Given The Function Defined On The Interval Below, Find The Value Of C C C Where F ( C F(c F ( C ] Equals The Average Value. Function: F ( X ) = Sin ⁡ X F(x) = \sin X F ( X ) = Sin X Interval: $\left[\frac{\pi}{2}, \pi\right] Find C = ? C = \, ? C = ? Round

by ADMIN 257 views

Introduction

In this article, we will explore the concept of finding the value of c where f(c) equals the average value of a given function defined on a specific interval. We will use the function f(x) = sin(x) and the interval [π/2, π] to find the value of c.

The Average Value of a Function

The average value of a function f(x) defined on the interval [a, b] is given by the formula:

1b−a∫abf(x)dx\frac{1}{b-a} \int_{a}^{b} f(x) dx

In our case, the interval is [π/2, π], so the average value of f(x) = sin(x) is:

1π−π2∫π2πsin⁡(x)dx\frac{1}{\pi - \frac{\pi}{2}} \int_{\frac{\pi}{2}}^{\pi} \sin(x) dx

Calculating the Average Value

To calculate the average value, we need to evaluate the integral:

∫π2πsin⁡(x)dx\int_{\frac{\pi}{2}}^{\pi} \sin(x) dx

Using the antiderivative of sin(x), which is -cos(x), we get:

∫π2πsin⁡(x)dx=−cos⁡(x)∣π2π\int_{\frac{\pi}{2}}^{\pi} \sin(x) dx = -\cos(x) \Big|_{\frac{\pi}{2}}^{\pi}

Evaluating the antiderivative at the limits of integration, we get:

−cos⁡(x)∣π2π=−cos⁡(π)+cos⁡(π2)-\cos(x) \Big|_{\frac{\pi}{2}}^{\pi} = -\cos(\pi) + \cos\left(\frac{\pi}{2}\right)

Since cos(π) = -1 and cos(π/2) = 0, we get:

−cos⁡(π)+cos⁡(π2)=−(−1)+0=1-\cos(\pi) + \cos\left(\frac{\pi}{2}\right) = -(-1) + 0 = 1

So, the average value of f(x) = sin(x) on the interval [π/2, π] is:

1π−π2∫π2πsin⁡(x)dx=1π2⋅1=2π\frac{1}{\pi - \frac{\pi}{2}} \int_{\frac{\pi}{2}}^{\pi} \sin(x) dx = \frac{1}{\frac{\pi}{2}} \cdot 1 = \frac{2}{\pi}

Finding the Value of c

Now that we have the average value of f(x) = sin(x) on the interval [π/2, π], we need to find the value of c where f(c) equals the average value.

To do this, we need to solve the equation:

f(c)=2πf(c) = \frac{2}{\pi}

Since f(x) = sin(x), we can rewrite the equation as:

sin⁡(c)=2π\sin(c) = \frac{2}{\pi}

Solving the Equation

To solve the equation sin(c) = 2/π, we can use the inverse sine function, which is denoted by arcsin(x).

So, we can rewrite the equation as:

c=arcsin⁡(2π)c = \arcsin\left(\frac{2}{\pi}\right)

Using a calculator or a trigonometric table, we can find that:

arcsin⁡(2π)≈0.588\arcsin\left(\frac{2}{\pi}\right) \approx 0.588

However, this is the only solution to the equation. Since the sine function is periodic with period 2π, we can add multiples of 2π to the solution to get other solutions.

So, the general solution to the equation sin(c) = 2/π is:

c=arcsin⁡(2π)+2kπc = \arcsin\left(\frac{2}{\pi}\right) + 2k\pi

where k is an integer.

Conclusion

In this article, we found the value of c where f(c) equals the average value of the function f(x) = sin(x) on the interval [π/2, π].

We calculated the average value of the function using the formula for the average value of a function, and then solved the equation f(c) = 2/π to find the value of c.

The value of c is given by the equation c = arcsin(2/π) + 2kπ, where k is an integer.

Discussion

The concept of finding the value of c where f(c) equals the average value of a function is an important one in mathematics.

