You May Use Your Calculator For This Question.The Function G ( X ) = 12 X 2 − Sin ⁡ X G(x) = 12x^2 - \sin X G ( X ) = 12 X 2 − Sin X Is The First Derivative Of F ( X F(x F ( X ]. If F ( 0 ) = − 2 F(0) = -2 F ( 0 ) = − 2 , What Is The Value Of F ( 2 Π F(2\pi F ( 2 Π ]?A. 2 Π 2 + 5 2\pi^2 + 5 2 Π 2 + 5 B. 32 Π 3 − 2 32\pi^3 - 2 32 Π 3 − 2 C.

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. Given the derivative of a function, we can find the original function by integrating the derivative. In this article, we will explore how to find the value of a function at a specific point using its derivative.

The Given Function and Its Derivative

We are given the function $g(x) = 12x^2 - \sin x$, which is the first derivative of the function $f(x)$. We are also given that $f(0) = -2$. Our goal is to find the value of $f(2\pi)$.

Understanding the Relationship Between a Function and Its Derivative

To find the original function $f(x)$, we need to integrate the given derivative $g(x)$. The integral of $g(x)$ with respect to $x$ will give us the original function $f(x)$ up to a constant.

Integrating the Derivative

To find the original function $f(x)$, we need to integrate the given derivative $g(x) = 12x^2 - \sin x$ with respect to $x$. We can do this using the power rule of integration and the fact that the integral of $\sin x$ is $-\cos x$.

$$\int g(x) dx = \int (12x^2 - \sin x) dx$$

$$= 12 \int x^2 dx - \int \sin x dx$$

$$= 12 \cdot \frac{x^3}{3} + \cos x + C$$

$$= 4x^3 + \cos x + C$$

where $C$ is the constant of integration.

Finding the Original Function

Since we are given that $f(0) = -2$, we can use this information to find the value of the constant $C$. Substituting $x = 0$ into the original function $f(x)$, we get:

$$f(0) = 4(0)^3 + \cos(0) + C$$

$$= 1 + C$$

Since we are given that $f(0) = -2$, we can set up the equation:

$$1 + C = -2$$

$$C = -3$$

Now that we have found the value of the constant $C$, we can write the original function $f(x)$ as:

$$f(x) = 4x^3 + \cos x - 3$$

Finding the Value of the Function at $2\pi$

Now that we have the original function $f(x)$, we can find the value of the function at $2\pi$ by substituting $x = 2\pi$ into the function.

$$f(2\pi) = 4(2\pi)^3 + \cos(2\pi) - 3$$

$$= 4(8\pi^3) + 1 - 3$$

$$= 32\pi^3 - 2$$

Therefore, the value of the function $f(x)$ at $2\pi$ is $32\pi^3 - 2$.

Conclusion

In this article, we have shown how to find the value of a function at a specific point using its derivative. We were given the derivative of the function $f(x)$ and the value of the function at $0$. We used this information to find the original function $f(x)$ and then evaluated the function at $2\pi$ to find the final answer.

Answer

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Introduction

In our previous article, we explored how to find the value of a function at a specific point using its derivative. We were given the derivative of the function $f(x)$ and the value of the function at $0$. We used this information to find the original function $f(x)$ and then evaluated the function at $2\pi$ to find the final answer.

Q&A Session

Q: What is the relationship between a function and its derivative?

A: The derivative of a function represents the rate of change of the function with respect to its input. Given the derivative of a function, we can find the original function by integrating the derivative.

Q: How do we find the original function from its derivative?

A: To find the original function, we need to integrate the given derivative with respect to $x$. We can do this using the power rule of integration and the fact that the integral of $\sin x$ is $-\cos x$.

Q: What is the power rule of integration?

A: The power rule of integration states that if $f(x) = x^n$, then $\int f(x) dx = \frac{x^{n+1}}{n+1} + C$.

Q: How do we find the constant of integration?

A: We can find the constant of integration by using the given information about the function. For example, if we are given that $f(0) = -2$, we can substitute $x = 0$ into the original function to find the value of the constant.

Q: What is the significance of the constant of integration?

A: The constant of integration represents the value of the function at a specific point. In our example, the constant of integration represents the value of the function at $0$.

Q: How do we evaluate the function at a specific point?

A: To evaluate the function at a specific point, we need to substitute the value of $x$ into the original function. For example, to find the value of the function at $2\pi$, we need to substitute $x = 2\pi$ into the original function.

Q: What is the final answer to the problem?

A: The final answer to the problem is $32\pi^3 - 2$.

Conclusion

In this Q&A article, we have answered some common questions about finding the value of a function using its derivative. We have explored the relationship between a function and its derivative, how to find the original function from its derivative, and how to evaluate the function at a specific point.

Frequently Asked Questions

• 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 its input.
• Q: How do we find the original function from its derivative?
  • A: We need to integrate the given derivative with respect to $x$.
• Q: What is the power rule of integration?
  • A: The power rule of integration states that if $f(x) = x^n$, then $\int f(x) dx = \frac{x^{n+1}}{n+1} + C$.
• Q: How do we find the constant of integration?
  • A: We can find the constant of integration by using the given information about the function.

Additional Resources

• Calculus Textbook: A comprehensive textbook on calculus that covers the basics of differentiation and integration.
• Online Calculus Course: An online course that covers the basics of calculus, including differentiation and integration.
• Calculus Calculator: A calculator that can be used to evaluate functions and their derivatives.