Find The Equation Of The Line Tangent To The Graph Of F ( X ) = − 2 Csc ⁡ ( X ) − 4 3 3 − 4 Π 9 − 1 F(x)=-2 \csc (x)-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1 F ( X ) = − 2 Csc ( X ) − 3 4 3 − 9 4 Π − 1 At X = − Π 3 X=-\frac{\pi}{3} X = − 3 Π . Provide Your Answer Below:

Mar 13, 2025 by ADMIN 277 views

**Finding the Equation of a Tangent Line to a Trigonometric Function**

Introduction

In calculus, finding the equation of a tangent line to a function at a given point is a fundamental concept. This involves using the concept of limits and derivatives to determine the slope of the tangent line. In this article, we will explore how to find the equation of the tangent line to the graph of the function $f(x)=-2 \csc (x)-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$ at the point $x=-\frac{\pi}{3}$ .

Understanding the Function

The given function is $f(x)=-2 \csc (x)-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$ . This function involves the cosecant function, which is the reciprocal of the sine function. The cosecant function is defined as $\csc (x)=\frac{1}{\sin (x)}$ . Therefore, the given function can be rewritten as $f(x)=-2 \frac{1}{\sin (x)}-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$ .

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 $f(x)=-2 \frac{1}{\sin (x)}-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$ .

Using the chain rule and the fact that the derivative of $\csc (x)$ is $-\csc (x) \cot (x)$ , we can find the derivative of the function as follows:

$f'(x)=2 \csc (x) \cot (x)$

Evaluating the Derivative at the Given Point

Now that we have the derivative of the function, we need to evaluate it at the given point $x=-\frac{\pi}{3}$ . Substituting $x=-\frac{\pi}{3}$ into the derivative, we get:

$f'(-\frac{\pi}{3})=2 \csc (-\frac{\pi}{3}) \cot (-\frac{\pi}{3})$

Using the fact that $\csc (-x)=-\csc (x)$ and $\cot (-x)=-\cot (x)$ , we can simplify the expression as follows:

$f'(-\frac{\pi}{3})=-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})$

Finding the Slope of the Tangent Line

The slope of the tangent line is given by the value of the derivative at the given point. In this case, the slope of the tangent line is:

$m=f'(-\frac{\pi}{3})=-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})$

Finding the Equation of the Tangent Line

Now that we have the slope of the tangent line, we can find the equation of the tangent line using the point-slope form of a linear equation. The point-slope form of a linear equation is given by:

$y-y_1=m(x-x_1)$

In this case, the point $(x_1,y_1)$ is $(-\frac{\pi}{3},f(-\frac{\pi}{3}))$ . Substituting the values of $m$ and $(x_1,y_1)$ into the point-slope form, we get:

$y-f(-\frac{\pi}{3})=-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})(x+\frac{\pi}{3})$

Simplifying the Equation

To simplify the equation, we need to find the value of $f(-\frac{\pi}{3})$ . Substituting $x=-\frac{\pi}{3}$ into the original function, we get:

$f(-\frac{\pi}{3})=-2 \csc (-\frac{\pi}{3})-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$

Using the fact that $\csc (-x)=-\csc (x)$ , we can simplify the expression as follows:

$f(-\frac{\pi}{3})=2 \csc (\frac{\pi}{3})-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$

Final Equation

Substituting the value of $f(-\frac{\pi}{3})$ into the equation, we get:

$y-2 \csc (\frac{\pi}{3})+\frac{4 \sqrt{3}}{3}+\frac{4 \pi}{9}+1=-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})(x+\frac{\pi}{3})$

Simplifying the equation, we get:

$y=-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})x-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})\frac{\pi}{3}+2 \csc (\frac{\pi}{3})-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$

Conclusion

In this article, we found the equation of the tangent line to the graph of the function $f(x)=-2 \csc (x)-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$ at the point $x=-\frac{\pi}{3}$ . The equation of the tangent line is given by:

$y=-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})x-2 \csc (\frac{\pi}{3}) \cot (\frac{\pi}{3})\frac{\pi}{3}+2 \csc (\frac{\pi}{3})-\frac{4 \sqrt{3}}{3}-\frac{4 \pi}{9}-1$

This equation represents the tangent line to the graph of the function at the given point.

Q: What is the tangent line to a function?

A: The tangent line to a function at a given point is a line that just touches the graph of the function at that point. It represents the instantaneous rate of change of the function at that point.