It is used in many areas of mathematics, including calculus, differential equations, and statistics.

In this article, we used the function f(x) = sin(x) and the interval [π/2, π] to find the value of c.

However, the concept can be applied to any function and any interval.

References

  • [1] "Calculus" by Michael Spivak
  • [2] "Differential Equations" by Morris Tenenbaum
  • [3] "Statistics" by James E. Gentle

Appendix

The following is a list of formulas and theorems used in this article:

  • The formula for the average value of a function: $\frac{1}{b-a} \int_{a}^{b} f(x) dx$
  • The antiderivative of sin(x): $-\cos(x)$
  • The inverse sine function: $\arcsin(x)$

Q: What is the average value of a function?

A: The average value of a function f(x) defined on the interval [a, b] is given by the formula:

1b−a∫abf(x)dx\frac{1}{b-a} \int_{a}^{b} f(x) dx

Q: How do I calculate the average value of a function?

A: To calculate the average value, you need to evaluate the integral:

∫abf(x)dx\int_{a}^{b} f(x) dx

Using the antiderivative of f(x), you can then plug in the limits of integration and simplify.

Q: What is the antiderivative of sin(x)?

A: The antiderivative of sin(x) is -cos(x).

Q: How do I find the value of c where f(c) equals the average value?

A: To find the value of c, you need to solve the equation:

f(c)=1b−a∫abf(x)dxf(c) = \frac{1}{b-a} \int_{a}^{b} f(x) dx

Using the inverse function of f(x), you can rewrite the equation as:

c=arcsin⁡(2π)+2kπc = \arcsin\left(\frac{2}{\pi}\right) + 2k\pi

where k is an integer.

Q: What is the inverse sine function?

A: The inverse sine function is denoted by arcsin(x) and is defined as:

arcsin⁡(x)=sin⁡−1(x)\arcsin(x) = \sin^{-1}(x)

Q: Can I use the inverse sine function to find the value of c?

A: Yes, you can use the inverse sine function to find the value of c. However, you need to be careful when using the inverse sine function, as it is only defined for x in the interval [-1, 1].

Q: What is the general solution to the equation sin(c) = 2/π?

A: The general solution to the equation sin(c) = 2/π is:

c=arcsin⁡(2π)+2kπc = \arcsin\left(\frac{2}{\pi}\right) + 2k\pi

where k is an integer.

Q: Can I use the general solution to find the value of c for any interval?

A: Yes, you can use the general solution to find the value of c for any interval. However, you need to be careful when choosing the interval, as the general solution may not be valid for all intervals.

Q: What are some common applications of finding the value of c where f(c) equals the average value?

A: Finding the value of c where f(c) equals the average value has many applications in mathematics, including:

  • Calculus: Finding the average value of a function is an important concept in calculus, and is used to solve many problems.
  • Differential equations: Finding the value of c where f(c) equals the average value is used to solve differential equations.
  • Statistics: Finding the average value of a function is used in statistics to calculate the mean of a dataset.

Q: Can I use a calculator to find the value of c?

A: Yes, you can use a calculator to find the value of c. However, you need to be careful when using a calculator, as it may not always give you the correct answer.

Q: What are some common mistakes to avoid when finding the value of c?

A: Some common mistakes to avoid when finding the value of c include:

  • Not using the correct formula for the average value of a function.
  • Not evaluating the integral correctly.
  • Not using the inverse function of f(x) correctly.
  • Not being careful when choosing the interval.

Q: Can I use the value of c to solve other problems?

A: Yes, you can use the value of c to solve other problems. For example, you can use the value of c to find the average value of a function on a different interval.

Q: What are some common applications of the value of c?

A: The value of c has many applications in mathematics, including:

  • Calculus: The value of c is used to solve many problems in calculus.
  • Differential equations: The value of c is used to solve differential equations.
  • Statistics: The value of c is used to calculate the mean of a dataset.

Q: Can I use the value of c to find the average value of a function on a different interval?

A: Yes, you can use the value of c to find the average value of a function on a different interval. However, you need to be careful when choosing the interval, as the value of c may not be valid for all intervals.