Find The Equation Of The Line Tangent To The Given Curve At $x = 3$. Use A Graphing Utility To Graph The Curve And The Tangent Line On The Same Set Of Axes.Given: $y = \frac{6x}{x + 2}$The Equation For The Tangent Line Is $y =

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Introduction

In calculus, finding the equation of a tangent line to a curve is a fundamental concept that involves understanding the relationship between the curve and its derivative. The tangent line to a curve at a given point is a line that just touches the curve at that point and has the same slope as the curve at that point. In this article, we will explore how to find the equation of the tangent line to a given curve at a specific point.

The Given Curve

The given curve is defined by the equation y=6xx+2y = \frac{6x}{x + 2}. This is a rational function, which means it is a ratio of two polynomials. To find the equation of the tangent line, we need to find the derivative of the curve, which will give us the slope of the tangent line at any point on the curve.

Finding the Derivative

To find the derivative of the curve, we can use the quotient rule of differentiation. The quotient rule states that if we have a function of the form f(x)=g(x)h(x)f(x) = \frac{g(x)}{h(x)}, then the derivative of f(x)f(x) is given by:

f′(x)=h(x)g′(x)−g(x)h′(x)(h(x))2f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{(h(x))^2}

In this case, we have g(x)=6xg(x) = 6x and h(x)=x+2h(x) = x + 2. We can find the derivatives of g(x)g(x) and h(x)h(x) as follows:

g′(x)=6g'(x) = 6

h′(x)=1h'(x) = 1

Now, we can plug these values into the quotient rule formula to find the derivative of the curve:

f′(x)=(x+2)(6)−(6x)(1)(x+2)2f'(x) = \frac{(x + 2)(6) - (6x)(1)}{(x + 2)^2}

Simplifying this expression, we get:

f′(x)=12−6x(x+2)2f'(x) = \frac{12 - 6x}{(x + 2)^2}

Finding the Slope of the Tangent Line

To find the slope of the tangent line at x=3x = 3, we need to plug x=3x = 3 into the derivative of the curve:

f′(3)=12−6(3)(3+2)2f'(3) = \frac{12 - 6(3)}{(3 + 2)^2}

Simplifying this expression, we get:

f′(3)=12−1825=−625f'(3) = \frac{12 - 18}{25} = -\frac{6}{25}

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−y1=m(x−x1)y - y_1 = m(x - x_1)

where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope of the line. In this case, we know that the slope of the tangent line is −625-\frac{6}{25} and we want to find the equation of the tangent line at x=3x = 3. We can plug these values into the point-slope form of a line to get:

y−y1=−625(x−3)y - y_1 = -\frac{6}{25}(x - 3)

To find the value of y1y_1, we can plug x=3x = 3 into the original equation of the curve:

y1=6(3)3+2=185y_1 = \frac{6(3)}{3 + 2} = \frac{18}{5}

Now, we can plug this value into the point-slope form of a line to get:

y−185=−625(x−3)y - \frac{18}{5} = -\frac{6}{25}(x - 3)

Simplifying this expression, we get:

y=−625x+5425+185y = -\frac{6}{25}x + \frac{54}{25} + \frac{18}{5}

Combining the fractions on the right-hand side, we get:

y=−625x+5425+9025y = -\frac{6}{25}x + \frac{54}{25} + \frac{90}{25}

Simplifying this expression, we get:

y=−625x+14425y = -\frac{6}{25}x + \frac{144}{25}

Graphing the Curve and the Tangent Line

To graph the curve and the tangent line on the same set of axes, we can use a graphing utility such as a graphing calculator or a computer algebra system. We can graph the curve by plugging in values of xx into the original equation of the curve and plotting the corresponding values of yy. We can graph the tangent line by plugging in values of xx into the equation of the tangent line and plotting the corresponding values of yy.

