Consider F ( X ) = 5 E X − 6 F(x) = 5e^x - 6 F ( X ) = 5 E X − 6 . Find The Equation Of The Line Tangent To The Curve At The Point ( 0 , 1 (0, 1 ( 0 , 1 ].

Introduction

In calculus, the concept of a tangent line to a curve is crucial in understanding the behavior of functions. Given a curve defined by a function, the tangent line at a specific point on the curve represents the instantaneous rate of change of the function at that point. In this article, we will explore how to find the equation of the tangent line to a curve defined by the function $f(x) = 5e^x - 6$ at the point $(0, 1)$.

Understanding the Function

The given function is $f(x) = 5e^x - 6$. This is an exponential function, where the base is $e$ (approximately 2.71828). The function has a constant coefficient of 5, which affects the rate of growth of the function. The term $-6$ is a vertical shift, indicating that the function is shifted down by 6 units.

Finding the Derivative

To find the equation of the tangent line, we need to find the derivative of the function, which represents the rate of change of the function with respect to $x$. The derivative of $f(x) = 5e^x - 6$ can be found using the chain rule and the fact that the derivative of $e^x$ is $e^x$.

$$f'(x) = \frac{d}{dx} (5e^x - 6) = 5 \cdot e^x = 5e^x$$

Evaluating the Derivative at the Given Point

Now that we have the derivative, we need to evaluate it at the point $(0, 1)$ to find the slope of the tangent line. Substituting $x = 0$ into the derivative, we get:

$$f'(0) = 5e^0 = 5 \cdot 1 = 5$$

Finding the Equation of the Tangent Line

The equation of a line in point-slope form 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. In this case, the point is $(0, 1)$, and the slope is $5$. Substituting these values into the equation, we get:

$$y - 1 = 5(x - 0)$$

Simplifying the equation, we get:

$$y - 1 = 5x$$

Adding 1 to both sides, we get:

$$y = 5x + 1$$

Conclusion

In this article, we found the equation of the tangent line to the curve defined by the function $f(x) = 5e^x - 6$ at the point $(0, 1)$. We first found the derivative of the function, which represents the rate of change of the function with respect to $x$. We then evaluated the derivative at the given point to find the slope of the tangent line. Finally, we used the point-slope form of a line to find the equation of the tangent line.

Key Takeaways

• The derivative of a function represents the rate of change of the function with respect to $x$.
• The equation of a tangent line to a curve can be found using the point-slope form of a line.
• The slope of the tangent line is equal to the derivative of the function evaluated at the given point.

Further Reading

For more information on derivatives and tangent lines, see the following resources:

• Derivative Rules
• Tangent Line
• Calculus

References

• Calculus by Michael Spivak
• Calculus by James Stewart

Glossary

• Derivative: A measure of the rate of change of a function with respect to $x$.
• Tangent Line: A line that touches a curve at a single point.
• Point-Slope Form: A form of a line that uses the point on the line and the slope to define the line.
  Q&A: Finding the Equation of a Tangent Line to a Curve =====================================================

Introduction

In our previous article, we explored how to find the equation of the tangent line to a curve defined by the function $f(x) = 5e^x - 6$ at the point $(0, 1)$. In this article, we will answer some common questions related to finding the equation of a tangent line to a curve.

Q: What is the purpose of finding the equation of a tangent line to a curve?

A: The purpose of finding the equation of a tangent line to a curve is to understand the behavior of the function at a specific point. The tangent line represents the instantaneous rate of change of the function at that point.

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

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

1. Find the derivative of the function, which represents the rate of change of the function with respect to $x$.
2. Evaluate the derivative at the given point to find the slope of the tangent line.
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 $(x_1, y_1)$ is the point on the line, and $m$ is the slope.

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. The derivative represents the rate of change of the function with respect to $x$.

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

A: The derivative is a crucial component in finding the equation of a tangent line. It represents the rate of change of the function with respect to $x$, which is essential in determining the slope of the tangent line.

Q: Can I find the equation of a tangent line to a curve using other methods?

A: Yes, you can find the equation of a tangent line to a curve using other methods, such as using the limit definition of a derivative or using implicit differentiation. However, the point-slope form of a line is a common and efficient method for finding the equation of a tangent line.

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

A: Finding the equation of a tangent line to a curve has numerous applications in various fields, including physics, engineering, and economics. Some common applications include:

• Modeling the behavior of physical systems, such as the motion of an object under the influence of gravity or friction.
• Analyzing the behavior of economic systems, such as the relationship between supply and demand.
• Optimizing the performance of systems, such as the design of a mechanical system or the optimization of a financial portfolio.

Conclusion

In this article, we answered some common questions related to finding the equation of a tangent line to a curve. We hope that this article has provided you with a better understanding of the concept and its applications. If you have any further questions or need additional clarification, please don't hesitate to ask.

Key Takeaways

• The derivative of a function represents the rate of change of the function with respect to $x$.
• The equation of a tangent line to a curve can be found using the point-slope form of a line.
• The slope of the tangent line is equal to the derivative of the function evaluated at the given point.

Further Reading

For more information on derivatives and tangent lines, see the following resources:

• Derivative Rules
• Tangent Line
• Calculus

References

• Calculus by Michael Spivak
• Calculus by James Stewart

Glossary

• Derivative: A measure of the rate of change of a function with respect to $x$.
• Tangent Line: A line that touches a curve at a single point.
• Point-Slope Form: A form of a line that uses the point on the line and the slope to define the line.