Writing the Equation of the Tangent Line

Oct 13, 2025 by windailys.com 109 views

Introduction

In calculus, finding the equation of a tangent line to a curve at a given point is a fundamental concept. This involves using the concept of derivatives to determine the slope of the tangent line. In this article, we will explore how to find the equation of the line tangent to the graph of $G(x) = 6e^{-x}$ at the point $(0, 6)$ .

Understanding the Problem

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

Find the derivative of the function $G(x) = 6e^{-x}$ .
Evaluate the derivative at the point $x = 0$ to find the slope of the tangent line.
Use the point-slope form of a line to write the equation of the tangent line.

Finding the Derivative of $G(x)$

The derivative of a function $f(x)$ is denoted as $f'(x)$ and represents the rate of change of the function with respect to $x$ . To find the derivative of $G(x) = 6e^{-x}$ , we will use the chain rule.

The chain rule states that if we have a composite function of the form $f(g(x))$ , then the derivative of the composite function is given by:

\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)

In this case, we have $G(x) = 6e^{-x}$ , which can be written as $G(x) = 6 \cdot e^{-x}$ . Here, $f(u) = 6u$ and $u = e^{-x}$ .

Using the chain rule, we get:

G'(x) = f'(u) \cdot u'

G'(x) = 6 \cdot e^{-x} \cdot (-e^{-x})

G'(x) = -6e^{-2x}

Evaluating the Derivative at $x = 0$

Now that we have found the derivative of $G(x)$ , we need to evaluate it at the point $x = 0$ to find the slope of the tangent line.

G'(0) = -6e^{-2(0)}

G'(0) = -6e^{0}

G'(0) = -6 \cdot 1

G'(0) = -6

Writing the Equation of the Tangent Line

Now that we have found the slope of the tangent line, we can use the point-slope form of a line to write 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 a point on the line and $m$ is the slope.

In this case, we have the point $(0, 6)$ and the slope $m = -6$ . Plugging these values into the point-slope form, we get:

y - 6 = -6(x - 0)

y - 6 = -6x

y = -6x + 6

Conclusion

In this article, we found the equation of the line tangent to the graph of $G(x) = 6e^{-x}$ at the point $(0, 6)$ . 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 write the equation of the tangent line.

The equation of the tangent line is given by:

y = -6x + 6

This equation represents the line that is tangent to the graph of $G(x) = 6e^{-x}$ at the point $(0, 6)$ .

Applications of Tangent Lines

Tangent lines have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of tangent lines include:

Optimization: Tangent lines are used to find the maximum or minimum value of a function.
Physics: Tangent lines are used to describe the motion of objects and to find the velocity and acceleration of an object.
Engineering: Tangent lines are used to design and optimize systems, such as bridges and buildings.
Economics: Tangent lines are used to analyze the behavior of economic systems and to make predictions about future trends.

Final Thoughts

In conclusion, finding the equation of a tangent line to a curve at a given point is a fundamental concept in calculus. 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 write the equation of the tangent line. The equation of the tangent line is given by:

y = -6x + 6

This equation represents the line that is tangent to the graph of $G(x) = 6e^{-x}$ at the point $(0, 6)$ .

We hope that this article has provided a clear and concise explanation of how to find the equation of a tangent line to a curve at a given point. If you have any questions or need further clarification, please don't hesitate to ask.

Introduction

In our previous article, we explored how to find the equation of a tangent line to a curve at a given point. 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 write the equation of the tangent line.

In this article, we will answer some frequently asked questions about finding the equation of a tangent line to a curve.

Q1: What is the significance of the derivative in finding the equation of a tangent line?

A1: The derivative of a function represents the rate of change of the function with respect to the variable. In the context of finding the equation of a tangent line, the derivative is used to determine the slope of the tangent line.

Q2: How do I find the derivative of a function?

A2: To find the derivative of a function, you can use the power rule, the product rule, and the chain rule. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x). The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).

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

A3: The point-slope form of a line is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Q4: How do I use the point-slope form to write the equation of a tangent line?

A4: To use the point-slope form to write the equation of a tangent line, you need to know the slope of the tangent line and a point on the line. The slope of the tangent line can be found by evaluating the derivative of the function at the point of tangency. Once you have the slope and a point on the line, you can plug these values into the point-slope form to write the equation of the tangent line.

Q5: What are some common applications of tangent lines?

A5: Tangent lines have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of tangent lines include optimization, physics, engineering, and economics.

Q6: How do I find the equation of a tangent line to a curve at a given point?

A6: To find the equation of a tangent line to a curve at a given point, you need to follow these steps:

Find the derivative of the function.
Evaluate the derivative at the point of tangency to find the slope of the tangent line.
Use the point-slope form of a line to write the equation of the tangent line.

Q7: What is the equation of a tangent line to a curve at a given point?

A7: The equation of a tangent line to a curve at a given point is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Q8: How do I determine the slope of a tangent line?

A8: To determine the slope of a tangent line, you need to evaluate the derivative of the function at the point of tangency.

Q9: What is the significance of the point of tangency in finding the equation of a tangent line?

A9: The point of tangency is the point on the curve where the tangent line touches the curve. The point of tangency is used to find the equation of the tangent line.

Q10: How do I use the equation of a tangent line to make predictions about the behavior of a function?

A10: To use the equation of a tangent line to make predictions about the behavior of a function, you need to analyze the slope of the tangent line. If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing.

Conclusion

In this article, we have answered some frequently asked questions about finding the equation of a tangent line to a curve. We have discussed the significance of the derivative in finding the equation of a tangent line, how to find the derivative of a function, and how to use the point-slope form to write the equation of a tangent line.

We hope that this article has provided a clear and concise explanation of how to find the equation of a tangent line to a curve. If you have any questions or need further clarification, please don't hesitate to ask.

Final Thoughts

Finding the equation of a tangent line to a curve at a given point is a fundamental concept in calculus. We have used the concept of derivatives to determine the slope of the tangent line and then used the point-slope form of a line to write the equation of the tangent line.

Additional Resources

If you are interested in learning more about finding the equation of a tangent line to a curve, we recommend the following resources:

Calculus textbooks: There are many excellent calculus textbooks that cover the topic of finding the equation of a tangent line to a curve.
Online resources: There are many online resources, such as Khan Academy and MIT OpenCourseWare, that provide video lectures and practice problems on finding the equation of a tangent line to a curve.
Calculus courses: If you are interested in learning more about finding the equation of a tangent line to a curve, we recommend taking a calculus course.