Let $f$ Be A Function Given By $f(x)=\frac{1}{4} X^4-\frac{2}{3} X^3+\frac{3}{2} X^2-x+1$.What Is The Instantaneous Rate Of Change Of $f^{\prime}$ At $x=-1$?A. $-10$ B. $-7$ C. $7$ D.

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Introduction

In calculus, the instantaneous rate of change of a function is a measure of how fast the function changes at a given point. This is also known as the derivative of the function. In this article, we will explore the concept of instantaneous rate of change and how to calculate it for a given function.

The Function

The function given is f(x)=14x4−23x3+32x2−x+1f(x)=\frac{1}{4} x^4-\frac{2}{3} x^3+\frac{3}{2} x^2-x+1. This is a polynomial function of degree 4.

Derivative of the Function

To find the instantaneous rate of change of the function, we need to find its derivative. The derivative of a function is a measure of how fast the function changes at a given point. It is denoted by f′(x)f^{\prime}(x).

To find the derivative of the function, we will use the power rule of differentiation, which states that if f(x)=xnf(x)=x^n, then f′(x)=nxn−1f^{\prime}(x)=nx^{n-1}.

Using this rule, we can find the derivative of the function as follows:

f′(x)=ddx(14x4−23x3+32x2−x+1)f^{\prime}(x)=\frac{d}{dx}(\frac{1}{4} x^4-\frac{2}{3} x^3+\frac{3}{2} x^2-x+1)

f′(x)=14⋅4x3−23⋅3x2+32⋅2x−1f^{\prime}(x)=\frac{1}{4} \cdot 4x^3-\frac{2}{3} \cdot 3x^2+\frac{3}{2} \cdot 2x-1

f′(x)=x3−x2+3x−1f^{\prime}(x)=x^3-x^2+3x-1

Instantaneous Rate of Change

Now that we have found the derivative of the function, we can find the instantaneous rate of change of the function at a given point. In this case, we want to find the instantaneous rate of change of the function at x=−1x=-1.

To do this, we will substitute x=−1x=-1 into the derivative of the function:

f′(−1)=(−1)3−(−1)2+3(−1)−1f^{\prime}(-1)=(-1)^3-(-1)^2+3(-1)-1

f′(−1)=−1−1−3−1f^{\prime}(-1)=-1-1-3-1

f′(−1)=−6f^{\prime}(-1)=-6

Conclusion

In conclusion, the instantaneous rate of change of the function f(x)=14x4−23x3+32x2−x+1f(x)=\frac{1}{4} x^4-\frac{2}{3} x^3+\frac{3}{2} x^2-x+1 at x=−1x=-1 is −6-6.

Answer

The correct answer is D. −6-6.

Discussion

The instantaneous rate of change of a function is an important concept in calculus. It is used to measure how fast the function changes at a given point. In this article, we have seen how to calculate the instantaneous rate of change of a function using the power rule of differentiation.

Related Topics

  • Derivative of a function
  • Power rule of differentiation
  • Instantaneous rate of change of a function

References

  • Calculus by Michael Spivak
  • Calculus by James Stewart

Frequently Asked Questions

  • What is the instantaneous rate of change of a function?
  • How do you calculate the instantaneous rate of change of a function?
  • What is the power rule of differentiation?

Answer to Frequently Asked Questions

  • The instantaneous rate of change of a function is a measure of how fast the function changes at a given point.
  • To calculate the instantaneous rate of change of a function, you need to find its derivative using the power rule of differentiation.
  • The power rule of differentiation states that if f(x)=xnf(x)=x^n, then f′(x)=nxn−1f^{\prime}(x)=nx^{n-1}.
    Instantaneous Rate of Change of a Function: Q&A =====================================================

Q: What is the instantaneous rate of change of a function?

A: The instantaneous rate of change of a function is a measure of how fast the function changes at a given point. It is denoted by f′(x)f^{\prime}(x) and is an important concept in calculus.

Q: How do you calculate the instantaneous rate of change of a function?

A: To calculate the instantaneous rate of change of a function, you need to find its derivative using the power rule of differentiation. The power rule of differentiation states that if f(x)=xnf(x)=x^n, then f′(x)=nxn−1f^{\prime}(x)=nx^{n-1}.

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a rule in calculus that states that if f(x)=xnf(x)=x^n, then f′(x)=nxn−1f^{\prime}(x)=nx^{n-1}. This rule is used to find the derivative of a function.

Q: How do you find the derivative of a function using the power rule of differentiation?

A: To find the derivative of a function using the power rule of differentiation, you need to follow these steps:

  1. Identify the function and its exponent.
  2. Multiply the exponent by the coefficient of the term.
  3. Subtract 1 from the exponent.
  4. Write the result as the derivative of the function.

Q: What is the difference between the instantaneous rate of change and the average rate of change of a function?

A: The instantaneous rate of change of a function is a measure of how fast the function changes at a given point, while the average rate of change of a function is a measure of how fast the function changes over a given interval.

Q: How do you calculate the average rate of change of a function?

A: To calculate the average rate of change of a function, you need to follow these steps:

  1. Identify the function and the interval over which you want to calculate the average rate of change.
  2. Find the difference between the function values at the endpoints of the interval.
  3. Divide the difference by the length of the interval.
  4. Write the result as the average rate of change of the function.

Q: What is the significance of the instantaneous rate of change of a function?

A: The instantaneous rate of change of a function is an important concept in calculus because it is used to measure how fast the function changes at a given point. This is useful in many real-world applications, such as physics, engineering, and economics.

Q: How do you use the instantaneous rate of change of a function in real-world applications?

A: The instantaneous rate of change of a function is used in many real-world applications, such as:

  • Physics: to measure the acceleration of an object
  • Engineering: to design and optimize systems
  • Economics: to model and analyze economic systems

Q: What are some common mistakes to avoid when calculating the instantaneous rate of change of a function?

A: Some common mistakes to avoid when calculating the instantaneous rate of change of a function include:

  • Failing to identify the function and its exponent
  • Making errors in the power rule of differentiation
  • Failing to simplify the derivative

Q: How do you check your work when calculating the instantaneous rate of change of a function?

A: To check your work when calculating the instantaneous rate of change of a function, you can:

  • Use a calculator or computer software to verify your answer
  • Check your work by plugging in values and checking the result
  • Use a second method to calculate the instantaneous rate of change of the function

Conclusion

In conclusion, the instantaneous rate of change of a function is an important concept in calculus that is used to measure how fast the function changes at a given point. By understanding the power rule of differentiation and how to calculate the instantaneous rate of change of a function, you can apply this concept to real-world problems and make informed decisions.