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

A: To find the equation of the tangent line to a function, you need to find the derivative of the function, evaluate it at the given point, and use the point-slope form of a linear equation.

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

A: The point-slope form of a linear equation 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.

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 derivative of a function?

A: The derivative of a function represents the rate of change of the function with respect to the variable. It is denoted by $f'(x)$ .

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

A: To find the derivative of a function, you need to use the power rule, product rule, and quotient rule of differentiation.

Q: What is the power rule of differentiation?

A: The power rule of differentiation states that if $f(x)=x^n$ , then $f'(x)=nx^{n-1}$ .

Q: What is the product rule of differentiation?

A: The product rule of differentiation states that if $f(x)=u(x)v(x)$ , then $f'(x)=u'(x)v(x)+u(x)v'(x)$ .

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation states that if $f(x)=\frac{u(x)}{v(x)}$ , then $f'(x)=\frac{u'(x)v(x)-u(x)v'(x)}{v(x)^2}$ .

Q: How do I evaluate the derivative at a given point?

A: To evaluate the derivative at a given point, you need to substitute the value of the variable into the derivative.

Q: What is the significance of the tangent line to a function?

A: The tangent line to a function represents the instantaneous rate of change of the function at a given point. It is an important concept in calculus and has many applications in physics, engineering, and economics.

Q: How do I use the tangent line to a function in real-world applications?

A: The tangent line to a function can be used to model real-world situations such as the motion of an object, the growth of a population, and the behavior of a system.

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:

Not evaluating the derivative at the given point
Not using the point-slope form of a linear equation
Not simplifying the equation
Not checking the units of the slope and the point

Q: How do I check my work when finding the equation of a tangent line?

A: To check your work, you need to:

Verify that the equation is in the point-slope form
Check that the slope is correct
Check that the point is correct
Simplify the equation and check that it is correct

Q: What are some common applications of the tangent line to a function?

A: Some common applications of the tangent line to a function include:

Modeling the motion of an object
Modeling the growth of a population
Modeling the behavior of a system
Finding the maximum or minimum value of a function
Finding the rate of change of a function

Q: How do I find the maximum or minimum value of a function using the tangent line?

A: To find the maximum or minimum value of a function using the tangent line, you need to:

Find the derivative of the function
Evaluate the derivative at the given point
Use the point-slope form of a linear equation to find the equation of the tangent line
Find the value of the function at the point where the tangent line is horizontal

Q: What are some common mistakes to avoid when finding the maximum or minimum value of a function?

A: Some common mistakes to avoid when finding the maximum or minimum value of a function include:

Not evaluating the derivative at the given point
Not using the point-slope form of a linear equation
Not simplifying the equation
Not checking the units of the slope and the point

Q: How do I check my work when finding the maximum or minimum value of a function?

A: To check your work, you need to:

Verify that the equation is in the point-slope form
Check that the slope is correct
Check that the point is correct
Simplify the equation and check that it is correct

Q: What are some common applications of the maximum or minimum value of a function?

A: Some common applications of the maximum or minimum value of a function include:

Finding the maximum or minimum value of a function
Finding the rate of change of a function
Modeling the behavior of a system
Finding the optimal solution to a problem

Q: How do I find the optimal solution to a problem using the maximum or minimum value of a function?

A: To find the optimal solution to a problem using the maximum or minimum value of a function, you need to:

Define the function that represents the problem
Find the derivative of the function
Evaluate the derivative at the given point
Use the point-slope form of a linear equation to find the equation of the tangent line
Find the value of the function at the point where the tangent line is horizontal

Q: What are some common mistakes to avoid when finding the optimal solution to a problem?

A: Some common mistakes to avoid when finding the optimal solution to a problem include:

Not evaluating the derivative at the given point
Not using the point-slope form of a linear equation
Not simplifying the equation
Not checking the units of the slope and the point

Q: How do I check my work when finding the optimal solution to a problem?

A: To check your work, you need to:

Verify that the equation is in the point-slope form
Check that the slope is correct
Check that the point is correct
Simplify the equation and check that it is correct

Q: What are some common applications of the optimal solution to a problem?

A: Some common applications of the optimal solution to a problem include:

Finding the optimal solution to a problem
Modeling the behavior of a system
Finding the maximum or minimum value of a function
Finding the rate of change of a function

Q: How do I use the optimal solution to a problem in real-world applications?

A: The optimal solution to a problem can be used to model real-world situations such as the motion of an object, the growth of a population, and the behavior of a system.