a function represents the rate of change of the function with respect to the variable. In this case, we need to find the derivative of $f(x)=\frac{(x-1)}{(x+1)}$.

To find the derivative, we can use the quotient rule, which states that if $f(x)=\frac{g(x)}{h(x)}$, then $f'(x)=\frac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}$. In this case, $g(x)=x-1$ and $h(x)=x+1$.

Applying the Quotient Rule

Using the quotient rule, we can find the derivative of $f(x)=\frac{(x-1)}{(x+1)}$.

f′(x)=(x+1)(1)−(x−1)(1)(x+1)2f'(x)=\frac{(x+1)(1)-(x-1)(1)}{(x+1)^2}

Simplifying the expression, we get:

f′(x)=x+1−x+1(x+1)2f'(x)=\frac{x+1-x+1}{(x+1)^2}

f′(x)=2(x+1)2f'(x)=\frac{2}{(x+1)^2}

Finding the Slope

Now that we have the derivative, we can find the slope of the tangent line at $x=1$. The slope of the tangent line is equal to the derivative of the function at that point.

f′(1)=2(1+1)2f'(1)=\frac{2}{(1+1)^2}

f′(1)=24f'(1)=\frac{2}{4}

f′(1)=12f'(1)=\frac{1}{2}

Finding the Equation of the Tangent Line

Now that we have the slope, we can find the equation of the tangent line. The equation of a line is given by $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.

We know that the slope of the tangent line is $\frac{1}{2}$, and we want to find the equation of the tangent line at $x=1$. To do this, we need to find the y-coordinate of the point on the curve at $x=1$.

y=f(1)=(1−1)(1+1)y=f(1)=\frac{(1-1)}{(1+1)}

y=0y=0

Now that we have the y-coordinate, we can find the equation of the tangent line.

y=12(x−1)+0y=\frac{1}{2}(x-1)+0

y=12(x−1)y=\frac{1}{2}(x-1)

Conclusion

In this article, we explored how to find the equation of a tangent line to the graph of a given function. We used the quotient rule to find the derivative of the function, and then used the derivative to find the slope of the tangent line at a given point. We then used the slope and the point on the curve to find the equation of the tangent line.

The final answer is: C. y=12(x−1)y=\frac{1}{2}(x-1)

Introduction

In our previous article, we explored how to find the equation of a tangent line to the graph of a given function. In this article, we will answer some common questions related to tangent line equations.

Q: What is the tangent line equation?

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

Q: How do I find the tangent line equation?

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

  1. Find the derivative of the function.
  2. Evaluate the derivative at the given point.
  3. Use the point-slope form of a line to find the equation of the tangent line.

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

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

Q: What is the significance of the tangent line equation?

A: The tangent line equation is significant because it allows us to approximate the behavior of a function at a given point. It is used in many applications, including physics, engineering, and economics.

Q: Can I use the tangent line equation to find the maximum or minimum of a function?

A: Yes, you can use the tangent line equation to find the maximum or minimum of a function. If the slope of the tangent line is zero, then the function has a maximum or minimum at that point.

Q: How do I find the equation of the tangent line to a curve at a given point?

A: To find the equation of the tangent line to a curve at a given point, you need to follow these steps:

  1. Find the derivative of the function.
  2. Evaluate the derivative at the given point.
  3. Use the point-slope form of a line to find the equation of the tangent line.

Q: Can I use the tangent line equation to find the equation of a curve?

A: No, you cannot use the tangent line equation to find the equation of a curve. The tangent line equation is used to approximate the behavior of a function at a given point, but it is not used to find the equation of a curve.

Q: What is the difference between the tangent line equation and the equation of a curve?

A: The tangent line equation is used to approximate the behavior of a function at a given point, while the equation of a curve is used to describe the entire curve.

Conclusion

In this article, we answered some common questions related to tangent line equations. We hope that this article has been helpful in understanding the concept of tangent line equations and how to use them to approximate the behavior of a function at a given point.

Frequently Asked Questions

  • Q: What is the tangent line equation? A: The tangent line equation is a line that touches a curve at a single point.
  • Q: How do I find the tangent line equation? A: To find the tangent line equation, you need to follow these steps: 1. Find the derivative of the function. 2. Evaluate the derivative at the given point. 3. Use the point-slope form of a line to find the equation of the tangent line.
  • Q: What is the significance of the tangent line equation? A: The tangent line equation is significant because it allows us to approximate the behavior of a function at a given point.

Common Mistakes

  • Mistake 1: Not evaluating the derivative at the given point.
  • Mistake 2: Not using the point-slope form of a line to find the equation of the tangent line.
  • Mistake 3: Not checking if the slope of the tangent line is zero.

Tips and Tricks

  • Tip 1: Make sure to evaluate the derivative at the given point.
  • Tip 2: Use the point-slope form of a line to find the equation of the tangent line.
  • Tip 3: Check if the slope of the tangent line is zero.