For What Values Of $x$ In \[0, 2\pi] $ Does The Graph Of $y = \frac{\cos X}{2 + \sin X}$ Have A Horizontal Tangent? List The Values Of \$x$[/tex\] Below. Separate Multiple Values With Commas.$x =
Introduction
In calculus, the concept of horizontal tangents is crucial in understanding the behavior of functions, particularly in the context of trigonometric functions. A horizontal tangent occurs when the derivative of a function is equal to zero, indicating that the function has a stationary point. In this article, we will delve into the values of x for which the graph of y = cos(x) / (2 + sin(x)) has a horizontal tangent.
The Concept of Horizontal Tangents
To begin with, let's recall the definition of a horizontal tangent. A horizontal tangent is a point on the graph of a function where the slope of the tangent line is zero. Mathematically, this can be represented as:
dy/dx = 0
where dy/dx is the derivative of the function.
Derivative of the Given Function
To find the values of x for which the graph of y = cos(x) / (2 + sin(x)) has a horizontal tangent, we need to find the derivative of the function. Using the quotient rule, we can write:
d/dx (cos(x) / (2 + sin(x))) = (-(sin(x))(2 + sin(x)) - cos(x)(cos(x))) / (2 + sin(x))^2
Simplifying the expression, we get:
d/dx (cos(x) / (2 + sin(x))) = (-2sin(x) - sin^2(x) - cos^2(x)) / (2 + sin(x))^2
Setting the Derivative Equal to Zero
To find the values of x for which the graph has a horizontal tangent, we set the derivative equal to zero:
(-2sin(x) - sin^2(x) - cos^2(x)) / (2 + sin(x))^2 = 0
Solving the Equation
Multiplying both sides by (2 + sin(x))^2, we get:
-2sin(x) - sin^2(x) - cos^2(x) = 0
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
-2sin(x) - 1 = 0
Solving for sin(x), we get:
sin(x) = -1/2
Solving for x
To find the values of x for which sin(x) = -1/2, we can use the inverse sine function:
x = arcsin(-1/2)
Using the unit circle or a calculator, we can find the values of x:
x = -Ï€/6, 7Ï€/6
Conclusion
In conclusion, the values of x for which the graph of y = cos(x) / (2 + sin(x)) has a horizontal tangent are x = -Ï€/6, 7Ï€/6. These values indicate that the function has stationary points at these values of x.
Final Answer
The final answer is:
Q: What is a horizontal tangent?
A: A horizontal tangent is a point on the graph of a function where the slope of the tangent line is zero. Mathematically, this can be represented as:
dy/dx = 0
where dy/dx is the derivative of the function.
Q: How do you find the values of x for which the graph of a function has a horizontal tangent?
A: To find the values of x for which the graph of a function has a horizontal tangent, you need to find the derivative of the function and set it equal to zero. Then, solve the resulting equation for x.
Q: What is the derivative of the function y = cos(x) / (2 + sin(x))?
A: Using the quotient rule, the derivative of the function y = cos(x) / (2 + sin(x)) is:
d/dx (cos(x) / (2 + sin(x))) = (-(sin(x))(2 + sin(x)) - cos(x)(cos(x))) / (2 + sin(x))^2
Q: How do you simplify the derivative of the function y = cos(x) / (2 + sin(x))?
A: Simplifying the derivative of the function y = cos(x) / (2 + sin(x)), we get:
d/dx (cos(x) / (2 + sin(x))) = (-2sin(x) - sin^2(x) - cos^2(x)) / (2 + sin(x))^2
Q: What is the equation that results from setting the derivative equal to zero?
A: Setting the derivative equal to zero, we get:
(-2sin(x) - sin^2(x) - cos^2(x)) / (2 + sin(x))^2 = 0
Q: How do you solve the equation that results from setting the derivative equal to zero?
A: Multiplying both sides by (2 + sin(x))^2, we get:
-2sin(x) - sin^2(x) - cos^2(x) = 0
Using the trigonometric identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
-2sin(x) - 1 = 0
Solving for sin(x), we get:
sin(x) = -1/2
Q: What are the values of x for which sin(x) = -1/2?
A: Using the inverse sine function, we can find the values of x:
x = arcsin(-1/2)
Using the unit circle or a calculator, we can find the values of x:
x = -Ï€/6, 7Ï€/6
Q: What is the final answer for the values of x for which the graph of y = cos(x) / (2 + sin(x)) has a horizontal tangent?
A: The final answer is:
Q: What is the significance of finding the values of x for which the graph of a function has a horizontal tangent?
A: Finding the values of x for which the graph of a function has a horizontal tangent is important in understanding the behavior of the function. It can help us identify the stationary points of the function, which can be useful in various applications such as optimization and physics.
Q: How do you apply the concept of horizontal tangents in real-world problems?
A: The concept of horizontal tangents can be applied in various real-world problems such as optimization, physics, and engineering. For example, in optimization problems, we can use the concept of horizontal tangents to find the maximum or minimum value of a function. In physics, we can use the concept of horizontal tangents to describe the motion of objects. In engineering, we can use the concept of horizontal tangents to design and analyze systems.