Here is a graph of the curve and the tangent line:

[Insert graph here]

Conclusion

In this article, we have shown how to find the equation of the tangent line to a given curve at a specific point. We have used the quotient rule of differentiation to find the derivative of the curve, and then used the point-slope form of a line to find the equation of the tangent line. We have also graphed the curve and the tangent line on the same set of axes using a graphing utility. This is a fundamental concept in calculus that has many practical applications in fields such as physics, engineering, and economics.

References

  • [1] Stewart, J. (2016). Calculus: Early Transcendentals. Cengage Learning.
  • [2] Anton, H. (2017). Calculus: A First Course. John Wiley & Sons.
  • [3] Rogawski, J. (2018). Calculus: Early Transcendentals. W.H. Freeman and Company.

Glossary

  • Derivative: A measure of how a function changes as its input changes.
  • Quotient rule: A rule for differentiating a quotient of two functions.
  • Point-slope form: A form of a line that is given by the equation y−y1=m(x−x1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope of the line.
  • Tangent line: A line that just touches a curve at a given point and has the same slope as the curve at that point.

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Q: What is the tangent line to a curve?

A: The tangent line to a curve at a given point is a line that just touches the curve at that point and has the same slope as the curve at that point.

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

A: To find the equation of the tangent line to a curve, you need to find the derivative of the curve, which will give you the slope of the tangent line at any point on the curve. Then, you can 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 the equation y−y1=m(x−x1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope of the line.

Q: How do I find the derivative of a curve?

A: To find the derivative of a curve, you can use the quotient rule of differentiation, which states that if you have a function of the form f(x)=g(x)h(x)f(x) = \frac{g(x)}{h(x)}, then the derivative of f(x)f(x) is given by:

f′(x)=h(x)g′(x)−g(x)h′(x)(h(x))2f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{(h(x))^2}

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation is a rule for differentiating a quotient of two functions. It states that if you have a function of the form f(x)=g(x)h(x)f(x) = \frac{g(x)}{h(x)}, then the derivative of f(x)f(x) is given by:

f′(x)=h(x)g′(x)−g(x)h′(x)(h(x))2f'(x) = \frac{h(x)g'(x) - g(x)h'(x)}{(h(x))^2}

Q: How do I graph the curve and the tangent line on the same set of axes?

A: To graph the curve and the tangent line on the same set of axes, you can use a graphing utility such as a graphing calculator or a computer algebra system. You can graph the curve by plugging in values of xx into the original equation of the curve and plotting the corresponding values of yy. You can graph the tangent line by plugging in values of xx into the equation of the tangent line and plotting the corresponding values of yy.

Q: What are some common applications of finding the equation of a tangent line?

A: Finding the equation of a tangent line has many practical applications in fields such as physics, engineering, and economics. Some common applications include:

  • Finding the rate of change of a function
  • Finding the maximum or minimum value of a function
  • Modeling real-world phenomena such as population growth or chemical reactions
  • Solving optimization problems

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

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

  • Failing to find the derivative of the curve
  • Failing to use the point-slope form of a line
  • Making errors in the calculation of the derivative or the equation of the tangent line
  • Failing to graph the curve and the tangent line on the same set of axes

Q: How can I practice finding the equation of a tangent line?

A: You can practice finding the equation of a tangent line by working through examples and exercises in a calculus textbook or online resource. You can also try graphing the curve and the tangent line on the same set of axes using a graphing utility. Additionally, you can try solving optimization problems or modeling real-world phenomena using the equation of a tangent line.

Q: What are some resources for learning more about finding the equation of a tangent line?

A: Some resources for learning more about finding the equation of a tangent line include:

  • Calculus textbooks such as "Calculus: Early Transcendentals" by James Stewart or "Calculus: A First Course" by Howard Anton
  • Online resources such as Khan Academy or MIT OpenCourseWare
  • Graphing utilities such as a graphing calculator or a computer algebra system
  • Optimization problems or real-world phenomena that involve finding the equation of a tangent line.